Post on 27-Jun-2020
Subdivision Schemes and theirApplications to Solve Differential
Equations
By
Syeda Tehmina Ejaz
Ph. D. DISSERTATION
Session 2012-2015
DEPARTMENT OF MATHEMATICS
The Islamia University of Bahawalpur
Bahawalpur 63100, PAKISTAN
2016
Subdivision Schemes and theirApplications to Solve Differential
Equations
By
Syeda Tehmina Ejaz
Supervised By
Prof. Dr. Ghulam Mustafa
Department of Mathematics
The Islamia University of Bahawalpur
Bahawalpur 63100, PAKISTAN
2016
Subdivision Schemes and theirApplications to Solve Differential
Equations
By
Syeda Tehmina Ejaz
A dissertation submitted to the department of Mathematics,
The Islamia University of Bahawalpur
in the partial fulfillment for the degree of
Doctor of Philosophy
in
Mathematics
Supervised By
Prof. Dr. Ghulam Mustafa
Department of Mathematics
The Islamia University of Bahawalpur
Bahawalpur 63100, PAKISTAN
2016
Scanned by CamScanner
Scanned by CamScanner
Scanned by CamScanner
Dedication
I WOULD LIKE TO DEDICATE MY THESIS TO MY
Beloved Fatherand
Sweet Mother
WHO ALWAYS PICKED ME UP ON TIME AND
ENCOURAGED ME TO GO ON EVERY ADVENTURE
ESPECIALLY THIS ONE
iv
Acknowledgments
I offer all the praises and deepest gratitude to Almighty ALLAH, the most
gracious, the most merciful and to His Holy Prophet Muhammad (Peace be
upon him), a teacher of the whole humanity and a source of inspiration and
guidance throughout my life.
I owe a scholarly debt of gratitude to Prof. Dr. Ghulam Mustafa, my super-
visor & Chairman Department of Mathematics, whose charisma, skill and con-
cern surpassed all understanding. This task would not have been accomplished
without his brilliant and devoted supervision. I extend my deepest thanks and
felicitation for his monumentally scholarly enterprise and giving me the chance
to make an enchanting voyage into the conglomerates of the present study.
I am grateful to my respectable teacher Prof. Dr. Tahir Mahmood, Ex-Chairman
Department of Mathematics for his care and cooperation throughout my stud-
ies.
Of course, I am grateful to my beloved parents, brothers and sisters for their
infinite patience and love; unconditional support. I owe everything I accom-
plished, indeed, without them this work would never have come into existence
(literally).
I am deeply indebted to my friends and well wishers especially, Nargis Khan
and Madiha Sana, who profusely lauded and encouraged me during this re-
search project. Their encouragement stimulated me to undertake this long and
onerous journey into the realms of our skewed culture of research.
v
I acknowledge that this research work is supported by Indigenous Ph. D 5000
Fellowship Program and National Research Program for Universities (NRPU)
Project No. 3183 of Higher Education Commission (HEC) of Pakistan.
Syeda Tehmina Ejaz
vi
Abstract
Subdivision schemes are important for the generation of smooth curves and
surfaces through an iterative process from a finite set of points. The subdivision
schemes have been considered well-regarded in many fields of computational
sciences. In this dissertation, we have used subdivision schemes for the numer-
ical solution of different types of boundary value problems. In literature three
methods such as spline based methods, finite difference methods and finite el-
ement methods are commonly used to find the numerical solution of boundary
value problems. Subdivision based algorithms for the numerical solution of
second order boundary value problems have also been used in the literature.
In this dissertation, we develop subdivision based collocation algorithms for
the numerical solution of linear and non linear boundary value problems of or-
der three and four. Subdivision based collocation algorithms for the solution
of second and third order singularly perturbed boundary value problems are
also presented in this dissertation. These algorithms are developed by using ba-
sis functions of subdivision schemes. Convergence analysis of these collocation
algorithms are also discussed. Accuracy and efficiency of the developed algo-
rithms are shown through comparison with the existing numerical algorithms.
vii
Contents
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Approval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Certificate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii
1 Introduction 1
1.1 Computer aided geometric design . . . . . . . . . . . . . . . . . . 1
1.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Boundary value problems . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 Numerical methods for BVPs . . . . . . . . . . . . . . . . . 5
1.2.2 Contribution of this dissertation . . . . . . . . . . . . . . . 13
1.3 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Outlines of the dissertation . . . . . . . . . . . . . . . . . . . . . . 17
2 Numerical Solution of Two Point Boundary Value Problems by Inter-
polating Subdivision Schemes 19
2.1 Interpolating schemes for curve design . . . . . . . . . . . . . . . 20
viii
2.1.1 8-point interpolating scheme . . . . . . . . . . . . . . . . . 21
2.2 Numerical interpolating collocation algorithm . . . . . . . . . . . 24
2.2.1 The collocation algorithm . . . . . . . . . . . . . . . . . . . 26
2.2.2 Adjustment of boundary conditions . . . . . . . . . . . . . 29
2.2.3 Existence of the solution . . . . . . . . . . . . . . . . . . . . 33
2.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4 Numerical examples and discussions . . . . . . . . . . . . . . . . . 38
2.5 Conclusion and future work . . . . . . . . . . . . . . . . . . . . . . 41
3 Subdivision Schemes Based Collocation Algorithms for the Solution
of Fourth Order Boundary Value Problems 45
3.1 Basic properties of the schemes . . . . . . . . . . . . . . . . . . . . 47
3.1.1 Subdivision matrices . . . . . . . . . . . . . . . . . . . . . 47
3.1.2 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2 Description of numerical algorithms . . . . . . . . . . . . . . . . . 53
3.2.1 Collocation algorithms . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Interpolating collocation algorithm . . . . . . . . . . . . . . 56
3.2.3 Approximating collocation algorithm . . . . . . . . . . . . 57
3.2.4 Boundary conditions at end points . . . . . . . . . . . . . 61
3.2.5 Approximation of derivative boundary conditions . . . . 62
3.2.6 Adjustment of boundary conditions . . . . . . . . . . . . . 63
3.2.7 Stable systems of linear equations . . . . . . . . . . . . . . 65
3.2.8 Stable system for interpolating collocation algorithm . . . 65
3.2.9 Stable system for approximating collocation algorithm . . 68
3.2.10 Existence of the solution . . . . . . . . . . . . . . . . . . . . 71
3.2.11 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3 Numerical examples and comparison . . . . . . . . . . . . . . . . 74
3.3.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 74
3.3.2 Comparison: . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
ix
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4 A Subdivision Based Iterative Collocations Algorithm for Nonlinear
Third Order Boundary Value Problems 85
4.1 Existence and uniqueness of the solution . . . . . . . . . . . . . . 86
4.2 Subdivision scheme and basis function . . . . . . . . . . . . . . . . 87
4.2.1 Interpolating subdivision scheme . . . . . . . . . . . . . . 87
4.2.2 Basis function and their derivatives . . . . . . . . . . . . . 87
4.3 Subdivision based iterative algorithm . . . . . . . . . . . . . . . . 89
4.3.1 The collocation algorithm . . . . . . . . . . . . . . . . . . . 89
4.3.2 Unstable nonlinear system . . . . . . . . . . . . . . . . . . 92
4.3.3 Stable nonlinear system . . . . . . . . . . . . . . . . . . . . 92
4.3.4 Approximated boundary condition . . . . . . . . . . . . . 93
4.3.5 Imposed boundary conditions . . . . . . . . . . . . . . . . 93
4.3.6 Non-singularity of a matrix . . . . . . . . . . . . . . . . . . 95
4.3.7 Iterative algorithm and its convergence . . . . . . . . . . . 97
4.3.8 Iterative algorithm based on basis function . . . . . . . . . 97
4.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.5 Examples, comparison and conclusion . . . . . . . . . . . . . . . . 104
4.5.1 Numerical examples . . . . . . . . . . . . . . . . . . . . . . 104
4.5.2 Comparison and discussion . . . . . . . . . . . . . . . . . . 107
4.5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5 A Numerical Approach Based on Subdivision Schemes for Solving
Nonlinear Fourth Order Boundary Value Problems 122
5.1 Basis functions and their derivatives . . . . . . . . . . . . . . . . . 123
5.1.1 Interpolating subdivision scheme . . . . . . . . . . . . . . 123
5.1.2 Basis functions . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.2 Description of iterative numerical algorithm . . . . . . . . . . . . 125
x
5.2.1 The collocation algorithm . . . . . . . . . . . . . . . . . . . 125
5.2.2 Boundary conditions at end points . . . . . . . . . . . . . 130
5.2.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . 130
5.2.4 Extrapolation method . . . . . . . . . . . . . . . . . . . . . 131
5.2.5 Non-singularity of a matrix . . . . . . . . . . . . . . . . . . 134
5.2.6 Iterative algorithm and its convergence . . . . . . . . . . . 134
5.2.7 Iterative algorithm based on basis function . . . . . . . . . 135
5.3 Error estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
5.4 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6 Subdivision Based Collocation Algorithm for Singularly Perturbed Bound-
ary Value Problems 147
6.1 Derivation of numerical algorithm . . . . . . . . . . . . . . . . . . 148
6.1.1 First and second derivatives of Ψ(x) . . . . . . . . . . . . . 149
6.1.2 The subdivision based collocation algorithm . . . . . . . . 149
6.1.3 Singularly perturbed linear system of equations . . . . . . 152
6.1.4 Compelled conditions . . . . . . . . . . . . . . . . . . . . . 153
6.1.5 Stable singularly perturbed linear system of equations . . 156
6.1.6 Existence of the solution . . . . . . . . . . . . . . . . . . . . 157
6.2 Convergence of the method . . . . . . . . . . . . . . . . . . . . . . 158
6.3 Numerical examples and discussions . . . . . . . . . . . . . . . . . 161
6.3.1 Results and discussion . . . . . . . . . . . . . . . . . . . . . 162
6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
7 A Subdivision Collocation Algorithm for Solving Two point Boundary
value Problems of Order Three 180
7.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.1.1 Third order singularly perturbed BVP . . . . . . . . . . . . 181
xi
7.1.2 Subdivision scheme and derivatives . . . . . . . . . . . . . 181
7.2 Subdivision collocation algorithm . . . . . . . . . . . . . . . . . . 182
7.2.1 Singularly perturbed system . . . . . . . . . . . . . . . . . 184
7.2.2 Approximation of derivative conditions . . . . . . . . . . . 185
7.2.3 Necessitated conditions . . . . . . . . . . . . . . . . . . . . 186
7.2.4 Stable linear system of equations . . . . . . . . . . . . . . . 187
7.3 Convergence of collocation algorithm . . . . . . . . . . . . . . . . 188
7.4 Numerical results and discussion . . . . . . . . . . . . . . . . . . . 191
7.5 Concluding remark . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8 Comparison, Conclusions, Limitations and Future Work 203
8.1 Comparison and Conclusion . . . . . . . . . . . . . . . . . . . . . . 203
8.1.1 Comparison with existing methods . . . . . . . . . . . . . 204
8.1.2 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.2 Limitations of algorithms . . . . . . . . . . . . . . . . . . . . . . . 205
8.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Bibliography 207
Publications of Syeda Tehmina Ejaz 222
xii
List of Tables
2.1 Determinants of the matrices: . . . . . . . . . . . . . . . . . . . . . 34
2.2 Solutions and error estimation of Example 2.4.1: . . . . . . . . . . 40
2.3 Solutions and error estimation of Example 2.4.2: . . . . . . . . . . 42
3.1 Eigenvalues and eigenvectors of the matrix S1 . . . . . . . . . . . 50
3.2 Eigenvalues and eigenvectors of the matrix S2 . . . . . . . . . . . 51
3.3 Derivatives of ϕ at cardinal data by (3.8) . . . . . . . . . . . . . . . 54
3.4 Derivatives of Φ at cardinal data by (3.11) . . . . . . . . . . . . . . 55
3.5 Determinants of the matrices . . . . . . . . . . . . . . . . . . . . . 73
3.6 Numerical results of Example 3.3.1 . . . . . . . . . . . . . . . . . . 78
3.7 Numerical results of Example 3.3.2 . . . . . . . . . . . . . . . . . . 79
3.8 Numerical results of Example 3.3.3 . . . . . . . . . . . . . . . . . . 80
3.9 Maximum absolute errors of Examples 3.3.1, 3.3.2 and 3.3.3 . . . . 83
3.10 Comparison of Example 3.3.1 with different methods . . . . . . . 84
4.1 Numerical results of Example 4.5.1: h = 10−1 . . . . . . . . . . . . 109
4.2 Numerical results of Example 4.5.2: h = 10−1 . . . . . . . . . . . . 110
4.3 Numerical results of Example 4.5.3 : h = 10−1 . . . . . . . . . . . . 111
4.4 Numerical results of Example 4.5.4 : k = 0 and h = 10−1 . . . . . . 112
4.5 Numerical results of Example 4.5.4 : k = 12
and h = 10−1 . . . . . . 113
4.6 Numerical results of Example 4.5.4 : k = 2 and h = 10−1 . . . . . . 114
5.1 Numerical results of Example 5.4.1 . . . . . . . . . . . . . . . . . . 143
xiii
5.2 Numerical results of Example 5.4.2 . . . . . . . . . . . . . . . . . . 145
6.1 First and second derivatives of Ψ . . . . . . . . . . . . . . . . . . . 149
6.2 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 164
6.3 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 165
6.4 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 166
6.5 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 167
6.6 Maximum absolute errors of Example 6.3.1 . . . . . . . . . . . . . 167
6.7 Maximum absolute errors of Example 6.3.2 . . . . . . . . . . . . . 168
6.8 Maximum absolute errors of Example 6.3.2 . . . . . . . . . . . . . 169
6.9 Maximum absolute errors of Example 6.3.2 . . . . . . . . . . . . . 170
6.10 Maximum absolute errors of Example 6.3.2 . . . . . . . . . . . . . 170
6.11 Maximum absolute errors of Example 6.3.3 . . . . . . . . . . . . . 171
6.12 Maximum absolute errors of Example 6.3.3 . . . . . . . . . . . . . 172
6.13 Maximum absolute errors of Example 6.3.3 . . . . . . . . . . . . . 173
7.1 Maximum absolute errors for N = 10 of Example 7.4.1 . . . . . . 193
7.2 Maximum absolute errors of Example 7.4.1 . . . . . . . . . . . . . 194
7.3 Maximum absolute errors of Example 7.4.2 . . . . . . . . . . . . . 195
7.4 Maximum absolute errors for N = 10 of Example 7.4.2 . . . . . . 196
7.5 Maximum absolute errors of Example 7.4.3 . . . . . . . . . . . . . 197
7.6 Maximum absolute errors of Example 7.4.4 . . . . . . . . . . . . . 198
xiv
List of Figures
1.1 (a) represents primal binary and (b) represents dual binary scheme. . . 16
2.1 Interpolatory basis function ϕ3(i) is shown in (a), first derivative of
ϕ3(i) i-e. ϕ′3(i) shown in (b), second derivative of ϕ3(i) i-e. ϕ′′
3(i) shown
in (c) and third derivative of ϕ3(i) i-e. ϕ′′′3 (i) shown in (d) respectively . 25
2.2 Comparison between analytic and approximating solutions Example 2.4.1. 43
2.3 Comparison between analytic and approximating solutions of Example
2.4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1 Comparison between analytic and approximate solutions of Example
3.3.1 obtained by interpolating and approximating collocation algorithm-
s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.2 Comparison between analytic and approximate solutions of Example
3.3.2 obtained by interpolating and approximating collocation algorithm-
s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.3 Comparison between analytic and approximate solutions of Example
3.3.3 obtained by interpolating and approximating collocation algorithm-
s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
xv
4.1 Comparison of the analytic and approximate solution of Example 4.5.1
by proposed algorithm and Caglar et al. (1999). In this figure solid line
shows exact solution, dotted lines show approximate solution by pro-
posed algorithm and dashed lines show the solution obtained by Caglar
et al. (1999). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
4.2 Comparison of the analytic and approximate solution of Example 4.5.2
by proposed algorithm and Hasan (2012). In this figure solid line shows
exact solution, dotted lines show approximate solution by proposed al-
gorithm and dashed lines show the solution obtained by Hasan (2012). . 116
4.3 Comparison of the analytic and approximate solution of Example 4.5.3
by proposed algorithm. In this figure solid line shows exact solution and
dashed lines show approximate solution by proposed algorithm. . . . . . 117
4.4 Comparison between exact and approximate solutions of Example 4.5.4
for k = 0. Solid line represents exact solution and dash line represents
approximate solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.5 Comparison between exact and approximate solutions of Example 4.5.4
for k = 12. Solid line represents exact solution and dash line represents
approximate solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.6 Comparison between exact and approximate solutions of Example 4.5.4
for k = 2. Solid line represents exact solution and dash line represents
approximate solution. . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.7 Approximate solutions of Examples 4.5.1, 4.5.2 and 4.5.3 at different
step sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.1 Graphical representation of basis functions is shown in figure (a), and
first, second, third and fourth derivatives of basis function are shown in
figure (b), (c), (d) and (e). . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2 Comparison of the analytic and approximate solution of Example 5.4.1. 144
5.3 Comparison of the analytic and approximate solution of Example 5.4.2. 146
xvi
6.1 Physical behavior of analytic and approximate solutions of Example
6.3.1 for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c)
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.2 Physical behavior of analytic and approximate solutions of Example
6.3.1 for N = 32 and ε = 2−25. . . . . . . . . . . . . . . . . . . . . . 175
6.3 Physical behavior of analytic and approximate solutions of Example
6.3.1 for N = 32 and ε = (2−20)2. . . . . . . . . . . . . . . . . . . . . 175
6.4 Physical behavior of analytic and approximate solutions of Example
6.3.2 for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c)
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.5 Physical behavior of analytic and approximate solutions of Example
6.3.2 for N = 16 and ε = 10−8. . . . . . . . . . . . . . . . . . . . . . 177
6.6 Physical behavior of analytic and approximate solutions of Example
6.3.2 for N = 32 and ε = 10−9. . . . . . . . . . . . . . . . . . . . . . 177
6.7 Physical behavior of analytic and approximate solutions of Example
6.3.3 for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c)
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
6.8 Physical behavior of analytic and approximate solutions of Example
6.3.3 for N = 16 and ε = 10−5. . . . . . . . . . . . . . . . . . . . . . 179
6.9 Physical behavior of analytic and approximate solutions of Example
6.3.3 for N = 16 and ε = 10−8. . . . . . . . . . . . . . . . . . . . . . 179
7.1 Comparability of analytic and approximate solutions of Example 7.4.1
for N = 100 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 193
7.2 Comparability of analytic and approximate solutions of Example 7.4.1
for N = 200 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 196
7.3 Comparability of analytic and approximate solutions of Example 7.4.2
for N = 100 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 199
xvii
7.4 Comparability of analytic and approximate solutions of Example 7.4.2
for N = 200 with ε = 10−4 . . . . . . . . . . . . . . . . . . . . . . . . 199
7.5 Comparability of analytic and approximate solutions of Example 7.4.3
for N = 250 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 200
7.6 Comparability of analytic and approximate solutions of Example 7.4.3
for N = 300 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 200
7.7 Comparability of analytic and approximate solutions of Example 7.4.4
for N = 250 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 201
7.8 Comparability of analytic and approximate solutions of Example 7.4.4
for N = 300 with ε = 10−5 . . . . . . . . . . . . . . . . . . . . . . . . 201
xviii
Chapter 1
Introduction
This chapter provides a brief introduction to the subject and historical develop-
ments in the fields of computer aided geometric design, subdivision schemes
and different methods for the numerical solutions of boundary value problems.
1.1 Computer aided geometric design
Computer Aided Geometric Design (CAGD) is a branch of computational math-
ematics that studies methods and algorithms for the mathematical description
of shapes. It is concerned with construction and representation of free form
curves and surfaces given by a set of points using polynomial, rational piece-
wise polynomial or piecewise rational methods. This branch is closely related
to several other branches, such as geometric modeling. For example NURBS ob-
ject represent the fundamental structures of modern computer system used in
the aircraft and car industry, or data fitting (interpolation, approximation of set
of points). CAGD is the basis for modern design in most branches of industry,
from naval and aeronautic to textile industry.
One of the main issues, in designing a geometric modeler is selecting a math-
ematical representation for curves and surfaces. Selecting a particular repre-
1
sentation is important because later manipulations and analysis depend greatly
on the specific representation. Different techniques have been used for curve
and surface designing. Subdivision schemes are the latest among these and are
being used, most commonly, in geometric modeling.
Subdivision schemes generate smooth curves and surfaces in an efficient way
from the discrete set of control points. This is consistent and efficient iterative
algorithm to be used for modeling of curve and surfaces. During the past two
decades, much research has been undertaken to construct and analyze new sub-
division algorithms for curves/surfaces. Moreover, a number of mathematical
and numerical procedures have been established to improve geometric analysis
of subdivision surfaces. Subdivision schemes can be classified into two impor-
tant branches approximating and interpolating ones. Approximating scheme
means that the limit curve approximates the initial polygon and that after sub-
division, only the newly generated control points are in the limit curve. While
interpolating scheme means that after subdivision, the control points of the o-
riginal control polygon and the newly generated control points both lie on the
limit curve. Following are the advantages and disadvantages of interpolating
and approximating subdivision schemes in the field of geometric modeling:
• Interpolating schemes are more useful for engineering applications, espe-
cially the schemes with the shape control but approximating schemes do
not satisfy the shape control property.
• Interpolating subdivision schemes have the drawback that in order to cre-
ate smoother curves, it is necessary to enlarge the support of the mask.
The designers in geometric modeling require subdivision schemes to have
their masks with a possibly smaller support and to create good smooth
curves.
2
• Approximating schemes yield smoother curves with smaller support as
compare to the interpolating schemes.
In modern era, subdivision techniques can be consider as an integral part of
image processing, graphic visualization, engineering and medical science etc. It
can also be used to develop an iterative algorithms for the solution of different
types of boundary value problems.
1.1.1 Literature review
It was de Rham (1956) who initiated the idea of subdivision techniques. This
idea was further proceeded by Dyn et al. (1987). He introduced a family of
schemes with the complexity of size four. This family was index by a tension
parameter to control the shape of curves. The number of points inserted be-
tween two consecutive levels of subdivision rules is called arity of the scheme.
Even-ary (odd-ary) schemes insert the even (odd) number of points between t-
wo consecutive levels of iterations. A succinct review of higher arity even-point
and odd-point schemes is presented below.
Even point binary and ternary interpolating symmetric subdivision schemes
are introduced by Ko et. al. (2007). Mustafa and Khan (2009) introduced a
new 4-point C3 quaternary approximating subdivision scheme. Even- and odd
-schemes for curve design were offered by Lian (2009). He introduced 2m-point
non-parametric interpolating schemes. A ternary even symmetric 2n- point
subdivision scheme was introduced by Zheng et al. (2009b). The generaliza-
tion of B-splines into p-ary subdivision schemes was introduced by Zheng et al.
(2009c). The unification of existing even-point interpolating and approximat-
ing schemes was offered by Mustafa and Rehman (2010). They offered general
formula to generate the mask of (2m + 4)-point even-ary schemes. Mustafa et.
al. (2014b) offered two families of (2n)-point and (2n − 1)-point p-ary interpo-
lating subdivision schemes, for any integers n > 2 and p > 3. These schemes
3
were originated from Lagrange polynomial. The ternary and three point uni-
variate schemes were offered by Hassan and Dodgson (2003). A 4-point ternary
interpolating subdivision scheme was introduced by Hassan et al. (2002). Lian
(2008) introduced two a-ary schemes with complexity 3 and 4. A family of non-
parametric interpolating odd-ary schemes with complexity (2m + 1) for curve
design was offered by Lian (2009). A family of (2n − 1)-point ternary interpo-
latory subdivision schemes was also introduced by Zheng et al. (2009a). The
generation of mask of (2n − 1)-point interpolating as well as approximating
schemes by an explicit formula was offered by Aslam et al. (2011). Mustafa et
al. (2012) also introduced an explicit formula which help out to find the mask of
odd-points n-ary interpolating schemes.
Different techniques have been used for the construction of subdivision schemes.
Baccou and Liandrat (2013) developed a new type of binary interpolatory sub-
division scheme through a stochastic approach. Their construction combines
position dependent multi scale approximation and kriging theory.
Zheng et al. (2014) presented class of convergent binary subdivision schemes
with high continuity based on eigenvalues of their difference matrix and the re-
lation between subdivision schemes and difference schemes. Rehan and Siddiqi
(2015) presented a combined binary 6-point interpolating and approximating
subdivision scheme with tension parameters.
Deng and Ma (2016) presented an efficient algorithm for the construction of
binary and ternary subdivision schemes with polynomial reproduction proper-
ty. Si et al. (2016) presented a new binary scheme called penalized Lagrange.
Their construction is based on an original reformulation for the construction of
the coefficients of the mask associated to the classical 4-points Lagrange inter-
polatory subdivision scheme. The main purpose of their work is to introduced
a new approach that allows to transform locally an interpolatory scheme into a
non-interpolatory once.
4
Deng et al. (2016) presented the m−ary 2N−point Deslauriers and Dubuc
(1989) subdivision scheme (DDSS) using a series of repeated local operations.
Nowadays subdivision schemes are constructed by using statistical techniques.
Dyn et al. (2015) presented univariate subdivision schemes based on least square
minimization to deal noisy data. Mustafa et al. (2015) used l1-regression to con-
struct subdivision schemes. Their schemes gives best fit to any type of data with
and without added noise and outliers in high dimensional spaces.
1.2 Boundary value problems
A boundary value problem is a problem (BVP), typically an ordinary differen-
tial equation or a partial differential equation, which has values assigned on the
physical boundary of the domain in which the problem is specified. Bound-
ary value problems arise in several branches of physics and engineering. For
the duration of the past period there has been a significant progress of interest
in problems associated with system of linear and nonlinear ordinary differential
equations with split boundary conditions. All over engineering and applied sci-
ence, we are challenged with two point higher order boundary value problems
that cannot be solved by analytic methods. With this interest in finding the so-
lutions of linear and nonlinear boundary value problems, different algorithms
have become an increasing need for researchers.
1.2.1 Numerical methods for BVPs
In modern development of Mathematics there are so many research problems
occurs in the form of differential equation with some conditions and researchers
work on them day by day. Numerical solutions of various problems described
by differential equations involving parameters have become increasingly com-
plex. Therefore we require the use of asymptotic methods. Several numerical
5
methods have been introduced to find the solution of boundary value problems
such as spline based method, finite difference method, finite element method
etc.
The theory of spline functions is a very interesting and active field of approx-
imation theory and boundary value problems, when numerical characteristics
are considered. Many researchers have used spline functions to construct the
algorithms for the solutions of boundary value problems. There are different
types of spline functions such as linear, quadratic, cubic, quartic, quintic, sextic,
septic, octic, nonic etc. known as polynomial spline function.
Spline functions used in the context of boundary value problems was first s-
tudied by Bickley (1968) for the solution of linear two point second order bound-
ary value problems. After that Ablasiny and Hoskins (1969), Fyfe (1969, 1970),
Sakai (1971) developed spline based method both for the linear and nonlinear
two point boundary value problems.
Caglar et al. (1999) presented a collocation algorithm based on fourth degree
B-spline for the numerical solution of third order linear and nonlinear boundary
value problems. They tested their algorithm on linear and nonlinear boundary
value problems and showed that their algorithm achieved first order accuracy.
A numerical approach based on quartic spline known as quartic spline inter-
polation to approximate the solution of second order boundary value presented
by Hamid et al. (2012). They tested their approach on several examples and
the results were compared with those of cubic B-spline and extended cubic B-
spline. They observed that their approach gives more accurate numerical results
comparative to the others.
Pandey (2016) constructed an algorithm by using quartic splines for the solu-
tion of third order boundary value problems. He also test the proposed algo-
rithm on some problems and illustrated numerical results to show the efficiency
of algorithm.
6
El-Danaf (2008) developed quartic non polynomial spline method for the nu-
merical solution of third order two point boundary value problems. They shown
that their method gives approximations better than those produced by other s-
pline methods. Convergence analysis of the method is discussed through stan-
dard procedures.
Islam and Shirin (2011) presented a method for the numerical solution of
linear and nonlinear differential equation with Dirichlet, Neumann and Robin
boundary conditions. They approximated the solution by Galerkin approxima-
tion method using bernoulli polynomials.
Rahman et al. (2012) solved numerically a second order linear boundary val-
ue problem, by the technique of Galerkin method. For this, they derived a sim-
ple and efficient matrix formulation using Hermite polynomials as trial func-
tions. The proposed method is tested on several numerical examples of sec-
ond order linear boundary value problems with Neumann and Cauchy types
boundary conditions. Wazwaz (2000, 2001a) displayed Adomian strategy to
take care of boundary value problems with Dirichlet and Neumann condition-
s. He (2001b) likewise introduced a dependable calculation for getting posi-
tive answers for nonlinear boundary value problems. Wazwaz (2001c) has fur-
ther defended the legitimacy of utilizing the disintegration strategy when mixed
boundary conditions were utilized to acquire explode arrangements. Wazwaz
(2001d) introduced numerical aftereffects of fifth order boundary value prob-
lems by utilizing the disintegration strategy. He additionally contrasted mis-
takes acquired by their technique and the errors got by utilizing the 6th degree
B-spline strategy. Their outcomes demonstrates that the deterioration technique
was more precise and simple than B-spline strategy.
Duan and Rach (2011) proposed a new modified recursion scheme for the
resolution of multi-order and multi-point boundary value problems for nonlin-
ear ordinary and partial differential equations by the Adomian decomposition
7
method.
Hasan (2012) presented a new modification of the Adomian decomposition
method to overcome difficulties occurred in the standard Adomian decompo-
sition method for solving three point boundary value problems, named as the
modified Adomian decomposition method. They developed their technique
for the numerical solution of linear and nonlinear third order boundary value
problems. They showed the efficiency of their modified method by illustrating
numerical examples.
Abdullah et al. (2013) presented a method for the solution of third order non-
linear boundary value problem using fourth order block method. Convergence
of their algorithm is checked by Newton’s method. Their method is simple,
efficient and numerical results shows that their method gives better results as
compared to others.
Kalyani and Rao (2013) have used non-polynomial spline functions to con-
struct a numerical algorithm to obtained the solution of second order boundary
value problem. They have shown that non-polynomial spline produced more
accurate results in comparison with the results obtained by the finite difference
method and B-spline method.
A detail discussion on the numerical and asymptotic study of some third or-
der ordinary differential equations relevant to draining and coating flows is giv-
en by Tuck and Schwartz (1990). Duffy and Wilson (1997) described the prop-
erties of the nonlinear third order differential equation y′′′ = y−2 relevant to
Tanner’s law. The solution of this type of problems first time given by Ford
(1992). Macroscopic thin liquid films are entities that are important in biophysic-
s, physics, and engineering, as well as in natural settings. Oron et al. (1997) re-
viewed, a unified mathematical theory that takes advantage of the disparity of
the length scales and is based on the asymptotic procedure of reduction of the
full set of governing equations and boundary conditions to a simplified, highly
8
nonlinear, evolution equation or to a set of equations.
Craster and Matar (2009) are reviewed these problems and exciting research
avenues in this thriving area of fluid mechanics. Numerical investigation of this
type of problem also presented by Momonait (2011), they solved the differential
equation from thin film flow by comparing two finite difference methods.
A mathematical analysis for boundary value problem which arises in the
model for flows draining down a dry vertical wall is presented by Wang and
Zhang (1998). They also discussed the existence and qualitative properties of
solutions. A new algorithm for solving the general nonlinear third order differ-
ential equation is developed by Bhrawy and Abd-Elhameed (2011) using shifted
Jacobi-Gauss collocation spectral method.
Yalcinbas et al. (2016) presented a numerical approach to approximate the
solution of nonlinear boundary value problems. Their technique is based on the
truncated Fermat series and its matrix representation with collocation points.
Their technique is easy to implement and produced accurate results.
El-Salam and Zaki (2010) developed a class of accurate methods based on
quartic non polynomial spline function at midknots for the numerical solution
of a fourth order two point boundary value problems associated with plate de-
flection theory.
Siddiqi and Akram (2008) used quintic spline to construct the algorithm for
the numerical solutions of the fourth order linear special case boundary value
problems. End conditions for the definition of spline are derived, consistent
with the fourth order boundary value problem. They also proved that their
method is a second order convergent.
Sakai and Usmani (1983) presented an algorithm for nonlinear fourth order
two point boundary value problems based on spline function. They developed
two algorithms of order two and four for the continuous approximation of the
solution of a nonlinear fourth order two point boundary value problems.
9
Gupta and Srivastava (2011) presented computational method using cubic B-
spline to solve fourth order boundary value problems. The proposed method
is first used for solution of fourth order boundary value problems and then
extended it for the solution of nonlinear and singular problems.
Usmani (1983) offered finite difference method of order two, four and six for
the numerical solution of fourth order linear boundary value problems. He
also described the sufficient conditions which guarantees the unique solution
of fourth order boundary value problems. He also compared his method with
shooting method and fourth order Runge Kutta method.
The above mention methods are mostly based on finite difference methods
and spline based methods. For finite differences methods, only discrete ap-
proximate values of the unknown y(x) can be obtained. We need further data
processing techniques to get accurate fitted curve to data. For spline interpo-
lation or approximation methods the unknown function y(x) is assumed to be
piecewise polynomial which in turn requires at least piecewise higher order d-
ifferentiability of the function f(x, y, u, v).
To overcome above disadvantages, Qu (1996a) presented a new technique for
the numerical differentiation and integration by using the interpolating subdi-
vision algorithm. The main advantage of his numerical formulae is that they
produce better numerical results if data comes from functions with fractal like
derivatives. Qu and Agarwal (1996b) introduced the subdivision based algo-
rithm for the solution of two point second order boundary value problems.
They used 6-point binary subdivision scheme to construct the algorithm for the
solution of second order linear boundary value problems. They also presented
the numerical implementation of their algorithm and obtained better results as
compared to the others. Qu and Agawal (1997a) formulated an iterative algo-
rithm for the solution of second order singular nonlinear two point boundary
value problems. Their method is basically a collocation method for nonlinear
10
second order two point boundary value problems with singularities at either
one or both of the boundary points. Qu and Agrwal (1997b) used 6-point bi-
nary subdivision scheme to construct an iterative scheme for solving nonlinear
two point boundary value problems. They presented their technique only for
second order boundary value problems. Qu and Agarwal (1998) also offered
a high accuracy algorithm based on subdivision scheme to compute numerical
solutions for two point boundary value problems of differential equations with
deviating arguments.
The numerical treatment of singularly perturbed problems is currently a field,
in which active research is going on these days. Singularly perturbed problem-
s in which the term containing the highest order derivative is multiplied by a
small parameter ε occur in a number of areas of applied mathematics, science
and engineering among them fluid mechanics (boundary layer problems) elas-
ticity (edge effort in shells) and quantum mechanics.
The solution of a singularly perturbed boundary value problem act like a
multi-scale character. The solution varies quickly near at thin transition lay-
er while away from the layer the solution performs regularly and varies slowly.
Therefore many obstacles are met in solving singularly perturbed boundary val-
ue problems using standard numerical methods. In recent development a large
number of methods for different purpose have been established to provide ac-
curate results.
Three standard approaches are common to solve numerically the singularly
perturbed boundary value problems in literature i.e. the finite element method,
the finite difference method and spline approximation method.
Aziz and Khan (2002) and Khan and Aziz (2005) solved second order sin-
gularly perturbed boundary value problems using cubic spline in compression
and in tension. The convergence of their method depends on the choice of the
parameter involved in the method. A numerical method for self adjoint sin-
11
gular perturbation problems based on quintic spline is presented by Bawa and
Natesan (2005). Kadalbajoo and patidar (2002) presented tension spline approx-
imation method, to solve the third order boundary value problems. Kadalbajoo
and Aggarwal (2005) presented fitted mesh B-spline collocation algorithm for
the solution of self adjoint singular perturbation problems. Tirmizi et al. (2008)
presented a generalized scheme based on non-polynomial spline functions for
the solution of singularly perturbed two point boundary value problems. A nu-
merical method based on uniform Haar wavelet for the solution of singularly
perturbed two point boundary value problems is presented by Pandit and Ku-
mar (2014). Kumar and Mehra (2009) designed a wavelet optimized difference
method for singularly perturbed problems. Their method is based on interpo-
lating wavelet transform. Lubuma and Patidar (2006) presented non standard
finite difference scheme for singularly perturbed problems. The finite difference
scheme based methods also presented by Niijima (1980a , 1980b).
Only few results are available for higher order singularly perturbed bound-
ary value problems. The class of third order singularly perturbed boundary
value problems has been solved by Kumar (2002). He used fourth order fi-
nite difference scheme based on non uniform mesh. Akram (2012) solved third
order self-adjoint singulary perturbed boundary problem by using fourth de-
gree spline. Cui and Geng (2008) presented a new numerical method for the
class of third order boundary value problems with a boundary layer at the left
of the underlying interval. Boundary value technique for the solution of class
of third order singulary perturbed boundary value problems is presented by
Valarmatht and Ramanujam (2002). Su-rang et al. (2001) presented a method to
solved singularly perturbed boundary value problem for quasi linear third or-
der ordinary differential equations involving two parameters. A uniform Haar
wavelets method is proposed by Haq et al. (2011) to find approximate solu-
tion of multi-point third order boundary value problems related to flow in a
12
wedge-shaped region, sandwich beam model, system of ordinary differential
equations, nonlinear third order initial value problems in thin film flow. To
check the performance of their method, they compared their method with the
finite difference method, Pade approximant, spline based methods, method of
superposition, method of chasing and method of adjoint operators. The main
advantage of their method is its efficiency and simple applicability as compared
to others.
1.2.2 Contribution of this dissertation
Subdivision based methods for the numerical solution of boundary value prob-
lems have also been used in the literature by Qu and Agarwal (1996, 1997a,
1997b). They developed subdivision based methods only for second order bound-
ary value problems. Higher order boundary value problems have not been
solved by subdivision methods. This motivates us to solve these type of prob-
lems by subdivision algorithms. The following problems have been solved in
this dissertation:
Mustafa and Ejaz (2014) solved linear third order boundary value problems
by using subdivision technique. They used 8-point binary subdivision scheme
to construct collection algorithm for the solution of linear third order boundary
value problems. They compared their technique with B-spline based method
introduced by Caglar et al. (1999).
Ejaz et al. (2015) solved two point fourth order linear boundary value prob-
lems by subdivision based algorithm. They construct two collocation algorithm-
s based on interpolating and approximating subdivision schemes for the solu-
tion of fourth order boundary value problems.
Ejaz and Mustafa (2016) presented an algorithm for the numerical solution of
nonlinear third order boundary value problems. Their algorithm is based on
eight point binary subdivision scheme. Proposed algorithm is stable, conver-
13
gent and give more accurate results than fourth degree B-spline algorithm.
Mustafa et al. (2017) presented an iterative collocation numerical approach
based on interpolating subdivision schemes for the solution of nonlinear fourth
order boundary value problems involving ordinary differential equations. Nu-
merical evidence suggested that their scheme converges to a smooth approxi-
mate solution of nonlinear fourth order boundary value problem. The conver-
gence of their approach is also discussed.
Mustafa and Ejaz presented subdivision based collocation algorithm for the
numerical solution of second order singularly perturbed boundary value prob-
lems. They also discussed convergence of their algorithm. They concluded
that their algorithm for the solutions of singularly perturbed boundary value
problems gives better results comparative to existing methods such as Aziz
and Khan (2002), Bawa and Natesan (2005), Khan and Aziz (2005), Kadalbajoo
and Aggarwal (2005) , Lubuma (2006), Miller (1979), Niijima (1980a), Niijima
(1980b), Kumar and Mehra (2002) and Pandit and Kumar (2014). (Article sub-
mitted )
Mustafa and Ejaz (2017) solved linear third order singularly perturbed bound-
ary value problems by subdivision based collocation algorithm. Convergence
properties of their algorithm is discussed. The comparison of the solutions ob-
tained by their algorithm with B-spline based method is also given.
1.3 Basic definitions
Definition 1.3.1. Parametric curve In a coordinate plane a curve is traced out
by the points (x, y) = (f(t), g(t)) as t varies, x and y are given as functions of a
variable t (called a parameter) by the equations
x = f(t), y = g(t),
which is called parametric equations.
14
Definition 1.3.2. Parametric continuity It is a concept applied to parametric
curve describing the smoothness of parametric value with distance along the
curve. Parametric continuity indicates the smoothness of the motion.
Definition 1.3.3. Order of parametric continuity
• C0: the segments meet at the nodes or joint points.
• C1: first derivatives are equal and continuous at joint points.
• C2: first and second derivatives are equal and continuous at joint points.
• Cn: first through n-th derivatives are equal and continuous at joint points.
Definition 1.3.4. Subdivision It means divide and rule. It is a set of subdivi-
sion rules called subdivision scheme for generating curves and surfaces as a
sequence of successively refined polygons.
Definition 1.3.5. Binary subdivision schemes A general form of univariate bi-
nary subdivision scheme S which maps a control polygon pk = {pki }i∈Z to a
refined polygon pk+1 = {pk+1i }i∈Z defined as
pk+12i+S =
∑j∈Z
a2j+Spki−j, S = 0, 1,
where the set {ai : i ∈ Z} of coefficient is called mask of the subdivision scheme.
The z-transform of the mask a of subdivision scheme can be given as
a(z) =∑i∈Z
aizi,
which is called the symbol or Laurent polynomial of the scheme.
Definition 1.3.6. Primal binary scheme The scheme which keeps or modifies
the old vertices and create new vertex at each old edge of the control polygon is
called primal binary scheme i.e. if edge maps into vertices then scheme is called
primal (see Figure 1.1(a)).
15
Definition 1.3.7. Dual binary scheme The scheme which creates two new ver-
tices at each old edge is called dual binary scheme i.e. if edges maps into edges
then scheme is called dual ( see Figure 1.1(b)).
(a) (b)
Figure 1.1: (a) represents primal binary and (b) represents dual binary scheme.
Definition 1.3.8. Approximating and interpolating subdivision scheme Approx-
imating scheme means that the limit curve approximates the initial polygon and
that after subdivision, only the newly generated control points are in the limit
curve. While interpolating scheme means that after subdivision, the control
points of the original control polygon and the newly generated control points
both lie on the limit curve.
Definition 1.3.9. Necessary condition of convergent A binary subdivision scheme,
with a mask a satisfies the necessary condition of convergence, if∑i∈Z
a2i =∑i∈Z
a2i+1 = 1.
Definition 1.3.10. Continuity of the scheme Continuity refers to the differentia-
bility of the limit curve/surface produced by subdivision process. Subdivision
16
schemes should be continuous of a certain order prior to construction i.e. Cm
continuity means that the first through m-th derivatives are equal and continues
at the joint points.
Definition 1.3.11. Degree of generation The generation degree of a subdivision
scheme is the maximum degree of polynomials that can potentially be generated
by the scheme, provided that the initial data is chosen correctly. Suppose p0 is
polynomial of degree d of initial data f 0i and symbol of the scheme is
a(z) = (1 + z + z2 + · · ·+ zn−1)d+1b(z),
then the limit curve of the refined data fki at any level k is polynomial of de-
gree d. So the condition is necessary and sufficient for the scheme being able to
generate polynomial of degree d.
Definition 1.3.12. Degree of reproduction A subdivision scheme Sa reproduces
polynomials of degree d if it is convergent and if S∞a f 0 = p for any polynomial
p ∈ πd and initial data f 0 = p(t0i ), i ∈ Z.
Definition 1.3.13. Approximation order A convergent subdivision scheme that
reproduces polynomial of degree d has an approximation order d+ 1.
1.4 Outlines of the dissertation
Rest of the dissertation is organized as follows:
In Chapter 2: A numerical interpolating collocation algorithm is formulated,
based on 8-point binary interpolating subdivision schemes for the generation of
curves, to solve the two point third order boundary value problems. Numerical
examples are given to illustrate the algorithm and its convergence.
In Chapter 3: We present two collocation algorithms based on interpolating and
approximating subdivision schemes for the solution of fourth order boundary
17
value problems. Main purpose of this chapter is to explore and seek the appli-
cations of interpolating and approximating subdivision schemes in the field of
boundary value problems along with intrinsic comparison of the results obtain
by subdivision based algorithms.
In Chapter 4: We construct an iterative algorithm for the numerical solution of
non linear third order boundary value problems. This algorithm is based on
eight point binary subdivision scheme. Proposed algorithm is stable, conver-
gent and give more accurate results than fourth degree B-spline algorithm.
In Chapter 5: We present an iterative collocation algorithm based on interpo-
lating subdivision schemes for the solution of non linear fourth order bound-
ary value problems. The convergence of the algorithm is also discussed. It is
proved that the iterative algorithm converges to a smooth approximate solution
provided the boundary value problem is well posed and algorithm is applied
appropriately.
In Chapter 6: Singularly perturbed second order boundary value problems fre-
quently arise in the various field of science and engineering. In this chapter,
we introduce subdivision based collocation algorithm for the solutions of these
types of problems. Numerical results show that the suggested algorithm for the
solutions of singularly perturbed boundary value problems is batter than spline
based methods, finite difference methods and Haar wavelet methods.
In Chapter 7: We propose an algorithm for the numerical solution of self ad-
joint singularly perturbed third order boundary value problems. Convergence
of the subdivision collocation algorithm is also discussed. Numerical examples
are presented to illustrate the proposed algorithm.
In Chapter 8: We present the comparison, conclusion, limitations of proposed
algorithms and future research directions.
18
Chapter 2
Numerical Solution of Two Point
Boundary Value Problems by
Interpolating Subdivision Schemes
In this chapter, we formulate the subdivision based collocation algorithm for the
approximate solution of two point boundary value problems of order three. Our
formulated subdivision based collocation algorithm treats the following types
of boundary value problems:
y′′′(x) = a(x)y(x) + b(x), 0 6 x 6 1
y(0) = α1, y(1) = α2, y′(0) = 0.(2.1)
where a(x) and b(x) are continuous and a(x) > 0 on [0, 1]. In Section 2.1, we
rewrite general form of interpolating subdivision scheme for curve design pre-
sented by Qu (1994) and some related results. The 8-point binary interpolating
scheme and derivatives of its basis function have been also discussed in this sec-
tion. In Section 2.2, a numerical interpolating algorithm of collocation to solve
(2.1) is formulated and its boundary treatments are discussed. In Section 2.3,
approximation properties of this algorithm are given. In Section 2.4, numerical
examples are presented.
19
2.1 Interpolating schemes for curve design
A general compact form of symmetric univariate binary interpolating subdi-
vision scheme presented by Qu (1994) which maps polygon pk = {pki }i∈Z to a
refined polygon pk+1 = {pk+1i }i∈Z is defined by
pk+12i = pki ,
pk+12i+1 =
n∑j=0
Ln,j
(pki−j + pki+j+1
),
(2.2)
where n is called the degree of the scheme and the constants are given by
Ln,j =((2n+ 1)!!)2
2(4n)(2n+ 1)!
(−1)j
(2j + 1)
2n+ 1
n− j
, j = 0, 1, 2, · · · , n, (2.3)
where
2n+ 1
n− j
denotes the binomial coefficient.
The boundary treatments are necessary to produce smooth curve segments by
scheme (2.2). Normally higher order approximation formulae should be used
near the ends of the segments and thus Lagrange formulae of order 2n + 1 are
recommended.
Remark 2.1.1. Let ϕ(x) be the limit curve generated from the cardinal data {pi =
(i, δ0)T}, that is, ϕ(x) is the fundamental solution of the subdivision scheme (2.2),
then
ϕ(i) =
1, i = 0,
0, i = 0.(2.4)
Furthermore, ϕ(x) satisfies the following two-scale equation
ϕ(x) = ϕn(x) = ϕ(2x) +n∑
j=−n
Ln,|j|ϕ(2x− 2j + 1), x ∈ R. (2.5)
20
Lemma 2.1.1. ( Qu (1994), Qu and Agarwal (1995)). The support of the fundamental
solution ϕn(x) to scheme (2.2) is finite. Explicitly, supp ϕn(x) = (−2n− 1, 2n+ 1).
Lemma 2.1.2. ( Qu and Agarwal (1996)). Given a square matrix A of order n, let the
normalized left and right (generalized) eigenvectors of A be denoted by {ηi, ξi}. Then
for any vector f ∈ Rn, then there exist following Fourier expansion
f =n∑
i=1
(fTηi)ξi.
Lemma 2.1.3. ( Qu and Agarwal (1996)). Suppose F (t), t ∈ R is a regular and C2n+2
curve in Rm, m ≥ 2. Let P (t), t ∈ R be the limit curve generated by (2.2) from the
initial data Pi = F (ih), i ∈ Z, 0 < h < 1. Then, on any finite interval [a, b], following
estimates hold
||F (ht)− p(t)||∞ ≤ M2n+2(F )
(2n+ 2)!h2n+2 = O(h2n+2), (2.6)
and
||hjF j(ht)− pj(t)||∞ = O(h2n+2−j), j = 0, 1, 2, · · · , n+ 2
2, (2.7)
where the number M2n+2(F ) depends only on the derivatives of F (t) and n.
2.1.1 8-point interpolating scheme
For n = 3 by (2.2) and (2.3), we have the following 8-point binary interpolating
subdivision scheme for curve design
pk+12i = pki
pk+12i+1 =
12252048
(pki + pki+1
)− 245
2048
(pki−1 + pki+2
)+ 49
2048
(pki−2 + pki+3
)− 5
2048
(pki−3 + pki+4
).
(2.8)
21
This scheme is C3-continuous by Dyn (2002) and reproduce polynomial curve
of degree seven by Conti and Hormann (2011). The local subdivision matrix of
(2.8) is denoted and defined by
S =
0 0 0 1 0 0 0 0 0 0 0 0 0
L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0
0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3 0
0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 L3,3 L3,2 L3,1 L3,0 L3,0 L3,1 L3,2 L3,3
0 0 0 0 0 0 0 0 0 1 0 0 0
where L3,0 = 1225
2048, L3,1 = − 245
2048, L3,2 = 49
2048and L3,3 = − 5
2048. The some of its
eigenvalues are
λ = 1, 12, 14, 18, 116, 132, 164, 1128
.
For an eigenvalue λ, the eigenvectors ξ and η that satisfies Sξ = λξ and ηST = ηλ
are called right and left eigenvectors of the matrix S respectively. Some of the
22
normalized left and right eigenvectors corresponding to eigenvalues are
ξ0 = (1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1)T ,
η0 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)T ,
ξ1 = (−6,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, 6)T ,
η1 =(
15946360
)(−5, 1024, 13225,−199680, 1141695,−4715520, 0, 4715520,−1141695,
199680,−13225,−1024, 5)T ,
ξ2 = (36, 25, 16, 9, 4, 1, 0, 1, 4, 9, 16, 25, 36)T ,
η2 =(
134546860
)(275,−28160,−182613, 2607616,−12053651, 45634048,−71955030,
45634048,−12053651, 2607616,−182613,−28160, 275)T ,
ξ3 = (−216,−125,−64,−27,−8,−1, 0, 1, 8, 27, 64, 125, 216)T ,
η3 =(
115039360
)(225,−11520, 10952, 476928,−3047987, 4677632, 0,−4677632,
3047987,−476928,−10952, 11520,−225)T .
Since ξTi ηj = 1 for i = j and 0 otherwise then by using Lemmas 2.1.1 and 2.1.2,
we get following result.
Lemma 2.1.4. The fundamental solution (Cardinal basis) ϕ(x) is thrice continuously
differentiable and supported on (−7, 7) and its derivatives at integers are given by
ϕ′(i) = 2sign(i)eT|i|η1, ϕ′′(i) = 22eT|i|η2, ϕ′′′(i) = 23sign(i)eT|i|η3, −6 ≤ i ≤ 6,
where
e0 = (0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0)T , e1 = (0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0)T ,
e2 = (0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0)T , e3 = (0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,
e4 = (0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T , e5 = (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,
e6 = (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0)T ,
23
and ϕ′(0) = 0, ϕ′(±1) = ∓78592
49553, ϕ′(±2) = ± 76113
198212,
ϕ′(±3) = ∓ 332849553
, ϕ′(±4) = ± 2645594636
, ϕ′(±5) = ± 256743295
,
ϕ′(±6) = ∓ 1594636
,
ϕ′′(0) = −342643
41124, ϕ′′(±1) = 5704256
1079505, ϕ′′(±2) = −12053651
8636040,
ϕ′′(±3) = 3259521079505
, ϕ′′(±4) = − 608712878680
, ϕ′′(±5) = − 704215901
,
ϕ′′(±6) = 551727208
,
ϕ′′′(0) = 0, ϕ′′′(±1) = ∓292352
117495, ϕ′′′(±2) = ±3047987
1879920,
ϕ′′′(±3) = ∓ 331213055
, ϕ′′′(±4) = ∓ 1369234990
, ϕ′′′(±5) = ± 162611
,
ϕ′′′(±6) = ∓ 541776
.
(2.9)
The graphical representations of the basis limit function defined on cardinal
data and its derivatives up to order three for n = 3 are shown in Figure 2.1. Fig-
ure 2.1(a) represents the basis limit function defined in (2.4). Graphical repre-
sentations of first, second and third derivatives of basis limit functions obtained
from (2.2) for n = 3 are shown in Figures 2.1(b), 2.1(c) and 2.1(d) at i = 0, 1,−1,
respectively. The numeric values of first, second and third derivative are given
in Lemma 2.1.4.
2.2 Numerical interpolating collocation algorithm
In this section, first we formulate a numerical interpolating collocation algorith-
m for linear third order two point boundary value problems. Then we settle
down the boundary conditions to get unique solution.
24
Figure 2.1: Interpolatory basis function ϕ3(i) is shown in (a), first derivative of ϕ3(i)
i-e. ϕ′3(i) shown in (b), second derivative of ϕ3(i) i-e. ϕ′′
3(i) shown in (c) and third
derivative of ϕ3(i) i-e. ϕ′′′3 (i) shown in (d) respectively
25
2.2.1 The collocation algorithm
Let N be a positive integer (N ≥ 6), h = 1/N and xi = i/N = ih, i = 0, 1, 2, · · ·N ,
and set ai = a(xi), bi = b(xi). Let
Z(x) =N+6∑i=−6
ziϕ(x−xi
h
), 0 ≤ x ≤ 1, (2.10)
be the approximate solution to (2.1) where {zi} are the unknown to be deter-
mined by (2.1). The collocation algorithm together with the boundary condi-
tions to be discussed, is given by setting
Z ′′′(xj) = a(xj)Z(xj) + b(xj), j = 0, 1, 2, · · · , N, (2.11)
where
Z ′(xj) =1h
N+6∑i=−6
ziϕ′ (xj−xi
h
),
Z ′′(xj) =1h2
N+6∑i=−6
ziϕ′′ (xj−xi
h
),
Z ′′′(xj) =1h3
N+6∑i=−6
ziϕ′′′ (xj−xi
h
).
(2.12)
Using (2.10) and (2.12) in (2.11), we get following N + 1 system of equationsN+6∑i=−6
ziϕ′′′ (xj−xi
h
)− h3aj
N+6∑i=−6
ziϕ(xj−xi
h
)= h3bj, j = 0, 1, 2, · · · , N. (2.13)
Now we simplify the above system in the following theorems.
Theorem 2.2.1. For j = 0 by (2.13), we get
z−6ϕ′′′6 + z−5ϕ
′′′5 + z−4ϕ
′′′4 + z−3ϕ
′′′3 + z−2ϕ
′′′2 + z−1ϕ
′′′1 + z0q0 + z1ϕ
′′′−1 + z2ϕ
′′′−2
+z3ϕ′′′−3 + z4ϕ
′′′−4 + z5ϕ
′′′−5 + z6ϕ
′′′−6 = h3b0, (2.14)
where ϕ′′′j = ϕ′′′(j) and q0 = ϕ′′′
0 − a0h3.
Proof. Substituting j = 0 in (2.13), we get{z−6ϕ
′′′(x0−x−6
h) + z−5ϕ
′′′(x0−x−5
h) + · · ·+ zN+5ϕ
′′′ (x0−xN+5
h
)+ zN+6ϕ
′′′ (x0−xN+6
h
)}−a0h
3{z−6ϕ(
x0−x−6
h) + z−5ϕ(
x0−x−5
h) + · · ·+ zN+5ϕ(
x0−xN+5
h) + zN+6ϕ(
x0−xN+6
h)}
= h3b0.
26
For xj = jh, j = 0, 1, 2 · · ·N , this implies
z−6ϕ′′′(6) + z−5ϕ
′′′(5) + · · ·+ zN+5ϕ′′′(−N − 5) + zN+6ϕ
′′′(−N − 6)
−a0h3 {z−6ϕ(6) + z−5ϕ(5) + · · ·+ zN+5ϕ(−N − 5) + zN+6ϕ(−N − 6)} = h3b0.
Since the support of basis function ϕ(x) is (−7, 7), ϕ′(x), ϕ′′(x) and ϕ′′′(x) are zero
out side the interval (−7, 7), also by (2.4) and (2.9), we get
z−6ϕ′′′(6) + z−5ϕ
′′′(5) + z−4ϕ′′′(4) + z−3ϕ
′′′(3) + z−2ϕ′′′(2) + z−1ϕ
′′′(1) + z0ϕ′′′(0)
+z1ϕ′′′(−1) + z2ϕ
′′′(−2) + z3ϕ′′′(−3) + z4ϕ
′′′(−4) + z5ϕ′′′(−5) + z6ϕ
′′′(−6)
−a0h3z0ϕ(0) = h3b0.
If ϕ′′′j = ϕ′′′(j), then
z−6ϕ′′′6 + z−5ϕ
′′′5 + z−4ϕ
′′′4 + z−3ϕ
′′′3 + z−2ϕ
′′′2 + z−1ϕ
′′′1 + z0(ϕ
′′′0 − a0h
3) + z1ϕ′′′−1
+z2ϕ′′′−2 + z3ϕ
′′′−3 + z4ϕ
′′′−4 + z5ϕ
′′′−5 + z6ϕ
′′′−6 = h3b0.
For q0 = ϕ′′′0 − a0h
3, we get (2.14). This completes the proof.
Theorem 2.2.2. For j = 1, 2, 3, · · · , N the system (2.13) is equivalent to
z−6ϕ′′′j+6 + z−5ϕ
′′′j+5 + · · ·+ z0ϕ
′′′j + z1(ϕ
′′′j−1 − ajh
3ϕj−1) + z2(ϕ′′′j−2 − ajh
3ϕj−2)
+ · · ·+ zN−1(ϕ′′′j−N+1 − ajh
3ϕj−N+1) + zN(ϕ′′′j−N − ajh
3ϕj−N) + zN+1ϕ′′′j−N−1
+ · · ·+ zN+6ϕ′′′j−N−6 = h3bj. (2.15)
Proof. By expanding equation (2.13), we get
z−6ϕ′′′(
xj−x−6
h) + z−5ϕ
′′′(xj−x−5
h) + · · ·+ zN+5ϕ
′′′(xj−xN+5
h) + zN+6ϕ
′′′(xj−xN+6
h)
−ajh3{z−6ϕ(
xj−x−6
h) + z−5ϕ(
xj−x−5
h) + · · ·+ zN+5ϕ(
xj−xN+5
h) + zN+6ϕ(
xj−xN+6
h)}
= h3bj.
27
For xj = jh, j = 1, 2 · · ·N , we get
z−6ϕ′′′(j + 6) + z−5ϕ
′′′(j + 5) + · · ·+ zN+5ϕ′′′(j −N − 5) + zN+6ϕ
′′′((j −N − 6)
−ajh3 {z−6ϕ(j + 6) + z−5ϕ(j + 5) + · · ·+ zN+5ϕ(j −N − 5) + zN+6ϕ(j −N − 6)}
= h3bj.
If ϕ′′′j = ϕ′′′(j), for j = 1, 2 · · ·N then
z−6(ϕ′′′j+6 − ajh
3ϕj+6) + z−5(ϕ′′′j+5 − ajh
3ϕj+5) + · · ·+ zN+5(ϕ′′′j−N−5 − ajh
3ϕj−N−5)
+zN+6(ϕ′′′j−N−6 − ajh
3ϕj−N−6) = h3bj.
Since ϕ′(x), ϕ′′(x) and ϕ′′′(x) are zero out side the interval (−7, 7) then by (2.4)
and (2.9), we get (2.15).
From (2.14) and (2.15), we get following un-determine system of (N + 1) e-
quations with (N + 13) unknowns {zi}
AZ = D, (2.16)
where the matrices A, Z, and D of orders (N + 1)× (N + 13), (N + 13)× 1 and
(N + 1)× 1 respectively are given by
A =
ϕ′′′6 ϕ′′′
5 ϕ′′′4 ϕ′′′
3 ϕ′′′2 ϕ′′′
1 q0 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3 ϕ′′′
−4 ϕ′′′−5 ϕ′′′
−6
0 ϕ′′′6 ϕ′′′
5 ϕ′′′4 ϕ′′′
3 ϕ′′′2 ϕ′′′
1 q1 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3 ϕ′′′
−4 ϕ′′′−5
0 0 ϕ′′′6 ϕ′′′
5 ϕ′′′4 ϕ′′′
3 ϕ′′′2 ϕ′′′
1 q2 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3 ϕ′′′
−4
0 0 0 ϕ′′′6 ϕ′′′
5 ϕ′′′4 ϕ′′′
3 ϕ′′′2 ϕ′′′
1 q3 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
28
· · · 0 0 0
· · · 0 0 0
· · · 0 0 0
· · · 0 0 0
· · · · · · · · · · · ·
· · · ϕ′′′−5 ϕ′′′
−6 0
· · · ϕ′′′−4 ϕ′′′
−5 ϕ′′′−6
, (2.17)
Z = (z−6, z−5, z−4, z−3, z−2, · · · , zN+6)T , D = (b0h
3, b1h3, b2h
3, b3h3, · · · , bNh3)T ,
where ϕ′′′j = ϕ′′′(j) and qj = ϕ′′′
0 − ajh3.
2.2.2 Adjustment of boundary conditions
The order of the coefficient matrix (2.17) is (N + 1) × (N + 13). In order to get
unique solution of the system, we need twelve more conditions. Here we con-
sider only two different cases. In coming section we will show that the approx-
imate solution can be improved by adjusting different boundary conditions.
Case:-1 If we assume z′0 = 0 (equivalently z0 = yr = finite) then two conditions
can be achieved by using following given boundary conditions i.e.
z0 = yr, z′0 = 0, zN = yl. (2.18)
Still we need ten more conditions to get stable system. Since subdivision scheme
reproduces seven degree polynomials, we define boundary conditions of order
eight for solution of (2.16). For simplicity only the left end points are discussed
and the values of right end points zN+1, zN+2, zN+3, zN+4, zN+5 can be treated
similarly.
29
The values z−5, z−4, z−3, z−2, z−1 can be determined by the septic polynomial q(x)
interpolating at (xi, zi), 0 ≤ i ≤ 7. Precisely, we have
z−i = q(−xi), i = 1, 2, 3, 4, 5,
where
q(xi) =8∑
j=1
8
j
(−1)j+1Z(xi−j).
Since by (2.10) Z(xi) = zi for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we have
q(−xi) =8∑
j=1
8
j
(−1)j+1z−i+j.
Hence the following boundary conditions can be employed at the left end
8∑j=0
8
j
(−1)jz−i+j = 0, i = 5, 4, 3, 2, 1. (2.19)
Similarly for the right end, we can define zi = q(−xi), i = N + 1, N + 2, N + 3,
N + 4, N + 5 and
q(xi) =8∑
j=1
8
j
(−1)j+1zi−j.
So we have the following boundary conditions at the right end
8∑j=0
8
j
(−1)jzi−j = 0, i = N + 1, N + 2, N + 3, N + 4, N + 5. (2.20)
Finally, we get a following new system of (N + 13) linear equations with (N +
13) unknowns {zi}, in which N + 1 equations are obtained from (2.13), two
equations from boundary conditions (2.18) and ten from boundary conditions
(2.19) and (2.20)
BZ = R, (2.21)
30
where the coefficients matrix B = (BT0 , A
T , BT1 )
T , A is defined by (2.17), B0 and
B1 formed by (2.18), (2.19) and (2.20)
B0 =
0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 · · · 0 0
0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 · · · 0 0
0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 · · · 0 0
0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 · · · 0 0
0 0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 · · · 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 · · · 0 0
,
where the first five rows of B0 come from (2.19) and the sixth row comes from
(2.18) at z0 = yr. Consider
B1 =
0 0 · · · 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 · · · 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 0
0 0 · · · 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0
0 0 · · · 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0
0 0 · · · 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0
0 0 · · · 0 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0
,
where first row of B1 comes from (2.18) at zN = yl remaining rows come from
(2.20) and the matrices Z and R are defined below
Z = (z−6, z−5, · · · zN+5, zN+6)T ,
R = (0, 0, 0, 0, 0, yr, DT , yl, 0, 0, 0, 0, 0)
T .
Case:-2 In this case we express the given boundary condition z′0 = 0 in the follow-
ing way:
By using (2.12) we have
Z ′(xj) =1
h
N+6∑i=−6
ziϕ′(xj − xi
h
).
31
As we define earlier xj = jh if we put j = 0 we get x0 = 0, the above equation
can be written as
Z ′(0) =1
h
N+6∑i=−6
ziϕ′(−i).
Since by boundary condition z′0 = Z ′(0) = 0, so
N+6∑i=−6
ziϕ′(−i) = 0.
By using (2.17) we can express above equation as
1
594636z−6 +
256
743295z−5 +
2645
594636z−4 −
3328
49553z−3 +
76113
198212z−2 −
78592
49553z−1
+78592
49553z1 −
76113
198212z2 +
3328
49553z3 −
2645
594636z4 −
256
743295z5 −
1
594636z6 = 0.
(2.22)
Finally, we get a following new system of (N+13) linear equations with (N+13)
unknowns {zi}, in which N+1 equations are obtained from (2.14) and (2.15), two
equations from boundary conditions (2.18) and ten from boundary conditions
(2.19) for i = 5, 4, 3, 2, (2.20) and (2.22)
BZ = R, (2.23)
where the coefficients matrix B = (BT0 , A
T , BT1 )
T , A is defined by (2.17), B0 and
B1 formed by (2.18), (2.19), (2.20) and (2.22) are defined below
B0 =
0 1 −8 28 −56 70 −56 28 −8
0 0 1 −8 28 −56 70 −56 28
0 0 0 1 −8 28 −56 70 −56
0 0 0 0 1 −8 28 −56 70
1594636
256743295
2645594636 − 3328
4955376113198212 −78592
49553 0 7859249553
76113198212
0 0 0 0 0 0 1 0 0
32
1 0 0 0 0 · · · 0 0
−8 1 0 0 0 · · · 0 0
28 −8 1 0 0 · · · 0 0
−56 28 −8 1 0 · · · 0 0
332849553
− 2645594636
− 256743295
− 1594636
0 · · · 0 0
0 0 0 0 0 · · · 0 0
,
in B first four rows come from (2.19) for i = 5, 4, 3, 2, fifth row comes from (2.22)
and the last row comes from (2.18). The matrix B1 is same as defined in Case 1
and the matrices Z and R are defined below
Z = (z−6, z−5, · · · zN+5, zN+6)T ,
R = (0, 0, 0, 0, y′(0), yr, D
T , yl, 0, 0, 0, 0, 0)T .
The non-singularity of the coefficients matrix B has been discussed in next sec-
tion.
2.2.3 Existence of the solution
In this section, we discuss the non-singularity of the coefficients matrix B. We
observe that the coefficients matrix B is neither symmetric nor diagonally dom-
inant. However it can be shown that B is a non-singular. Since B is almost a
band matrix with half band width 7, numerical complexity for solving the linear
system using Gaussian elimination is about 49(N +9) multiplications. For large
N , the matrix is almost symmetric except the first and last six rows and columns
due to the boundary conditions. Therefore we first consider the symmetric part
33
of it i.e. square band matrix C of order N + 3 defined as
C =
ϕ′′′1 ϕ′′′
0 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3 ϕ′′′
−4 ϕ′′′−5 ϕ′′′
−6 · · · 0 0 0
ϕ′′′2 ϕ′′′
1 ϕ′′′0 ϕ′′′
−1 ϕ′′′−2 ϕ′′′
−3 ϕ′′′−4 ϕ′′′
−5 · · · 0 0 0
ϕ′′′3 ϕ′′′
2 ϕ′′′1 ϕ′′′
0 ϕ′′′−1 ϕ′′′
−2 ϕ′′′−3 ϕ′′′
−4 · · · 0 0 0
ϕ′′′4 ϕ′′′
3 ϕ′′′2 ϕ′′′
1 ϕ′′′0 ϕ′′′
−1 ϕ′′′−2 ϕ′′′
−3 · · · 0 0 0
0 0 0 0 0 0 0 0 · · · ϕ′′′4 ϕ′′′
3 ϕ′′′2
0 0 0 0 0 0 0 0 · · · ϕ′′′3 ϕ′′′
2 ϕ′′′1
.
It can be shown that C is always non-singular for each value of N . However,
B is non singular for N 6 1000. We have checked the non-singularity of matrix
B by different methods. In first method we observe the determinants of matrix
B increase as N increases and it is not zero for N 6 1000. The determinants of
B at some values of N are shown in Table 2.1. In second method we observe
that for N 6 1000, the eigenvalues of matrix B are non-zero so by Strang (2011)
matrix B is non-singular. However for N > 1000 matrix B may or may not be
non-singular. Therefore we claim that system (2.21) and (2.23) are stable for
N 6 1000.
Table 2.1: Determinants of the matrices:
N C B
10 −8667/56 1/1048870018371741
50 -177183 1/4981270309
100 -552709050 1/1964492
500 -5.033491471916955×1036 4.728852755761116×1021
1000 -4.477989536166907×1071 42069711017699999×1040
34
2.3 Error estimation
In this section, we discuss the approximation properties of the numerical in-
terpolating collocation algorithm. Since the scheme (2.8) reproduce polynomial
curve of degree seven so by Dyn (2002) scheme has approximation order eight.
So the collocation algorithm (2.10) with septic precision treatments at the end-
points has the power of approximation O(h2). Here we present our main result
for error estimation.
Proposition 2.3.1. Suppose the exact solution y(x) ∈ C8[0, 1] and {zi} are obtained
by solving either (2.21) or (2.23) with 8th order boundary condition at the end points,
then
||err(x)||∞ = ||Zj − yj||∞ = O(h2−j), j = 0, 1, 2, 3. (2.24)
here j denotes the order of derivative.
Proof. Since the order of approximation of subdivision scheme (2.8) is eight so
by eigenvector η3, we can write for smooth function y(x) and small h as
y′′′(xj) =23
15039360h3 {225y(xj − 6h)− 11520y(xj − 5h) + 10952y(xj − 4h)
+476928y(xj − 3h)− 3047987y(xj − 2h) + 4677632y(xj − h)− 4677632y(xj + h)
+3047987y(xj + 2h)− 476928y(xj + 3h)− 10952y(xj + 4h) + 11520y(xj + 5h)
−225y(xj + 6h)}+O(h8).
This can be written as
y′′′(xj) =23
15039360h3 {225yj−6 − 11520yj−5 + 10952yj−4 + 476928yj−3
−3047987yj−2 + 4677632yj−1 − 4677632yj+1 + 3047987yj+2
−476928yj+3 − 10952yj+4 + 11520yj+5 − 225yj+6}+O(h8).
(2.25)
Similarly, we have
Z ′′′(xj) =23
15039360h3 {225zj−6 − 11520zj−5 + 10952zj−4 + 476928zj−3
−3047987zj−2 + 4677632zj−1 − 4677632zj+1 + 3047987zj+2
−476928zj+3 − 10952zj+4 + 11520zj+5 − 225zj+6} .
(2.26)
35
If we define error function e(x) = Z(x)− y(x) and error vectors at the nodes by
e(xj) = Z(xj)− y(xj + jh), −6 ≤ j ≤ N + 6,
or equivalently ej = Zj − y(xj + jh), −6 ≤ j ≤ N + 6, then this implies
e′j = Z ′j − y′(x+ jh),
e′′j = Z ′′j − y′′(x+ jh),
e′′′j = Z ′′′j − y′′′(x+ jh).
By subtracting (2.25) from (2.26), we get
Z ′′′(xj)− y′′′(xj) =23
15039360h3 {225(zj−6 − yj−6)− 11520(zj−5 − yj−5) + 10952(zj−4
−yj−4) + 476928(zj−3 − yj−3)− 3047987(zj−2 − yj−2) + 4677632(zj−1 − yj−1)
−4677632(zj+1 − yj+1) + 3047987(zj+2 − yj+2)− 476928(zj+3 − yj+3)
−10952(zj+4 − yj+4) + 11520(zj+5 − yj+5)− 225(zj+6 − yj+6)} .
This implies
e′′′(xj) =1
15039360h3 {225ej−6 − 11520ej−5 + 10952ej−4 + 476928ej−3 − 3047987ej−2
+4677632ej−1 − 4677632ej+1 + 3047987ej+2 − 476928ej+3 − 10952ej+4
+11520ej+5 − 225ej+6} .
By Lemma 2.1.4, we get the following expression
e′′′j =1
h3{ϕ′′′
6 ej−6 + ϕ′′′5 ej−5 + ϕ′′′
4 ej−4 + ϕ′′′3 ej−3 + ϕ′′′
2 ej−2 + ϕ′′′1 ej−1 + ϕ′′′
0 ej
+ϕ′′′−1ej+1 + ϕ′′′
−2ej+2 + ϕ′′′−3ej+3 + ϕ′′′
−4ej+4 + ϕ′′′−5ej+5 + ϕ′′′
−6ej+6
}+O(h8),
(2.27)
where j = 0, 1, 2, · · · , N . By subtracting (2.1) from( 2.11), we get
Z ′′′j − y′′′j = aj (Zi − Yj) .
This implies
e′′′j = ajej, 0 ≤ i ≤ N.
36
Using (2.27), we get
ϕ′′′6 ej−6 + ϕ′′′
5 ej−5 + ϕ′′′4 ej−4 + ϕ′′′
3 ej−3 + ϕ′′′2 ej−2 + ϕ′′′
1 ej−1 + qjej + ϕ′′′−1ej+1
+ϕ′′′−2ej+2 + ϕ′′′
−3ej+3 + ϕ′′′−4ej+4 + ϕ′′′
−5ej+5 + ϕ′′′−6ej+6 = 0, (2.28)
where qj = ϕ′′′0 − h3aj and j = 0, 1, 2, · · · , N .
As 0 6 x 6 1 and xj = jh, j = 0, 1, 2, · · · , N so e0, e1, · · · , eN are non zero while
e−6, e−5, · · · , e−1 and eN+1, eN+2, · · · , eN+6 are zero because they lie outside the
interval [0, 1]. Let us define these (the left and right end) error values as
ej =
max0≤k≤7
{|ek|}O(h8), −6 ≤ j ≤ 0,
maxN−7≤k≤N
{|ek|}O(h8), N ≤ j ≤ N + 6.(2.29)
Thus system (2.28) is equivalent to(B +O(h6)
)E = 0,
where B + O(h6) is the matrix obtained by deleting the first and last six rows
and columns of the matrix B, where
E = (e−6, e−5, e−4, · · · , eN+4, eN+5, eN+6)T .
By using (2.6) and (2.7)(B +O(h6)
)E = O(h8)||Z(xj)− y(xj)|| = O(h8)||E|| = O(h8).
Hence, for small h, the coefficients matrix B +O(h6) will be invertible and thus
using the standard result from linear algebra and effect of ∥B−1∥, we have
||E|| 6(
||B−1||1−O(h6)
O(h8))= O(h2). (2.30)
This completes the proof.
The above discussion suggest that the approximations of the solution com-
puted by the method developed in pervious section are second order accurate
approximations. This suggestion is supported by the numerical experiments
given in the next section.
37
2.4 Numerical examples and discussions
In this section, the numerical collocation algorithm based on 8-point interpolat-
ing subdivision scheme described in Section 2.2, with the 8th order boundary
conditions at the end points, is tested on the two point third order boundary
value problems. Absolute errors in the analytical solutions are also calculated.
For the sake of comparisons, we also tabulated the results in this section.
Example 2.4.1. Consider the boundary value problem
y′′′(x) = y(x)− 3ex, 0 < x < 1,
with boundary conditions y′(0) = 0, y(1) = 0, y(0) = 1. The analytical solution
of this problem is
y(x) = (1− x)ex.
By using the collocation algorithm for N = 10, we get following solution of the
above problem: Zj =16∑
i=−6
ziϕ(j− i), where the values of {z−6, z−5, . . . , z5, z16} are
by using (2.21) are
{24.5275284967525, 0.690493373832105, 0.811531107670859, 0.901997043612006,
0.962835576958380, 0.995101658947808, 1, 0.978925359857389, 0.93350395624740,
0.865636023693632, 0.797539554103408, 0.771795250759200, 0.651392727341382,
0.519777983609805, 0.380902186129123, 0.199271785305208, 0,−0.1311402990079,
−0.247662275449558,−0.342309352611581,−0.406997945734322,
−0.432827137019005,−0.413625213339484},
38
and by using (2.23) are
{0.8188703235427, 0.8701325409827, 0.9139995186898, 0.9498836119009,
0.9768946175522, 0.9940000846665, 1, 0.9934991816499, 0.9728768806892,
0.9362530908960, 0.8814510666459, 0.8059555490326, 0.7068662002854,
0.5808456281783, 0.4240613657868, 0.2321210447258, 0,−0.2780394553965,
−0.6085378480019,−0.9989382153934,−1.457696393548, 1.993136716176,
−2.54764506053343}.
By using two different boundary treatments presented in Section 2.2, we ob-
tained two different solutions which are presented in Table 2.2 along with their
absolute errors. The graphical representation of the analytic and approximate
solutions of above problem is shown in Figure 2.2. Figure 2.2(a) represents the
the comparison of analytic and approximate solutions obtained by (2.21) while
analytic and approximate solutions by (2.23) are shown in Figure 2.2(b). From
this table and figure, we observe that the solution obtained by (2.23) is signif-
icantly better than the solution obtained by (2.21). So our claim, that is, the
approximate solution can be improved by adjusting boundary treatment, is jus-
tified. The maximum absolute errors in the solutions obtained by (2.21) and
(2.23) at step size 10 are 9.755× 10−2 and 2.328× 10−2 respectively.
Example 2.4.2. Consider the following third order boundary values problem
y′′′(x) = xy(x)− (x3 − 2x2 − 5x− 3)ex, 0 < x < 1,
with boundary conditions y(0) = 0 = y(1), y′(0) = 1. Its exact solution is
y(x) = x(1− x)ex.
By the homogeneous process of the boundary condition, let u(x) = y(x)−x(1−x)
then above problem can be transformed into its equivalent form
u′′′(x) = xy(x)+x2(1−x)+(x3−2x2−5x−3)ex, 0 < x < 1, u(0) = u(1) = u′(0) = 0.
39
Tabl
e2.
2:So
luti
ons
and
erro
res
tim
atio
nof
Exam
ple
2.4.
1:
xj
Ana
lyti
cso
luti
onY
App
roxi
mat
eso
luti
onA
ppro
xim
ate
solu
tion
Abs
olut
eer
ror
Abs
olut
eer
ror
Z1
by(2
.21)
Z2
by(2
.23)
by(2
.21)
by(2
.23)
0.0
11
0.00
000
0
0.1
0.99
4653
8262
0.97
8925
3598
5738
90.
9934
9918
780.
0157
2846
630.
0011
5463
837
0.2
0.97
7122
2064
0.93
3503
9562
4740
00.
9728
7689
360.
0436
1825
020.
0042
4531
280
0.3
0.94
4901
1656
0.86
5636
0236
9363
20.
9362
5311
050.
0792
6514
190.
0086
4805
513
0.4
0.89
5094
8188
0.79
7539
5541
0340
80.
8814
5109
030.
0975
5526
470.
0136
4372
850
0.5
0.82
4360
6355
0.77
1795
2507
5920
00.
8059
5557
840.
0525
6538
470.
0184
0505
71
0.6
0.72
8847
5200
0.65
1392
7273
4138
20.
7068
6623
360.
0774
5479
270.
0219
8128
64
0.7
0.60
4125
8121
0.51
9777
9836
0980
50.
5808
4566
790.
0843
4782
850.
0232
8014
41
0.8
0.44
5108
1856
0.38
0902
1861
2912
30.
4240
6141
196
0.06
4205
9995
90.
0210
4677
36
0.9
0.24
5960
3111
0.19
9271
7853
0520
80.
2321
2109
670
0.04
6688
5258
0.01
3839
2144
1.0
00
0.00
000.
0000
.000
00.
000
40
The solutions of this problem and their absolute errors obtained by two differ-
ent boundary treatments are shown in Table 2.3. The graphical representations
of the analytic and approximate solutions are shown in Figure 2.3. Figure 2.3(a)
represents the the comparison of analytic and approximate solutions obtained
by (2.21) and Figure 2.3(b) represents the the comparison of analytic and ap-
proximate solutions obtained by (2.23). We observe that the solution obtained
by (2.23) has less absolute error than that of the solution obtained by (2.21). A-
gain this supports our claim.
Comparison:- The maximum absolute errors in the solutions of Example 2.4.2
obtained by (2.21) and (2.23) at step size 10 are 2.143× 10−2 and 1.437× 10−2 re-
spectively. Caglar et al. (1999) obtained the same maximumabsolute errors but
at the step size 32 and 50 respectively. Therefore we conclude that our method
is more efficient than that of Caglar et al. (1999).
2.5 Conclusion and future work
In this work, we present an interpolatory symmetric subdivision algorithm for
the numerical solution of third order linear problems. Septic polynomials were
used for the adjustment of boundary conditions at the end points. We estab-
lished collocation algorithm and obtained stable system of linear equations which
can be solved by any well-known numerical method. The numerical result
showed that the adjustment of boundary conditions at the end points influence
the accuracy of approximate solution. That is, the accuracy of the solution can
be improved by the proper adjustment of boundary conditions. So our algorith-
m has flexibility to improve the results by adjusting boundary conditions. The
automatic selection and adjustment of the boundary conditions which improve
the approximation order of the solution is possible future research direction.
41
Tabl
e2.
3:So
luti
ons
and
erro
res
tim
atio
nof
Exam
ple
2.4.
2:
xj
Ana
lyti
cso
luti
onY
App
roxi
mat
eso
luti
onA
ppro
xim
ate
solu
tion
Abs
olut
eer
ror
Abs
olut
eer
ror
Z1
by(2
.21)
Z2
by(2
.23)
by(2
.21)
by(2
.23)
0.0
00
0.00
000.
000
0.00
0
0.1
0.09
9465
3826
20.
1142
7450
060.
0971
0744
670
0.01
4809
1179
80.
0023
5793
592
0.2
0.19
5424
4413
0.21
6856
1360
0.18
7602
4920
0.02
1431
6947
0.00
7821
9493
0.3
0.28
3470
3497
0.30
4520
3260
0.27
6779
2681
0.02
1049
9763
0.00
6691
0816
0.4
0.35
8037
9275
0.37
3200
2884
0.34
3710
7723
0.01
5162
3609
0.01
4327
1552
0.5
0.41
2180
3178
0.41
7799
6505
0.40
1831
6295
0.00
5619
3327
0.01
0348
6883
0.6
0.43
7308
5120
0.43
1980
8639
0.42
2936
8864
0.00
5327
6481
0.01
4371
6256
0.7
0.42
2888
0685
0.40
7925
8312
0.41
9957
5084
0.01
4962
2373
0.00
2930
5601
0.8
0.35
6086
5485
0.33
6064
7276
0.34
7344
2367
0.02
0021
8209
0.00
8742
3118
0.9
0.22
1364
2800
0.20
4768
4968
0.20
6999
8108
0.01
6595
7832
0.01
4364
4692
1.0
00
0.00
000.
000
0.00
0
42
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
X
Sol
utio
n
Analytic solution YApproximate solution Z
(a)
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
X
Sol
utio
n
Analytic solution YApproximate solution Z
(b)
Figure 2.2: Comparison between analytic and approximating solutions Example 2.4.1.
43
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
X
Sol
utio
n
Analytic solution YApproximate solution Z
(a)
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
X
Sol
utio
n
Analytic solution YApproximate solution Z
(b)
Figure 2.3: Comparison between analytic and approximating solutions of Example
2.4.2.
44
Chapter 3
Subdivision Schemes Based
Collocation Algorithms for the
Solution of Fourth Order Boundary
Value Problems
In this chapter, we present two collocation algorithms based on interpolating
and approximating subdivision schemes for the solution of fourth order bound-
ary value problems. Main purpose of this chapter is to explore and seek the ap-
plications of interpolating and approximating subdivision schemes in the field
of boundary value problems along with intrinsic comparison of the results ob-
tain by algorithms based on these schemes. We consider following interpolat-
ing Deng and Ma (2013) and approximating Mustafa et al. (2014a) subdivision
45
schemes
pk+12i = pki ,
pk+12i+1 =
3565536
(pki−4 + pki+5
)− 405
65536
(pki−3 + pki+4
)+ 567
16384
(pki−2 + pki+3
)− 2205
16384
(pki−3 + pki+4
)+ 19845
32768
(pki + pki+1
),
(3.1)
and
pk+12i = 23
16384(pki−3 + pki+3)− 285
8192(pki−2 + pki+2) +
207316384
(pki−1 + pki+1) +33334096
pki ,
pk+12i+1 = − 3
32768(pki−3 + pki+4)− 33
32768(pki−2 + pki+3)− 1931
32768(pki−1 + pki+2)
+1835132768
(pki + pki+1)
(3.2)
with order of continuity C4. The schemes (3.1) and (3.2) reproduce polynomial
curves of degree nine and three by Conti and Hormann (2011) and Mustafa et al.
(2014a) respectively. Cardinal supports of these schemes are [−8, 8] and [−6, 6]
respectively.
We construct collocation algorithms by using the basis functions of above inter-
polating and approximating subdivision schemes for the numerical solution of
linear fourth order boundary value problems. The mathematical form of fourth
order boundary value problems is given by
y(iv)(x) = a(x)y(x) + b(x), 0 6 x 6 1, (3.3)
subject to the boundary condition
y(0) = α, y′(0) = β, y(1) = γ, y
′(1) = ω (3.4)
where a(x) and b(x) are continuous and a(x) > 0 on [0, 1]. Analytic solution of
such type of boundary value problem is possible only in very rare cases.
46
The outline of this chapter is as follows. In Section 3.1, we construct subdi-
vision matrices of subdivision schemes (3.1) and (3.2) for the computation of
eigenvalues and their corresponding (right and left) eigenvectors. Basis func-
tions and their derivatives have been also discussed in this section. In Sec-
tion 3.2, subdivision based collocation algorithms for solution of fourth order
boundary value problems are formulated. Approximation properties of these
algorithms are also given in Section 3.2. In Section 3.3, numerical examples are
presented. Comparison of approximate solutions by interpolating and approxi-
mating schemes based collocation algorithms is also given. Conclusion is given
in Section 3.4.
3.1 Basic properties of the schemes
In this section, we construct subdivision matrices of the schemes defined in (3.1)
and (3.2) for the computation of eigenvalues and their corresponding eigenvec-
tors. Basis functions of these schemes and their derivatives have also been dis-
cussed in this section.
3.1.1 Subdivision matrices
If S1 and S2 are subdivision matrices of the schemes (3.1) and (3.2) then these
matrices are defined as
47
S1 =
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0
0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5 0
0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 L5 L4 L3 L2 L1 L1 L2 L3 L4 L5
0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0
,
where L1 =1984532768
, L2 = − 220532768
, L3 =567
16384, L4 = − 405
65536, L5 =
3565536
and
48
S2 =
s1 s2 s3 s4 s3 s2 s1 0 0 0 0 0 0
s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1 0 0 0 0 0
0 s1 s2 s3 s4 s3 s2 s1 0 0 0 0 0
0 s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1 0 0 0 0
0 0 s1 s2 s3 s4 s3 s2 s1 0 0 0 0
0 0 s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1 0 0 0
0 0 0 s1 s2 s3 s4 s3 s2 s1 0 0 0
0 0 0 s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1 0 0
0 0 0 0 s1 s2 s3 s4 s3 s2 s1 0 0
0 0 0 0 s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1 0
0 0 0 0 0 s1 s2 s3 s4 s3 s2 s1 0
0 0 0 0 0 s′1 s
′2 s
′3 s
′4 s
′4 s
′3 s
′2 s
′1
0 0 0 0 0 0 s1 s2 s3 s4 s3 s2 s1
,
where
s1 =23
16384, s2 = − 285
8192, s3 =
2073
16384, s4 =
3333
4096,
s′
1 = − 3
32768, s
′
2 = − 33
32768, s
′
3 = − 1931
32768, s
′
4 =18351
32768.
The first ten real eigenvalues of matrices S1 and S2 are same which are given
below
λi = 1, 12, 1
4, 1
8, 1
16, 1
32, 1
64, 1
128, 1
256, 1
512, i = 0, 1, 2, · · · , 9.
The remaining eigenvalues are complex which are not required. For above
eigenvalues λi, the eigenvectors υRiand υLi
that satisfies S1υRi= λiυRi
and
υLiST1 = υLi
λi are called right and left eigenvectors of the matrix S1 respectively.
We can also define the right νRiand left νLi
eigenvectors of S2 in a similar way.
The normalized left and right eigenvectors corresponding to first five eigenval-
ues of S1 and S2 are given in Table 3.1 and 3.2 respectively.
49
Tabl
e3.
1:Ei
genv
alue
san
dei
genv
ecto
rsof
the
mat
rixS1
Eige
nval
uesλi
Cor
resp
ondi
ngri
ghta
ndle
ftei
genv
ecto
rs
1υR
0=
(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)T,
υL0=
(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)T
1 2υR
1=
(−8,−7,−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6,7,8)
T
υL1=
(−1575,−
1474560,−315738080,−1397587968,43588613880,−
4311679549440,1336741045920,
−4824847319040,0,4824847319040,−1336741045920,4311679549440,−43588613880,
1397587968,315738080,1474560,1575)
T/5
8418
8424
5680
1 4υR
2=
(64,49,36,25,16,9,4,1,0,1,4,9,16,25,36,49,64)T
υL2=
(459375,215040000,47660030080,103151616000,−
3882427261296,23490017902592,−
84313449846912,
313469774708736,−497829885297150,313469774708736,−84313449846912,23490017902592,
−3882427261296,103151616000,47660030080,215040000,459375)
T/2
5962
4836
2131
201 8
υR
3=
(−512,−343,−216,−125,−64,−
27,−
8,−1,0,1,8,27,64,125,216,343,512)
T
υL3=
(−104125,−
24371200,1177382520,−5986263040,−
10571778214,207884427264,
−972244098856,
−1386160480256,0,1386160480256,972244098856,−
207884427264,10571778214,5986263040,
−1177382520,24371200,104125)T
/946
0416
8599
041 16
υR
4=
(4096,2401,1296,625,256,81,16,1,0,1,16,81,256,625,1296,2401,4096)T
υL4=
(392875,45977600,−
1296269280,5912719360,1180083476,−
86261280768,332951715808,−
67767008256,
850467338370,−
67767008256,332951715808,−
86261280768,1180083476,5912719360,−1296269280,
45977600,392875)
T/1
8376
8238
080
50
Tabl
e3.
2:Ei
genv
alue
san
dei
genv
ecto
rsof
the
mat
rixS2
Eige
nval
uesλi
Cor
resp
ondi
ngri
ghta
ndle
ftei
genv
ecto
rs
1ν R
0=
(1,1,1,1,1,1,1,1,1,1,1,1,1)T
,ν L
0=
(0,0,0,0,0,0,1,0,0,0,0,0,0)T
1 2ν R
1=
(−6,−5,−4,−3,−2,−1,0,1,2,3,4,5,6)
T
ν L1=
(−271,1475866,286711948,−3680077858,73546185237,
−507541369116,0,507541369116,−73546185237,3680077858,
−286711948,−1475866,271)
T/7
4067
0013
440
1 4ν R
2=
(36,25,16,9,4,1,0,1,4,9,16,25,36)T
ν L2=
(−3927,10663114,−582891346,−38170879262,32176934487,
2719571096148,−5426009838428,2719571096148,32176934487,
−38170879262,−582891346,10663114,−
3927)T
/499
1362
1913
601 8
ν R3=
(−216,−125,−64,−
27,−
8,−1,0,1,8,27,−
64,−
125,−216)
T
ν L3=
(23,−31050,−1351836,21998258,−
507201588,193168476,0,
−193168476,507201588,−21998258,1351836,31050,−
23)T
/635
3510
401 16
ν R4=
1 27(51867,25027,10267,3267,667,67,27,67,667,3267,10267,25027,51867)
T
ν L4=
(−6693,4466462,117537274,−
1293454266,6372836901,
−15842276196,21281793036,−15842276196,6372836901,−
1293454266,
117537274,4466462,−6693)T
/421
5029
7600
51
3.1.2 Basis functions
The basis functions for the convergent subdivision schemes (3.1) and (3.2) are
the limit curves ϕ(x) and Φ(x) generated from the cardinal data {pi = (i, δ0)T}.
The ϕ(x) and Φ(x) are also known as fundamental solutions of the subdivision
schemes, so
ϕ(i) = Φ(i) =
1, i = 0,
0, i = 0.(3.5)
Furthermore, ϕ(x) = Φ(x) satisfies the following two-scale equation
ϕ(x) =
p∑j=−p
aiϕ(2x− i), Sup(ϕ) = (−p− 1, p+ 1). (3.6)
The lth derivative of basis function at integers satisfy the relation
ϕ(l)(x) =
p∑j=−p
aiϕ(l)(2x− i), Sup(ϕ) = [−p, p]. (3.7)
Since υTRiυLj
= 1 and νTRiνLj
= 1 for i = j and 0 otherwise then by Qu (1996), we
get following result.
Lemma 3.1.1. The fundamental solution (Cardinal basis) ϕ(x) of the subdivision scheme
(3.1) is fourth times continuously differentiable, supported on [−8, 8] and its derivatives
at integers are given by
ϕ′(t) = 2sgn(t)eT|t|υL1 , ϕ′′(t) = 22eT|t|υL2 ,
ϕ′′′(t) = 23sgn(t)eT|t|υL3 , ϕiv(t) = 24eT|t|υL4 , −8 6 t 6 8, (3.8)
where υLi, 0 6 i 6 4 are defined in Table 3.1, the sgn function of a real number t is
defined as
sgn(t) =
−1, t < 0,
0, t = 0,
1, t > 0,
(3.9)
52
et’s are the column matrices defined as
et = (a8t, a7t, a6t, a5t, a4t, a3t, a2t, a1t, a0t, a−1t, a−2t, a−3t, a−4t, a−5t, a−6t, a−7t, a−8t)T ,
where 0 ≤ t ≤ 8 and
ait =
1, i = t,
0, i = t.(3.10)
Lemma 3.1.2. The fundamental solution Φ(x) of subdivision scheme (3.2) defined in
(3.6) is fourth times continuously differentiable, supported on [−6, 6] and its derivatives
at integers are defined as
Φ′(t) = 2sgn(t)eT|t|νL1 , Φ′′(t) = 22eT|t|νL2 ,
Φ′′′(t) = 23sgn(t)eT|t|νL3 , Φiv(t) = 24eT|t|νL4 − 6 6 t 6 6, (3.11)
where νLifor 0 6 i 6 4 are defined in Table 3.2, the sgn function of a real number t is
defined by (3.9) et’s are the column matrices defined as
et = (a6t, a5t, a4t, a3t, a2t, a1t, a0t, a−1t, a−2t, a−3t, a−4t, a−5t, a−6t)T , 0 6 t 6 6,
where ait are defined by (3.10).
From (3.8) and (3.11), we get values of derivatives at the integers given in
Table 3.3 and Table 3.4 respectively.
3.2 Description of numerical algorithms
In this section, first we formulate two collocation algorithms which are based on
interpolating (3.1) and approximating (3.2) subdivision schemes for the solution
of (3.3). Then we settle down the boundary conditions to get unique solution.
3.2.1 Collocation algorithms
Here we formulate two collocation algorithms based on two subdivision schemes.
These collocation algorithms are defined in coming subsections.
53
Table 3.3: Derivatives of ϕ at cardinal data by (3.8)
i ϕ′(i) ϕ′′(i) ϕ′′′(i) ϕiv(i)
0 0 −2370618501415309077185968
0 33869667457408
±1 ∓19146219521159104017
3265310153216676106344305
±4331751500815295995855
−529505475289730585
±2 ± 5304527961159104017
−878265102572676106344305
∓12153051235761183983420
10404741119358922340
±3 ∓ 147046413780629
7340630594562028319032915
±240606976566518365
−748795849970065
±4 ± 172970691159104017
− 808839012771352212688610
∓ 5285889107244735933680
2950208692871378720
±5 ∓ 27729925795520085
214899200135221268861
∓ 374141443059199171
923862417946117
±6 ∓ 112763610431936153
297875188405663806583
± 1090169453214692
− 9001877976052
±7 ∓ 40968113728119
6400019317324123
∓ 21760437028453
7184017946117
±8 ∓ 59272832136
4375618154371936
∓ 297513984910496
11225328157568
54
Table 3.4: Derivatives of Φ at cardinal data by (3.11)
i Φ′(i) Φ′′(i) Φ′′′(i) Φiv(i)
0 0 −1356502459607311960136960
0 1773482753219532800
±1 ∓4229511409330861250560
226630924679103986712320
±596199245120
−1320189683219532800
±2 ± 350219929717635000320
3575214943138648949760
∓1565437980480
708092989292710400
±3 ∓ 1840038929185167503360
− 19085439631623920273920
±1099912939709440
−215575711439065600
±4 ± 7167798792583751680
− 291445673623920273920
∓ 12517735360
587686371317196800
±5 ± 10541926452500480
5331557623920273920
∓ 115294144
22332311317196800
±6 ∓ 271370335006720
− 1309415946849280
± 2379418880
− 2231878131200
55
3.2.2 Interpolating collocation algorithm
The collocation algorithm based on interpolating scheme (3.1) say interpolating
collocating algorithm is given below. In this algorithm, we assume approximate
solution Z1(x) of (3.3) as
Z1(x) =N+8∑i=−8
ziϕ(x−xi
h
), 0 6 x 6 1, (3.12)
where N is the positive integer N > 8, h = 1/N and xi = i/N = ih, and {zi}
are the unknown to be determined for the solution of (3.3). The collocation
algorithm together with the boundary conditions to be discussed, is given by
Ziv1 (xj) = a(xj)Z1(xj) + b(xj), j = 0, 1, 2, · · · , N, (3.13)
with the following type of boundary conditions
Z1(0) = α, Z′
1(0) = β, Z1(N) = γ, Z′
1(N) = ω, (3.14)
where α, β, γ and ω are constants. Let aj = a(xj), bj = b(xj), then equation (3.13)
can be written as
Ziv1 (xj) = ajZ1(xj) + bj, j = 0, 1, 2, · · · , N, (3.15)
where
Ziv1 (xj) =
1
h4
N+8∑i=−8
ziϕiv
(xj − xi
h
). (3.16)
Using (3.12) and (3.16) in (3.15), we get following N + 1 system of equations
N+8∑i=−8
ziϕiv
(xj − xi
h
)− h4aj
N+8∑i=−8
ziϕ
(xj − xi
h
)= h4bj, j = 0, 1, 2, · · · , N.
(3.17)
56
3.2.3 Approximating collocation algorithm
In approximating collocating algorithm (i.e. algorithm based on approximating
scheme (3.2)), we assume approximate solution Z2(x) of (3.3) as
Z2(x) =N+6∑i=−6
ziΦ(x−xi
h
), 0 6 x 6 1, (3.18)
where N is the positive integer N > 6, h = 1/N and xi = i/N = ih and {zi}
are the unknown to be determined for the solution of (3.3). The collocation
algorithm together with the boundary conditions to be discussed, is given by
Ziv2 (xj) = a(xj)Z2(xj) + b(xj), j = 0, 1, 2, · · · , N, (3.19)
with the following type of boundary conditions
Z2(0) = α, Z′
2(0) = β, Z2(N) = γ, Z′
2(N) = ω. (3.20)
The equation (3.19) can be written as
Ziv2 (xj) = ajZ2(xj) + bj, j = 0, 1, 2, · · · , N, (3.21)
where
Ziv2 (xj) =
1
h4
N+6∑i=−6
ziΦiv
(xj − xi
h
). (3.22)
Using (3.18) and (3.22) in (3.21), we get following N + 1 system of equationsN+6∑i=−6
ziΦiv
(xj − xi
h
)− h4aj
N+6∑i=−6
ziΦ
(xj − xi
h
)= h4bj, j = 0, 1, 2, · · · , N.
(3.23)
Now we simplify the above systems (3.17) and (3.23) in following theorems.
Theorem 3.2.1. Interpolating collocation algorithm: For j = 0 by (3.17), we get
z−8ϕiv−8 + z−7ϕ
iv−7 + z−6ϕ
iv−6 + z−5ϕ
iv−5 + z−4ϕ
iv−4 + z−3ϕ
iv−3 + z−2ϕ
iv−2 + z−1ϕ
iv−1
+z0q0 + z1ϕiv1 + z2ϕ
iv2 + z3ϕ
iv3 + z4ϕ
iv4 + z5ϕ
iv5 + z6ϕ
iv6 + z7ϕ
iv7 + z8ϕ
iv8 = h4b0,
(3.24)
where ϕivj = ϕiv(j) and q0 = ϕiv
0 − a0h4.
57
Proof. Substituting j = 0 in (3.17), we get{z−8ϕ
iv(x0−x−8
h) + z−7ϕ
iv(x0−x−7
h) + · · ·+ zN+7ϕ
iv(x0−xN+7
h) + zN+8ϕ
iv(x0−xN+8
h)}
−a0h4{z−8ϕ(
x0−x−8
h) + z−7ϕ(
x0−x−7
h) + · · ·+ zN+7ϕ(
x0−xN+7
h) + zN+8ϕ(
x0−xN+8
h)}
= h4b0.
For xi = ih, i = 0, 1, 2 · · ·N , this implies
z−8ϕiv(8) + z−7ϕ
iv(7) + · · ·+ zN+7ϕiv(−N − 7) + zN+8ϕ
iv(−N − 8)− a0h4 {z−8ϕ(8)
+z−7ϕ(7) + · · ·+ zN+7ϕ(−N − 7) + zN+8ϕ(−N − 8)} = h4b0.
Since the cardinal support of basis function ϕ(x) is [−8, 8], so ϕ′(x), ϕ′′(x), ϕ′′′(x)
and ϕiv(x) are zero out side the interval [−8, 8], also by Table 3.3, we get
z−8ϕiv(8) + z−7ϕ
iv(7) + z−6ϕiv(6) + z−5ϕ
iv(5) + z−4ϕiv(4) + z−3ϕ
iv(3) + z−2ϕiv(2)
+z−1ϕiv(1) + z0ϕ
iv(0) + z1ϕiv(−1) + z2ϕ
iv(−2) + z3ϕiv(−3) + z4ϕ
iv(−4) + z5ϕiv(−5)
+z6ϕiv(−6) + z7ϕ
iv(−7) + z8ϕiv(−8)− a0h
4z0ϕ(0) = h4b0.
If ϕivi = ϕiv(i), then
z−8ϕiv8 + z−7ϕ
iv7 + z−6ϕ
iv6 + z−5ϕ
iv5 + z−4ϕ
iv4 + z−3ϕ
iv3 + z−2ϕ
iv2 + z−1ϕ
iv1 + z0(ϕ
iv0
−a0h4) + z1ϕ
iv−1 + z2ϕ
iv−2 + z3ϕ
iv−3 + z4ϕ
iv−4 + z5ϕ
iv−5 + z6ϕ
iv−6 + z7ϕ
iv−7 + z8ϕ
iv−8 = h4b0.
As we observe from Table 3.3, ϕiv−i = ϕiv
i , we have
z−8ϕiv−8 + z−7ϕ
iv−7 + z−6ϕ
iv−6 + z−5ϕ
iv−5 + z−4ϕ
iv−4 + z−3ϕ
iv−3 + z−2ϕ
iv−2 + z−1ϕ
iv−1 + z0(ϕ
iv0
−a0h4) + z1ϕ
iv1 + z2ϕ
iv2 + z3ϕ
iv3 + z4ϕ
iv4 + z5ϕ
iv5 + z6ϕ
iv6 + z7ϕ
iv7 + z8ϕ
iv8 = h4b0.
For q0 = ϕiv0 − a0h
4, we get (3.24). This completes the proof.
Theorem 3.2.2. Interpolating collocation algorithm: For j = 1, 2, 3, · · · , N the system
(3.17) is equivalent to
z−8ϕiv−j−8 + z−7ϕ
iv−j−7 + · · ·+ z0ϕ
iv−j + z1(ϕ
iv1−j − ajh
4ϕ1−j) + z2(ϕiv2−j − ajh
4ϕ2−j)
+ · · ·+ zN(ϕivN−j − ajh
4ϕN−j) + zN+1ϕivN+1−j + · · ·+ zN+8ϕ
ivN+8−j = h4bj. (3.25)
58
Proof. By expanding equation (3.17), we get
z−8ϕiv(
xj−x−8
h) + z−7ϕ
iv(xj−x−7
h) + · · ·+ zN+7ϕ
iv(xj−xN+7
h) + zN+8ϕ
iv(xj−xN+8
h)
−ajh4{z−8ϕ(
xj−x−8
h) + z−7ϕ(
xj−x−7
h) + · · ·+ zN+7ϕ(
xj−xN+7
h) + zN+8ϕ(
xj−xN+8
h)}
= h4bj.
For xj = jh, j = 1, 2 · · ·N , we get
z−8ϕiv(j + 8) + z−7ϕ
iv(j + 7) + · · ·+ zN+7ϕiv(j −N − 7) + zN+8ϕ
iv(j −N − 8)
−ajh4 {z−8ϕ(j + 8) + z−7ϕ(j + 7) + · · ·+ zN+7ϕ(j −N − 7) + zN+8ϕ(j −N − 8)}
= h4bj.
This implies
z−8(ϕiv(j + 8)− ajh
4ϕ(j + 8)) + z−7(ϕiv(j + 7)− ajh
4ϕ(j + 7)) + · · ·+ zN+7(ϕiv(j
−N − 7)− ajh4ϕ(j −N − 7)) + zN+8(ϕ
iv(j −N − 8)− ajh4ϕ(j −N − 8)) = h4bj.
If ϕivj = ϕiv(j), for j = 1, 2 · · ·N then
z−8(ϕivj+8 − ajh
4ϕj+8) + z−7(ϕivj+7 − ajh
4ϕj+7) + · · ·+ zN+7(ϕivj−N−7 − ajh
4ϕj−N−7)
+zN+8(ϕivj−N−8 − ajh
4ϕj−N−8) = h4bj.
As we observe from Table 3.3, ϕiv−j = ϕiv
j j = 1, 2, · · · , N , then above equation
can be written as
z−8(ϕiv−j−8 − ajh
4ϕ−j−8) + z−7(ϕiv−j−7 − ajh
4ϕ−j−7) + · · ·+ zN+7(ϕivN+7−j
−ajh4ϕN+7−j) + zN+8(ϕ
ivN+8−j − ajh
4ϕN+8−j) = h4bj.
Since ϕ′(x), ϕ′′(x), ϕ′′′(x) and ϕiv(x) are zero out side the interval [−8, 8] then by
Table 3.3, we get (3.25).
Theorem 3.2.3. Approximating collocation algorithm: For j = 0 by (3.23), we get
z−6Φiv−6 + z−5Φ
iv−5 + z−4Φ
iv−4 + z−5Φ
iv−5 + z−4Φ
iv−4 + z−3Φ
iv−3 + z−2Φ
iv−2
+z−1Φiv−1 + z0q0 + z1Φ
iv1 + z2Φ
iv2 + z3Φ
iv3 + z4Φ
iv4 + z5Φ
iv5 + z6Φ
iv6 = h4b0
, (3.26)
where Φivj = Φiv(j) and Υ0 = Φiv
0 − a0h4.
59
Proof. Substituting j = 0 in (3.23), we get{z−6Φ
iv(x0−x−6
h) + z−5Φ
iv(x0−x−5
h) + · · ·+ zN+5Φ
iv(x0−xN+5
h) + zN+6Φ
iv(x0−xN+6
h)}
−a0h4{z−6Φ(
x0−x−6
h) + z−5Φ(
x0−x−5
h) + · · ·+ zN+5Φ(
x0−xN+5
h) + zN+6Φ(
x0−xN+6
h)}
= h4b0.
For xi = ih, i = 0, 1, 2 · · ·N , this implies
z−6Φiv(6) + z−5Φ
iv(5) + · · ·+ zN+5Φiv(−N − 5) + zN+6Φ
iv(−N − 6)− a0h4 {z−6Φ(6)
+z−5Φ(5) + · · ·+ zN+5Φ(−N − 5) + zN+6Φ(−N − 6)} = h4b0.
Since the cardinal support of basis function Φ(x) is [−6, 6], so Φ′(x), Φ′′(x), Φ′′′(x)
and Φiv(x) are zero out side the interval [−6, 6], also by Table 3.4, we get
z−6Φiv(6) + z−5Φ
iv(5) + z−4Φiv(4) + z−3Φ
iv(3) + z−2Φiv(2) + z−1Φ
iv(1) + z0Φiv(0)
+z1Φiv(−1) + z2Φ
iv(−2) + z3Φiv(−3) + z4Φ
iv(−4) + z5Φiv(−5) + z6Φ
iv(−6)
−a0h4z0Φ(0) = h4b0.
If Φivi = Φiv(i), then
z−6Φiv6 + z−5Φ
iv5 + z−4Φ
iv4 + z−3Φ
iv3 + z−2Φ
iv2 + z−1Φ
iv1 + z0(Φ
iv0 − a0h
4) + z1Φiv−1
+z2Φiv−2 + z3Φ
iv−3 + z4Φ
iv−4 + z5Φ
iv−5 + z6Φ
iv−6 = h4b0.
As we observe from Table 3.4, Φiv−i = Φiv
i , we have
z−6Φiv−6 + z−5Φ
iv−5 + z−4Φ
iv−4 + z−3Φ
iv−3 + z−2Φ
iv−2 + z−1Φ
iv−1 + z0(Φ
iv0 − a0h
4) + z1Φiv1
+z2Φiv2 + z3Φ
iv3 + z4Φ
iv4 + z5Φ
iv5 + z6Φ
iv6 + z7Φ
iv7 + z8Φ
iv8 = h4b0.
For Υ0 = Φiv0 − a0h
4, we get (3.26). This completes the proof.
Theorem 3.2.4. Approximating collocation algorithm: For j = 1, 2, 3, · · · , N the sys-
tem (3.23) is equivalent to
z−6Φiv−j−6 + z−5Φ
iv−j−5 + · · ·+ z0Φ
iv−j + z1(Φ
iv1−j − ajh
4Φ1−j) + z2(Φiv2−j − ajh
4Φ2−j)
+ · · ·+ zN(ΦivN−j − ajh
4ΦN−j) + zN+1ΦivN+1−j + · · ·+ zN+6Φ
ivN+6−j = h4bj. (3.27)
60
Proof. By expanding equation (3.23), we get
z−6Φiv(
xj−x−6
h) + z−5Φ
iv(xj−x−5
h) + · · ·+ zN+5Φ
iv(xj−xN+5
h) + zN+6Φ
iv(xj−xN+6
h)
−ajh4{z−8Φ(
xj−x−8
h) + z−7Φ(
xj−x−7
h) + · · ·+ zN+5Φ(
xj−xN+5
h) + zN+6Φ(
xj−xN+6
h)}
= h4bj.
For xj = jh, j = 1, 2 · · ·N , we get
z−6Φiv(j + 6) + z−5Φ
iv(j + 5) + · · ·+ zN+5Φiv(j −N − 5) + zN+6Φ
iv(j −N − 6)
−ajh4 {z−6Φ(j + 6) + z−5Φ(j + 5) + · · ·+ zN+5Φ(j −N − 5) + zN+6Φ(j −N − 6)}
= h4bj.
This implies
z−6(Φiv(j + 6)− ajh
4Φ(j + 6)) + z−5(Φiv(j + 5)− ajh
4Φ(j + 5)) + · · ·+ zN+5(Φiv(j
−N − 5)− ajh4Φ(j −N − 5)) + zN+6(Φ
iv(j −N − 6)− ajh4Φ(j −N − 6)) = h4bj.
If Φivj = Φiv(j), for j = 1, 2 · · ·N then
z−6(Φivj+6 − ajh
4Φj+6) + z−5(Φivj+5 − ajh
4Φj+5) + · · ·+ zN+5(Φivj−N−5 − ajh
4Φj−N−5)
+zN+6(Φivj−N−6 − ajh
4Φj−N−6) = h4bj.
As we observe from Table 3.4, Φiv−j = Φiv
j j = 1, 2, · · · , N , then above equation
can be written as
z−6(Φiv−j−6 − ajh
4Φ−j−6) + z−5(Φiv−j−5 − ajh
4Φ−j−5) + · · ·+ zN+5(ΦivN+5−j
−ajh4ΦN+5−j) + zN+6(Φ
ivN+6−j − ajh
4ΦN+6−j) = h4bj.
Since Φ′(x), Φ′′(x), Φ′′′(x) and Φiv(x) are zero out side the interval [−6, 6] then by
Table 3.4, we get (3.27).
3.2.4 Boundary conditions at end points
We have two different systems of (N +1) equations defined by (3.17) and (3.23).
In order to get unique solution of these systems, we need sixteen more con-
ditions for the system (3.17) and twelve more conditions for the system (3.23).
61
Four conditions can be achieved from boundary conditions given in (3.4) for
both systems of linear equations in which first order derivatives are involved
and remaining conditions are achieved by some extrapolation method at the
end points. First we find the approximation of the first derivative by difference
operators and after that we define the extrapolation method at end points for
both system of linear equations.
3.2.5 Approximation of derivative boundary conditions
In this section, we approximate the derivative boundary conditions by differ-
ence operators. Since approximation order of interpolating scheme (3.1) and
approximating scheme (3.2) is ten and four respectively. So we approximate
derivative boundary conditions at end points with approximation orders ten
and four for interpolating and approximating collocation algorithms.
If we use interpolating collocation algorithm for the solution of (3.3) then ap-
proximation of derivative conditions at ends point is defined as
Z ′1(0) =
(N
2520
){−7381z0 + 25200z1 − 56700z2 + 100800z3 − 132300z4
+127008z5 − 88200z6 + 43200z7 − 14175z8 + 28800z9 − 252z10}
+O(h10), (3.28)
Z ′1(N) =
(N
2520
){7381zN − 25200zN−1 + 56700zN−2 − 100800zN−3 + 132300zN−4
−127008zN−5 + 88200zN−6 − 43200zN−7 + 14175zN−8 − 28800zN−9
+252zN−10}+O(h10), (3.29)
and if we use approximating collocation algorithm for the solution of (3.3) then
approximation of derivative conditions at end points is defined as
Z ′2(0) =
(N
12
){−25z0 + 48z1 − 36z2 + 16z3 − 3z4}+O(h4), (3.30)
Z ′2(N) =
(N
12
){25zN − 48zN−1 + 36zN−2 − 16zN−3 + 3zN−4}+O(h4). (3.31)
62
3.2.6 Adjustment of boundary conditions
Still we need twelve and eight more conditions for the systems (3.17) and (3.23)
respectively to get stable systems for the solution of (3.3). For this we made
some adjustment of boundary conditions for the system (3.17) and (3.23), which
are defined below.
Case 1:- If we use interpolating collocation algorithm for the approximate solu-
tion of (3.3) then we define six conditions at left end points and six conditions at
the right end points. Since subdivision scheme (3.1) reproduces nine degree (i.e.
tenth order) polynomials, so we define boundary conditions of order ten for so-
lution of (3.17). For simplicity only left end points z−7, z−6, z−5, z−4, z−3, z−2 are
discussed and the values of right end points zN+2, zN+3, zN+4, zN+5, zN+6, zN+7
can be treated similarly.
The values z−7, z−6, z−5, z−4, z−3, z−2 can be determined by the nonic polynomial
q(x) interpolating (xi, zi), 2 ≤ i ≤ 7. Precisely, we have
z−i = q(−xi), i = 2, 3, 4, 5, 6, 7,
where
q(xi) =10∑j=1
10
j
(−1)j+1Z1(xi−j).
Since by (3.22) Z1(xi) = zi for i = 2, 3, 4, 5, 6, 7 then by replacing xi by −xi, we
have
q(−xi) =10∑j=1
10
j
(−1)j+1z−i+j.
Hence the following boundary conditions can be employed at the left end
10∑j=0
10
j
(−1)jz−i+j = 0, i = 7, 6, 5, 4, 3, 2. (3.32)
63
Similarly for the right end, we can define zi = q(−xi), i = N + 2, N + 3, N + 4,
N + 5, N + 6, N + 7 and
q(xi) =10∑j=1
10
j
(−1)j+1zi−j.
So we have the following boundary conditions at the right end
10∑j=0
10
j
(−1)jzi−j = 0, i = N + 2, N + 3, N + 4, N + 5, N + 6, N + 7.
(3.33)
Finally, we get a following new system of (N+17) linear equations with (N+17)
unknowns {zi}, in which N + 1 equations are obtained from (3.24) and (3.25),
four equations from boundary conditions (3.14) and twelve from boundary con-
ditions (3.32) and (3.33).
Case 2:- If we use approximating collocation algorithm (3.23) for the solution of
(3.3) then we need eight more conditions. So in this case, we define four extra
conditions at the left end points and four conditions at the right end points by
some extrapolation method. Since the subdivision scheme reproduce cubic (i.e.
fourth order) polynomial, so we define quartic polynomial for the adjustment of
boundary treatment. The values z−4, z−3, z−2, z−1 are determined by the quartic
polynomial p(x) interpolating (xi, zi). This polynomial is defined as
z−i+1 = p(−xi+1), i = 1, 2, 3, 4,
where
p(xi+1) =4∑
j=1
4
j
(−1)j+1Z2(xi−j+1).
Since by (3.28) Z2(xi) = zi for i = 1, 2, 3, 4 then by replacing xi by −xi, we have
p(−xi+1) =4∑
j=1
4
j
(−1)j+1z−i+j+1.
64
Hence the following boundary conditions can be employed at the left end
4∑j=0
4
j
(−1)jz−i+j+1 = 0, i = 1, 2, 3, 4. (3.34)
Similarly for the right end, we can define zi+1 = p(xi+1), i = N + 1 N + 2, N + 3,
N + 4 and
p(xi+1) =4∑
j=1
4
j
(−1)j+1zi−j+1.
So we have the following boundary conditions at the right end
4∑j=0
4
j
(−1)jzi−j+1 = 0, i = N + 1, N + 2, N + 3, N + 4.
(3.35)
Finally, we get a following new system of (N+13) linear equations with (N+13)
unknowns {zi}, in which N + 1 equations are obtained from (3.26) and (3.27),
four equations from boundary conditions (3.20) and eight from boundary con-
ditions (3.34) and (3.35).
3.2.7 Stable systems of linear equations
In this section, we present stable systems of linear equations for both interpolat-
ing and approximating collocation algorithms.
3.2.8 Stable system for interpolating collocation algorithm
From (3.24) and (3.25), we get following un-determine system of (N + 1) equa-
tions with (N + 17) unknowns {zi}
A1Z1 = G1, (3.36)
65
where the matrices A1, Z1, and G1 of orders (N +1)× (N +17), (N +17)× 1 and
(N + 1)× 1 respectively are given by
A1 =
ϕiv−8 ϕiv
−7 ϕiv−6 ϕiv
−5 ϕiv−4 ϕiv
−3 ϕiv−2 ϕiv
−1 q0 ϕiv1 ϕiv
2 ϕiv3 ϕiv
4
0 ϕiv−8 ϕiv
−7 ϕiv−6 ϕiv
−5 ϕiv−4 ϕiv
−3 ϕiv−2 ϕiv
−1 q1 ϕiv1 ϕiv
2 ϕiv3
0 0 ϕiv−8 ϕiv
−7 ϕiv−6 ϕiv
−5 ϕiv−4 ϕiv
−3 ϕiv−2 ϕiv
−1 q2 ϕiv1 ϕiv
2
0 0 0 ϕiv−8 ϕiv
−7 ϕiv−6 ϕiv
−5 ϕiv−4 ϕiv
−3 ϕiv−2 ϕiv
−1 q3 ϕiv1
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0
ϕiv5 ϕiv
6 ϕiv7 ϕiv
8 · · · 0 0 0
ϕiv4 ϕiv
5 ϕiv6 ϕiv
7 · · · 0 0 0
ϕiv3 ϕiv
4 ϕiv5 ϕiv
6 · · · 0 0 0
ϕiv2 ϕiv
3 ϕiv4 ϕiv
5 · · · 0 0 0
· · · · · · · · · · · · · · · · · · · · · · · ·
· · · · · · · · · · · · · · · ϕ′′′7 ϕ′′′
8 0
· · · · · · · · · · · · · · · ϕ′′′6 ϕ′′′
7 ϕ′′′8
, (3.37)
Z1 = (z−8, z−7, z−6, z−5, z−4, · · · , zN+8)T , (3.38)
G1 = (b0h4, b1h
4, b2h4, b4h
3, · · · , bNh4)T , (3.39)
where ϕivj = ϕiv(j) and qj = ϕiv
0 − ajh4.
For obtaining the unique solution of (3.36), we made some adjustment of bound-
ary conditions in previous section which is defined in (3.28), (3.29), (3.32) and
(3.33). By using this adjustment, we get a following system of (N + 17) linear
equations with (N + 17) unknowns {zi}, defined as
D1Z1 = R1, (3.40)
66
where the coefficient matrix
D1 = (BT0 , A
T1 , B
T1 )
T , (3.41)
A1 is defined by (3.37), B0 and B1 are defined as
B0 =
0 1 −10 45 −120 210 −252 210 −120 45 −10
0 0 1 −10 45 −120 210 −252 210 −120 45
0 0 0 1 −10 45 −120 210 −252 210 −120
0 0 0 0 1 −10 45 −120 210 −252 210
0 0 0 0 0 1 −10 45 −120 210 −252
0 0 0 0 0 0 1 −10 45 −120 210
0 0 0 0 0 0 0 0 7381N2520
25200N2520
−56700N2520
0 0 0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 0 · · · 0
−10 1 0 0 0 0 0 0 · · · 0
45 −10 1 0 0 0 0 0 · · · 0
−120 45 −10 1 0 0 0 0 · · · 0
210 −120 45 −10 1 0 0 0 · · · 0
−252 210 −120 45 −10 1 0 0 · · · 0
100800N2520
−132300N2520
127008N2520
−88200N2520
43200N2520
−14175N2520
2800N2520
−252N2520
· · · 0
0 0 0 0 0 0 0 0 · · · 0
,
(3.42)
first six rows of B0 are obtained from (3.32), second last row is obtained from
(3.28) and last row is taken from given boundary conditions Z1(0) which is de-
67
fined in (3.14) and
B1 =
0 0 · · · 0 0 0 0 0 0 0 0 0 0
0 0 · · · N10 − 10N
945N8 − 120N
7 35N − 252N5
105N2 −40N 45N
2 −10N
0 0 · · · 0 0 1 −10 45 −120 210 −252 210 −120
0 0 · · · 0 0 0 1 −10 45 −120 210 −252 210
0 0 · · · 0 0 0 0 1 −10 45 −120 210 −252
0 0 · · · 0 0 0 0 0 1 −10 45 −120 210
0 0 · · · 0 0 0 0 0 0 1 −10 45 −120
0 0 · · · 0 0 0 0 0 0 0 1 −10 45
1 0 0 0 0 0 0 0 0
7381N2520 0 0 0 0 0 0 0 0
45 −10 1 0 0 0 0 0 0
−120 45 −10 1 0 0 0 0 0
210 −120 45 −10 1 0 0 0 0
−252 210 −120 45 −10 1 0 0 0
−120 210 −252 210 −120 45 −10 1 0
, (3.43)
first row of B1 is obtained from Z1(N) which is defined in (3.14), second row is
obtained from (3.29) and the last six rows are obtained from (3.33), Z1 which is
defined in (3.38) and R1 is defined as
R1 = (0, 0, 0, 0, 0, 0, β, α,GT1 , γ, ω, 0, 0, 0, 0, 0, 0)
T ,
where G1 is defined by (3.39).
3.2.9 Stable system for approximating collocation algorithm
From (3.26) and (3.27), we get following un-determine system of (N + 1) equa-
tions with (N + 13) unknowns {zi}
A2Z2 = G2, (3.44)
68
where the matrices A2, Z2, and G2 of orders (N +1)× (N +13), (N +13)× 1 and
(N + 1)× 1 respectively are given by
A2 =
Φiv−6 Φiv
−5 Φiv−4 Φiv
−3 Φiv−2 Φiv
−1 Υ0 Φiv1 Φiv
2 Φiv3 Φiv
4
0 Φiv−6 Φiv
−5 Φiv−4 Φiv
−3 Φiv−2 Φiv
−1 Υ1 Φiv1 Φiv
2 Φiv3
0 0 0 Φiv−6 Φiv
−5 Φiv−4 Φiv
−3 Φiv−2 Φiv
−1 Υ2 Φiv1
0 0 0 0 Φiv−6 Φiv
−5 Φiv−4 Φiv
−3 Φiv−2 Φiv
−1 Υ3
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0
Φiv5 Φiv
6 · · · 0 0 0
Φiv4 Φiv
5 · · · 0 0 0
Φiv3 Φiv
4 · · · 0 0 0
Φiv2 Φiv
3 · · · 0 0 0
· · · · · · · · · · · · · · · · · ·
· · · · · · · · · Φ′′′5 Φ′′′
6 0
· · · · · · · · · Φ′′′4 Φ′′′
5 Φ′′′6
, (3.45)
Z2 = (z−6, z−5, z−4, z−3, z−2, · · · , zN+6)T , (3.46)
G2 = (b0h4, b1h
4, b2h4, b4h
3, · · · , bNh4)T , (3.47)
where Φivj = Φiv(j) and Υj = Φiv
0 − ajh4.
In order to get the unique solution of system (3.44), we have defined some extra
conditions in (3.30), (3.31), (3.34) and (3.35). By using these extra conditions we
get a following system of (N + 13) linear equations with (N + 13) unknowns
{zi}
D2Z2 = R2, (3.48)
where the coefficient matrix
D2 = (BT0 , A
T2 ,BT
1 )T , (3.49)
69
A2 is defined by (3.45), B0 and B1 are defined as
B0 =
0 0 0 1 −4 6 −4 1 0 0 0 0 0
0 0 0 0 1 −4 6 −4 1 0 0 0 0
0 0 0 0 0 1 −4 6 −4 1 0 0 0
0 0 0 0 0 0 1 −4 6 4 1 0 0
0 0 0 0 0 0 −251N12
48N12
−36N12
16N12
−3N12
0 0
0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
0 0 · · · 0 0
, (3.50)
first four rows of B0 are obtained from (3.34), second last row is obtained from
(3.30) and the last row is taken from the given boundary conditions Z2(0) which
is defined in (3.20) and
B1 =
0 0 0 0 · · · 0 0 0 0 0 0 1 0 0 0
0 0 0 0 · · · 0 0 3N12
−16N12
36N12
−48N12
25N12
0 0 0
0 0 0 0 · · · 0 0 0 0 0 0 0 1 −4 6
0 0 0 0 · · · 0 0 0 0 0 0 0 0 1 −4
0 0 0 0 · · · 0 0 0 0 0 0 0 0 0 1
0 0 0 0 · · · 0 0 0 0 0 0 0 0 0 0
70
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
−4 1 0 0 0 0 0 0
6 −4 1 0 0 0 0 0
−4 6 −4 1 0 0 0 0
1 −4 6 −4 1 0 0 0
, (3.51)
first row of B1 is obtained from Z2(N) which is defined in (3.20), second row
is obtained from (3.31) and the last four rows are obtained from (3.35), Z2 is
defined in (3.46) and R2 is defined as
R2 = (0, 0, 0, 0, 0, 0, β, α,GT2 , γ, ω, 0, 0, 0, 0, 0, 0)
T ,
where G2 is defined by (3.47). Hence to obtain the approximate solution of the
fourth order boundary value problem (3.3) by interpolating and approximating
collocation algorithms we need to solve the systems (3.40) and (3.48) respective-
ly.
3.2.10 Existence of the solution
In this section, we discuss the non-singularity of the coefficient matrices D1 and
D2 defined in (3.41) and (3.49) respectively. We observe that the coefficient ma-
trices D1 and D2 are neither symmetric nor diagonally dominant. However it
can be shown that D1 and D2 are non-singular. Since D1 and D2 are band matri-
ces with half band width 9 and 7, numerical complexities for solving the linear
systems using Gaussian elimination are about 81(N + 17) and 49(N + 13) mul-
tiplications respectively. For large N , the matrices are almost symmetric except
the first and last eight rows and columns of D1 and first and last four rows and
columns of D2 due to the boundary conditions. Therefore we first consider their
71
symmetric part i.e. square symmetric matrices E1 and E2 of orders N+1 defined
as
E1 =
ϕiv0 ϕiv
1 ϕiv2 ϕiv
3 ϕiv4 ϕiv
5 ϕiv6 ϕiv
7 ϕiv8 · · · 0 0 0 0
ϕiv−1 ϕiv
0 ϕiv1 ϕiv
2 ϕiv3 ϕiv
4 ϕiv5 ϕiv
6 ϕiv7 · · · 0 0 0 0
ϕiv−2 ϕiv
−1 ϕiv0 ϕiv
1 ϕiv2 ϕiv
3 ϕiv4 ϕiv
5 ϕiv6 · · · 0 0 0 0
ϕiv−3 ϕiv
−2 ϕiv−1 ϕiv
0 ϕiv1 ϕiv
2 ϕiv3 ϕiv
4 ϕiv5 · · · 0 0 0 0
0 0 0 0 0 0 0 0 0 · · · ϕiv4 ϕiv
3 ϕiv2 ϕiv
1
0 0 0 0 0 0 0 0 0 · · · ϕiv3 ϕiv
2 ϕiv1 ϕiv
0
,
and
E2 =
Φiv0 Φiv
1 Φiv2 Φiv
3 Φiv4 Φiv
5 Φiv6 0 0 · · · 0 0 0 0
Φiv−1 Φiv
0 Φiv1 Φiv
2 Φiv3 Φiv
4 Φiv5 Φiv
6 0 · · · 0 0 0 0
Φiv−2 Φiv
−1 Φiv0 Φiv
1 Φiv2 Φiv
3 Φiv4 Φiv
5 Φiv6 · · · 0 0 0 0
Φiv−3 Φiv
−2 Φiv−1 Φiv
0 Φiv1 Φiv
2 Φiv3 Φiv
4 Φiv5 · · · 0 0 0 0
0 0 0 0 0 0 0 0 0 · · · Φiv4 Φiv
3 Φiv2 Φiv
1
0 0 0 0 0 0 0 0 0 · · · Φiv3 Φiv
2 Φiv1 Φiv
0
.
So E1 and E2 are symmetric matrices obtained from D1 and D2 respectively.
It can be shown that E1 and E2 are non-singular when N increase. The non-
singularity of the matrices E1 and E2 is shown in Table 3.5 by finding their
determinants. The matrices E1 and E2 remain non-singular for N 6 500 and
N 6 100 respectively. For large N the determinants of the matrices may or may
not be equals to zero. The non-singularity of the matrices D1 and D2 have been
checked by finding their eigenvalues for N 6 500 and for N 6 100 respectively.
Since the eigenvalues for both the matrices are non-zero, then by Strang (2011)
matrices D1 and D2 are non-singular. However the matrices D1 for N > 500
and D2 for N > 100 may or may not be non-singular. Therefore we claim that
systems of equations (3.40) and (3.48) are stable.
72
Table 3.5: Determinants of the matrices
N E1 E2
10 −8667/56 1.11401292× 103
50 -177183 3.2517495× 102
100 -552709050 0.753776508473953
200 -5.033491472×1036 · · ·
300 -4.477989536×1071 · · ·
400 3987757210720454 · · ·
500 3987757210720454 · · ·
3.2.11 Error estimation
In this section, we discuss the approximation properties of the interpolating and
approximating collocation algorithms. Since the scheme (3.1) and (3.2) repro-
duce polynomial curves of degree nine and three so by Dyn (2002) and Mustafa
et al. (2014a) schemes have approximation order ten and four respectively. Here
we present our main results for error estimation.
Proposition 3.2.5. Suppose the exact solution y(x) ∈ C4[0, 1] and {zi} are obtained
by (3.40) then absolute error by interpolating collocation algorithm is
||err1(x)||∞ = ||Z(l)1 (x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3,
where l denotes the order of derivative.
Proposition 3.2.6. Suppose the exact solution y(x) ∈ C4[0, 1] and {zi} are obtained
by (3.48) then absolute error by approximating collocation algorithm is
||err2(x)||∞ = ||Z(l)2 (x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.
Proof of these results are similar to the proof of proposition by Mustafa and
Ejaz (2014) and Qu and Agarwal (1996).
73
3.3 Numerical examples and comparison
In this section, the interpolating and approximating collocation algorithms de-
scribed in Section 3.2, are tested on the problems given below. Absolute errors
between exact and approximate solutions are also calculated. For the sake of
comparisons, we also tabulated the results in this section. Graphical illustra-
tions of solutions are presented.
3.3.1 Numerical examples
Here we find the numerical solutions of some of the boundary value problems
arising in the mathematical modeling of viscoelastic and inelastic flows etc.
Example 3.3.1. Consider the fourth order linear boundary value problem
yiv(x) + xy = −(8 + 7x+ x3)ex, 0 < x < 1, (3.52)
subject to the boundary conditions
y(0) = y(1) = 0, y′(0) = 1, y′(1) = −e. (3.53)
By comparing the above problem with equation (3.3), we have a(x) = −x and
b(x) = −(8 + 7x + x3)ex. The exact solution for the above problem is y =
x(1− x)ex.
Here we present the numerical solution of above problem by interpolating and
approximating collocation algorithms.
74
Solution by interpolating collocation algorithm:
In this method, by solving the system of linear equations (3.40) at N = 10, we
obtain the approximate solution (3.12) of (3.52) where {zi}, −8 ≤ i ≤ 18 are
{−0.736798818, −0.643536669, −0.554623724, −0.467529093, −0.379799398,
−0.289680649, −0.196215973, −0.099344276, 0.0000, 0.0997855, 0.196780435,
0.286445566, 0.362823930, 0.4184258296, 0.444109480, 0.428957387, 0.360149377,
0.222833881, 0.000, −0.327646133,−0.781795936, −1.386637856, −2.168934169,
−3.158065904, −4.386034900, −5.887410047,−7.698656102}.
Solution by approximating collocation algorithm:
In this method, we solve the system of linear equations (3.48) at N = 10 and get
solution (3.18) of (3.52) where {zi}, −6 ≤ i ≤ 16 are
{240576495.97007, 346838.29496, 518.9347067, −0.27530396, −0.192116155,
−0.098802082, 0.00, 0.099651832, 0.195515154, 0.282951708, 0.357323235,
0.412290577, 0.439106327, 0.426338357, 0.359800241, 0.223138586, 0,
−0.325968911, −0.771121538, −1.351811276, 759.3687002, 508693.855464,
352843031.1188856}.
Example 3.3.2. Consider the following fourth order linear boundary value prob-
lem
yiv(x) = (x4 + 14x3 + 49x2 + 32x− 12)ex 0 ≤ x ≤ 1 (3.54)
with
y(0) = y(1) = y′(0) = y′(1) (3.55)
corresponds to the bending of a thin beam clamped at both ends. The unique
solution of (3.54) is
y(x) = x2(1− x)2ex.
75
Solution by interpolating collocation algorithm:
By using this method, we solve system (3.40) at N = 10 and get solution (3.12)
of (3.54) where {zi}, −8 ≤ i ≤ 18 are
{−1.29218842, 0.538942241, 0.376822099, 0.25497974, 0.157378182, 0.084436111,
0.035355124, 0.008209932, 0, 0.006713642, 0.023450361, 0.044645810, 0.064445941,
0.077282873, 0.078714604, 0.066604028, 0.042729735, 0.014941769, 0, 0.027260994,
0.143411251, 0.418480429, 0.953407669, 1.889477424, 3.419988190, 5.826254985,
6.993262174}.
Solution by approximating collocation algorithm:
In this method, by solving system of linear equations (3.48) at N = 10, we obtain
the approximate solution (3.18) of (3.54) where {zi}, −6 ≤ i ≤ 16 are
{1.711850528× 1011,−2.4699609× 108, 3.560795701× 105,−5.33546354× 105,
0.06034270857, 0.013560666, 0, 0.01051064293, 0.03594252611, 0.067145581105,
0.09496973950, 0.11026493288, 0.10721651489, 0.08657152262, 0.05642424602,
0.02486897512, 0, 0.01008838960, 0.00269809658,−3.898592698× 103,
2.597091086× 106,−1.801597056× 109, 1.2486272867× 1012}.
Example 3.3.3. Consider the boundary value problem
y(iv) − y = −4(2x cos(x) + 3 sin(x)) (3.56)
with boundary conditions
y(0) = 0, y(1) = 0, y′(0) = −1, y′(1) = 2 sin(1). (3.57)
The exact solution of this problem is y = (x2 − 1) sin(x).
76
Solution by interpolating collocation algorithm:
Here, we solve system (3.40) at N = 10 and get solution (3.12) of (3.56) where
{zi}, −8 ≤ i ≤ 18 are
{0.0600152940, 0.18160335240, 0.25833493990, 0.29216208360, 0.2870204837,
0.1824477485, 0.09700293, 0,−0.1001836945,−0.1951709144,−0.2768975252,
−0.3717065517,−0.372918743,−0.3376558661,−0.3379193763,−0.26367852,
−0.1505934691, 0, 0.1844062122, 0.3967742453, 0.6290856272, 0.87118909250,
1.11088175850, 1.33403484280, 1.52476071733, 1.6661398769}.
Solution by approximating collocation algorithm:
By solving system of linear equations (3.48) at N = 10, we obtain the approxi-
mate solution (3.18) of (3.56) where {zi}, −6 ≤ i ≤ 16 are
{115585876195.8271, 166975210.2723412, 240725.36249556, 360.797052040,
0.1912504195, 0.0989171622, 0,−0.0988737231,−0.1910766633,−0.2699814765,
−0.328960819,−0.361387346,−0.3620271786,−0.3276693903,−0.2568529513,
−0.1481168313, 0, 0.1889585727, 0.4202199172, 648.2401207, 432617.012,
300071307.86, 207719711144.92}.
77
Tabl
e3.
6:N
umer
ical
resu
lts
ofEx
ampl
e3.
3.1
xi
Ana
lyti
cA
ppro
xim
ate
solu
tion
App
roxi
mat
eso
luti
on
solu
tion
Z1
byin
terp
olat
ing
Z2
byap
prox
imat
ing
||err
1(x
i)|| ∞
||err
2(x
i)|| ∞
yco
lloca
tion
algo
rith
mco
lloca
tion
algo
rith
m
0.0
00
00
0
0.1
0.09
9465
380.
0997
8551
520.
0996
5183
170.
0003
2013
0.00
0186
45
0.2
0.19
5424
440.
1967
8040
150.
1955
1515
400.
0013
5596
0.00
0090
71
0.3
0.28
3470
350.
2864
4551
080.
2829
5170
800.
0029
7516
0.00
0518
64
0.4
0.35
8037
930.
3628
2385
780.
3573
2323
450.
0047
8593
0.00
0714
69
0.5
0.41
2180
320.
4184
2575
120.
4122
9057
650.
0062
4543
0.00
0110
25
0.6
0.43
7308
510.
4441
0940
750.
4391
0632
740.
0068
0090
0.00
1797
81
0.7
0.42
2888
070.
4289
5733
140.
4263
3835
690.
0060
6926
0.00
3450
29
0.8
0.35
6086
550.
3601
4934
300.
3598
0024
080.
0040
6279
0.00
3713
69
0.9
0.22
1364
280.
2228
3386
920.
2231
3858
610.
0014
6959
0.00
1774
31
1.0
00
00
0
78
Tabl
e3.
7:N
umer
ical
resu
lts
ofEx
ampl
e3.
3.2
xi
Ana
lyti
cA
ppro
xim
ate
solu
tion
App
roxi
mat
eso
luti
on
solu
tion
Z1
byin
terp
olat
ing
Z2
byap
prox
imat
ing
||err
1(x
i)|| ∞
||err
2(x
i)|| ∞
yco
lloca
tion
algo
rith
mco
lloca
tion
algo
rith
m
0.0
00
00
0
0.1
0.00
8951
884
0.00
6713
642
0.01
0510
643
0.00
2238
243
0.00
1558
758
0.2
0.03
1267
9106
0.02
3450
361
0.03
5942
526
0.00
7817
550
0.00
4674
615
0.3
0.05
9528
773
0.04
4645
810
0.06
7145
581
0.01
4882
9636
0.00
7616
808
0.4
0.08
5929
102
0.06
4445
941
0.09
4969
740
0.02
1483
162
0.00
9040
637
0.5
0.10
3045
079
0.07
7282
873
0.11
0264
933
0.02
5762
206
0.00
7219
853
0.6
0.10
4954
043
0.07
8714
604
0.10
7216
515
0.02
6239
439
0.00
2262
472
0.7
0.08
8806
494
0.06
6604
028
0.08
6571
523
0.02
2202
467
0.00
2234
972
0.8
0.05
6973
848
0.04
2729
735
0.05
6424
256
0.01
4244
112
0.00
0549
602
0.9
0.01
9922
785
0.01
4941
769
0.02
4868
975
0.00
4981
016
0.00
4946
1810
1.0
00
00
0
79
Tabl
e3.
8:N
umer
ical
resu
lts
ofEx
ampl
e3.
3.3
xi
Ana
lyti
cA
ppro
xim
ate
solu
tion
App
roxi
mat
eso
luti
on
solu
tion
Z1
byin
terp
olat
ing
Z2
byap
prox
imat
ing
||err
1(x
i)|| ∞
||err
2(x
i)|| ∞
yco
lloca
tion
algo
rith
mco
lloca
tion
algo
rith
m
0.0
00
00
0
0.1
-0.0
9883
508
-0.1
0018
369
-0.0
9887
3723
0.00
1348
60.
0000
3864
1
0.2
-0.1
9072
256
-0.1
9517
091
-0.1
9107
6663
30.
0044
484
0.00
0354
11
0.3
-0.2
6892
339
-0.2
7689
752
-0.2
6998
1476
50.
0079
741
0.00
1058
0.4
-0.3
2711
141
-0.3
3791
938
-0.3
2896
0819
0.01
0808
0.00
1849
4
0.5
-0.3
5956
915
-0.3
7170
655
-0.3
6138
7346
0.01
2137
0.00
1818
2
0.6
-0.3
6137
118
-0.3
7291
874
-0.3
6202
7178
60.
0115
480.
0006
5600
0.7
-0.3
2855
102
-0.3
3765
587
-0.3
2766
9390
30.
0091
048
0.00
0881
63
0.8
-0.2
5824
819
-0.2
6367
852
-0.2
5685
2951
30.
0054
303
0.00
1395
2
0.9
-0.1
4883
211
-0.1
5059
347
-0.1
4811
6831
30.
0017
614
0.00
0715
28
1.0
00
00
0
80
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
X
Sol
utio
n
Analytic solution YApproximate solution Z
1
Approximate solution Z2
Figure 3.1: Comparison between analytic and approximate solutions of Example 3.3.1
obtained by interpolating and approximating collocation algorithms.
0 0.2 0.4 0.6 0.8 10
0.02
0.04
0.06
0.08
0.1
0.12
X
Sol
utio
n
Analytic solution YApproximate solution Z
1
Approximate solution Z2
Figure 3.2: Comparison between analytic and approximate solutions of Example 3.3.2
obtained by interpolating and approximating collocation algorithms.
81
0 0.2 0.4 0.6 0.8 1−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05
0
X
Sol
utio
n
Analytic solution YApproximate solution Z
1
Approximate solution Z2
Figure 3.3: Comparison between analytic and approximate solutions of Example 3.3.3
obtained by interpolating and approximating collocation algorithms.
3.3.2 Comparison:
The numerical results of Examples 3.3.1, 3.3.2 and 3.3.3 by interpolating and
approximating collocation algorithms are presented in Tables 3.6, 3.7 and 3.8
respectively. The maximum absolute errors in the solution of Examples 3.3.1,
3.3.2 and 3.3.3 obtained by interpolating and approximating collocation algo-
rithms are given in Table 3.9. Graphical representation of these results is shown
in Figures 3.1, 3.2 and 3.3. In these figures solid curve represents the exact so-
lutions, dashed lines represent approximate solutions obtained by (3.40) and
dotted lines represent the approximate solutions obtained by (3.48). Following
is the comparison of the numerical solutions obtained by proposed algorithms
and other approaches of this type of boundary value problems:
• From above results we see that approximating schemes based collocation
algorithms give better results then interpolating schemes based colloca-
tion algorithms.
82
• Example 3.3.1 is also solved by Usmani (1983). He solved this problem by
second order finite difference method and obtained the maximum abso-
lute errors at different step sizes h. We observe that the maximum absolute
error at the step size h = 110
by the proposed approximating collocation al-
gorithm is better than the maximum absolute error obtained by Usmani
(1983) at step size h = 116
. The comparison of proposed methods with
second order finite difference method of Usmani (1983) at difference step
sizes is shown in Table 3.10.
• Example 3.3.2 is also solved by Russell and Shampine (1972) by quintic
spline based collocation methods. We observe that order of error approx-
imation obtained by Russell and Shampine (1972) and proposed approxi-
mating collocation algorithm are same.
Table 3.9: Maximum absolute errors of Examples 3.3.1, 3.3.2 and 3.3.3
Example Max. absolute errors Max. absolute errors
by interpolating by approximating
collocation algorithm collocation algorithm
3.3.1 6.8010× 10−3 3.7137× 10−3
3.3.2 2.6239× 10−2 0.9041× 10−2
3.3.3 1.2137× 10−2 0.18494× 10−2
3.4 Conclusion
In this chapter, we have presented interpolating and approximating collocation
algorithms based on interpolating and approximating subdivision schemes for
the solution of linear fourth order boundary value problems. The proposed
83
Table 3.10: Comparison of Example 3.3.1 with different methods
Max. absolute errors Max. absolute errors Second order finite
h by interpolating by approximating difference method
collocation algorithm collocation algorithm by Usmani (1983)
14
· · · · · · 8.50× 10−2
18
· · · 1.4098× 10−2 2.09× 10−2
110
6.8010× 10−3 3.7137× 10−3 · · ·116
6.7469× 103 4.4572× 10−3 5.27× 10−3
algorithms have been applied on different linear fourth order boundary val-
ue problems. Results show that the approximating collocation algorithm gives
better results comparative to interpolating collocation algorithm. We have also
observed that the accuracy of the solution can be improved by choosing differ-
ent subdivision schemes with the proper adjustment of boundary conditions.
Approximating subdivision scheme based collocation algorithm gives better
results comparative to second order finite difference method. However, ap-
proximating subdivision scheme based collocation algorithm and quintic spline
based collocation algorithm have same order of approximation.
84
Chapter 4
A Subdivision Based Iterative
Collocations Algorithm for
Nonlinear Third Order Boundary
Value Problems
Many problems in physics, chemistry and engineering science are demonstrated
mathematically by third order boundary value problems. These boundary value
problems can be found in different areas of applied mathematics and physics as,
in the deflection of a curved beam having a constant or varying cross section, a
three layer beam, electromagnetic waves, or gravity driven flows. In this chap-
ter, we develop subdivision schemes based collocation iterative algorithm for
the solution of nonlinear third order boundary value problems.
An outline of this chapter is as follows: In Section 4.1, some results about ex-
istence and uniqueness of the solution of third order boundary value problem
are given. In Section 4.2, subdivision algorithm, basis function and their deriva-
tives are briefed. In Section 4.3, subdivision based iterative algorithm for the
solution of nonlinear third order BVPs using the derivatives of basis functions
85
is formulated. Convergence of the proposed algorithm is also discussed in this
section. Error analysis is given in Section 4.4. Numerical examples illustrating
the usefulness of our proposed algorithm are given in Section 4.5.
4.1 Existence and uniqueness of the solution
In this section, we present some results about the existence and uniqueness of
the solution of third order nonlinear boundary value problems. The detail of
these results can be found in Agarwal (1973).
The general third order nonlinear boundary value problem can be prescribed as
y′′′= f(x, y, y
′, y
′′) (4.1)
with boundary conditions define as
y(0) = α1, y′(0) = α2, y(1) = α3. (4.2)
where αi, i = 1, 2, 3 are constants.
Proposition 4.1.1. If the function f(x, y, y′, y′′) is continuous and satisfy the following
uniform Lipschitz condition∣∣f(x, y, y′, y′′)− f(x, y, y′, y′′)∣∣ ≤ M0 |y − y|+M1
∣∣y′ − y′∣∣+M2 |y′′ − y′′| ,
(4.3)
∀ (x, y, y′, y′′), (x, y, y′, y′′) ∈ [a, b]×R3
where the constants M0, M1 and M2 satisfy
2
81M0(b− a)3 +
1
6M1(b− a)2 +
2
3M2(b− a) < 1, (4.4)
or
3
160M0(b− a)3 +
17
150M1(b− a)2 +
2
3M2(b− a) < 1, (4.5)
86
then the boundary value problem has one and only one solution
y′′′= f(x, y, y
′, y
′′), a ≤ x ≤ b, y(a) = y1, y′(a) = y2, y(b) = y3. (4.6)
The existence and uniqueness of the differential equation (4.1) with boundary
conditions at two or three points are presented in Agarwal (1973).
Remark 4.1.1. Throughout this paper the function f(x, y, y′, y′′) satisfies Lipschitz
conditions (4.3) along with condition (4.4)-(4.6). So the existence and uniqueness
of the solutions of (4.1) is guaranteed.
4.2 Subdivision scheme and basis function
In this section, we define binary subdivision algorithm and their basis function
that are used to construct the approximate solutions of (4.1).
4.2.1 Interpolating subdivision scheme
We consider the following 8-point binary interpolating subdivision scheme in-
troduced in Deng and Ma (2013) and Deslauriers and Dubuc (1989)
pk+12i = pki
pk+12i+1 =
12252048
(pki + pki+1
)− 245
2048
(pki−1 + pki+2
)+ 49
2048
(pki−2 + pki+3
)− 5
2048
(pki−3 + pki+4
)(4.7)
where pki and pk+1i are points at kth and (k+1)th iterative level. The scheme (4.7)
is C3-continuous and support length (−7, 7) with 8th order approximation.
4.2.2 Basis function and their derivatives
The basis function is the limit function resulting from cardinal data, where all
vertices of the polygon have value zero except for one. Let g(x), x ∈ R be the
87
fundamental solution of (4.7) satisfies the two scale equation
g(x) = g(2x) +1
2048[1225{g(2x− 1) + g(2x+ 1)} − 245{g(2x− 3)
+g(2x+ 3)}+ 49{g(2x− 5) + g(2x+ 5)} − 5{g(2x− 7)
+g(2x+ 7)}] , x ∈ R
and
g(x) ∈ C3, g(x) = 0, if x /∈ [-6, 6], g(i) = δ0, i ∈ Z.
Proposition 4.2.1. The fundamental solution g(x) is three time continuously differen-
tiable over the interval [−6, 6]. Its derivatives at integers are given by
g′(i) = 2sign(i)ET|i|η1, g′′(i) = 22ET
|i|η2, g′′′(i) = 23sign(i)ET|i|η3,
where
sgn(t) =
−1, t < 0,
0, t = 0,
1, t > 0,
and Et’s for 0 ≤ t ≤ 6, are defined below
Et = (e6t, e5t, e4t, e3t, e2t, e1t, e0t, e−1t, e−2t, e−3t, e−4t, e−5t, e−6t)T ,
where
eit =
1, i = t,
0, i = t,
and ηj , 1 ≤ j ≤ 3 are defined in Mustafa and Ejaz (2014).
Furthermore, the numeric values of first, second and third derivatives of g(i)
at i ∈ [−6, 6] are
g′(0) = 0, g′(±1) = ∓7859249553
,
g′(±2) = ± 76113198212
, g′(±3) = ∓ 332849553
,
g′(±4) = ± 2645594636
, g′(±5) = ± 256743295
,
g′(±6) = ∓ 1594636
,
(4.8)
88
g′′(0) = −34264341124
, g′′(±1) = 57042561079505
,
g′′(±2) = −120536518636040
, g′′(±3) = 3259521079505
,
g′′(±4) = − 608712878680
, g′′(±5) = − 704215901
,
g′′(±6) = 551727208
,
(4.9)
g′′′(0) = 0, g′′′(±1) = ∓292352117495
,
g′′′(±2) = ±30479871879920
, g′′′(±3) = ∓ 331213055
,
g′′′(±4) = ∓ 1369234990
, g′′′(±5) = ± 162611
,
g′′′(±6) = ∓ 541776
.
(4.10)
The above derivative values are found by using the left eigenvectors of the sub-
division process (4.7). The detailed description about these left eigenvectors ηj ,
1 ≤ j ≤ 3 and derivatives can be found in Mustafa and Ejaz (2014).
4.3 Subdivision based iterative algorithm
In this section, we describe the algorithm for the numerical solution of nonlinear
boundary value problem (4.1).
4.3.1 The collocation algorithm
In this subsection, we construct the collocation method based on the interpolat-
ing subdivision scheme (4.7). Let U(x) be the assumed solution of (4.1)
U(x) =N+6∑i=−6
uig
(x− xi
h
), 0 6 x 6 1, (4.11)
where N(> 6) ∈ Z+, h = 1/N , xi = i/N = ih and {ui} are the unknown to be
determined for the solution of (4.1). The collocation algorithm together with the
boundary conditions is defined as follows:
U′′′(xj) = f(xj, U(xj), U
′(xj)), j = 0, 1, 2, · · · , N, (4.12)
89
with the following type of boundary conditions
U(0) = α1, U′(0) = α2, U(N) = α3, (4.13)
by taking the third derivative of (4.11) we get
U′′′(x) =
1
h3
N+6∑i=−6
uig′′′(x− xi
h
), 0 6 x 6 1, (4.14)
using (4.14) into (4.12), we get
N+6∑i=−6
uig′′′(xj − xi
h
)= h3f(xj, U(xj), U
′(xj)), j = 0, 1, 2, · · · , N.
This can be written asN+6∑i=−6
uig′′′
j−i = h3f(xj, U(xj), U′(xj)), j = 0, 1, 2, · · · , N.
As we know that g′′′i = −g
′′′−i, then above system of equations become
N+6∑i=−6
(−1)uig′′′
i−j = h3f(xj, U(xj), U′(xj)), j = 0, 1, 2, · · · , N. (4.15)
Now we simplify the nonlinear system of equations (4.15) in following theorem-
s.
Theorem 4.3.1. The nonlinear system of equations (4.15) for j = 0 becomes
6∑i=−6
(−1)uig′′′
i = h3f(x0, U(x0), U′(x0)). (4.16)
Proof. By expanding (4.15) for j = 0, we get
(−1){u−6g′′′
−6 + u−5g′′′
−5 + · · ·+ uN+5g′′′
N+5 + uN+6g′′′
N+6} = h3f(x0, U(x0), U′(x0)).
As we know that g′′′(i) exist only for the interval for i ∈ [−6, 6] and outside the
interval it will be zero. Then above equation can be written as
(−1){u−6g′′′
−6 + u−6g′′′
−6 + · · ·+ u5g′′′
5 + u6g′′′
6 } = h3f(x0, U(x0), U′(x0)).
This implies (4.16).
90
Theorem 4.3.2. For j = 1, 2, · · · , N , the nonlinear system of equations (4.15) becomes
6+j∑i=−6+j
(−1)uig′′′
i−j = h3f(xj, U(xj), U′(xj)). (4.17)
Proof. Substituting j = 1 in (4.15), it becomes
(−1){u−6g′′′
−7 + u−5g′′′
−6 · · ·+ u7g′′′
6 · · ·+ uN+6g′′′
N+5} = h3f(x1, U(x1), U′(x1)).
Since g′′′(i) is non-zero only for the interval for i ∈ [−6, 6] and outside the inter-
val it will be zero. Then above equation becomes
(−1){u−5g′′′
−6 + u−4g′′′
−5 + · · ·+ u4g′′′
5 + u5h′′′
6 } = h3f(x1, U(x1), U′(x1)). (4.18)
For j=2, (4.15) becomes
(−1){u−6g′′′
−8 + u−5g′′′
−7 + u−4g′′′
−6 + · · ·+ u4g′′′
2 + u5g′′′
3 + u6g′′′
4 + u7g′′′
5 u8g′′′
6
+ · · ·+ uN+5g′′′
N+3 + uN+6g′′′
N+4} = h3f(x2, U(x2), U′(x2)).
This implies
(−1){u−4g′′′
−6 + u−3g′′′
−5 + · · ·+ u7g′′′
5 + u8g′′′
6 } = h4f(x2, U(x2), U′(x2)). (4.19)
Similarly, we can find the expression for j = 3, 4, · · · , N , i-e.
(−1){u−3g′′′
−6 + u−4g′′′
−5 + · · ·+ u8g′′′
5 + u9g′′′
6 } = h3f(x3, U(x3), U′(x3)). (4.20)
(−1){u−2g′′′
−6 + u−1g′′′
−5 + · · ·+ u9g′′′
5 + u10g′′′
6 } = h3f(x4, U(x4), U′(x4)). (4.21)
...
(−1){uN−6g′′′
−6 + uN−5g′′′
−5 + · · ·+ uN+5g′′′
5 + uN+6g′′′
6 } = h3f(xN , U(xN), U′(xN)).
(4.22)
Hence by combining the equations (4.18)-(4.22) we get (4.17).
91
4.3.2 Unstable nonlinear system
The nonlinear system of (4.15) is equivalent to the following nonlinear system
of N + 1 equations with (N+13) unknowns {ui}:
A1U = F (u), (4.23)
where A1 is banded matrix of order (N +1)× (N +13), U is the unknown vector
of order N + 13 and F (u) is the vector of order N + 1 depends on u. The matrix
A, vectors U and F (u) are given explicitly by
A1 = (−1)(g′′′
pq(q − p− 6))(N+1)×(N+13), (4.24)
where p and q represent the row and column respectively i.e. p = 1, 2, 3 · · · , N+1
and q = 1, 2, 3, · · · , N + 13,
F (u) =(h3f(x0, U(x0), U
′(x0)), · · · , h3f(xN , U(xN), U
′(xN))
)T
, (4.25)
U = (u−6, u−5, z−4, · · · , uN+4, uN+5, uN+6)T , (4.26)
U′(xj) =
N+6∑i=−6
ujg′(xj − xi
h
),
g′(i) is defined in (4.8) and g(i) = gi. The system (4.23) is unstable and we
need to make it stable to get unique solution. The detail for the stable system of
nonlinear equations is given in next section.
4.3.3 Stable nonlinear system
For unique solution of nonlinear systems (4.23), we need twelve more condi-
tions. Three conditions can be attained from given boundary conditions for
nonlinear systems of equations and remaining nine conditions are attained by
setting some extrapolation method. The detail of the given boundary conditions
and extrapolation method are given below:
92
4.3.4 Approximated boundary condition
The given boundary conditions are :
U(0) = α1, U′(0) = α2, U(N) = α3. (4.27)
We see that first derivative involve in the given boundary conditions, since ap-
proximation order of interpolating scheme (4.7) is eight, so we approximate
derivative boundary conditions at end point with approximation order eight.
The approximation of derivative conditions at end point is defined as
U ′(0) =
(N
840
){−2283u0 + 6720u1 − 11760u2 + 15680u3 − 14700u4 + 9408u5
−3920u6 + 960u7 − 105u8}+O(h8). (4.28)
4.3.5 Imposed boundary conditions
Remaining nine conditions for the nonlinear systems (4.23) to get stable sys-
tems for the solution of (4.1) are obtained by setting the following extrapolation
method.
We define five conditions at left end points and four conditions at the right
end points. Since subdivision scheme reproduces seven degree polynomials, so
we define boundary conditions of order eight for solution of (4.23). For simplic-
ity only the left end points u−5, u−4, u−3, u−2, u−1are discussed and the values of
right end points uN+2, uN+3, uN+4, uN+5 can be treated similarly.
The values u−5, u−4, u−3, u−2, u−1 can be determined by the septic polynomial
R(x) interpolating (xi, ui), 0 ≤ i ≤ 5. Precisely, we have
u−i = R(−xi), i = 1, 2, 3, 4, 5,
where
R(xi) =8∑
r=1
8
r
(−1)r+1U(xi−r).
93
Since by (4.11), U(xi) = ui for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we
have
R(−xi) =8∑
r=1
8
r
(−1)r+1u−i+r.
Hence the following boundary conditions can be employed at the left end
8∑r=0
8
r
(−1)ru−i+r = 0, i = 5, 4, 3, 2, 1. (4.29)
Similarly for the right end, we can define ui = R(−xi), i = N + 2, N + 3, N + 4,
N + 5 and
R(xi) =8∑
r=0
8
r
(−1)r+1ui−r.
So we have the following boundary conditions at the right end
8∑r=0
8
r
(−1)rui−r = 0, i = N + 2, N + 3, N + 4, N + 5. (4.30)
Finally, we get a following new system of (N + 13) linear equations with (N +
13) unknowns {ui}, in which N + 1 equations are obtained from (4.15), three
equations from boundary conditions (4.27) and nine from boundary conditions
(4.29) and (4.30).
Hence the stable nonlinear system of equations define as:
AU = G(u), (4.31)
where the matrix A is given by
A = (AT0 , A
T1 , A
T2 )
T , (4.32)
the matrix A1 is defined in (4.24). The matrix (A0)7×(N+13) is constructed as by
taking first five rows from (4.29), sixth row taking from (4.28) and the last row
U(0) from (4.27). Hence
94
A0 =
0 1 −8 28 −56 70 −56 28 −8
0 0 1 −8 28 −56 70 −56 28
0 0 0 1 −8 28 −56 70 −56
0 0 0 0 1 −8 28 −56 70
0 0 0 0 0 1 −8 28 −56
0 0 0 0 0 0 −2283N840
6720N840 − 11760N
840
0 0 0 0 0 0 1 0 0
1 0 0 0 0 0 0 · · · 0 0
−8 1 0 0 0 0 0 · · · 0 0
28 −8 1 0 0 0 0 · · · 0 0
−56 28 −8 1 0 0 0 · · · 0 0
70 −56 28 −8 1 0 0 · · · 0 0
15680N840 − 14700N
8409408N840 − 3920N
840960N840 −105N
840 0 · · · 0 0
0 0 0 0 0 0 0 · · · 0 0
.
The matrix (A2)6×(N+13) is constructed as by taking first row U(N) from (4.27)
and last five rows obtained from (4.30). Hence
A2 =
0 0 · · · 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 · · · 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0 0
0 0 · · · 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0 0
0 0 · · · 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0 0
0 0 · · · 0 0 0 1 −8 28 −56 70 −56 28 −8 1 0 0
.
The column vectors U is defined in (4.26) and G is defined as
G(u) = (0, 0, 0, 0, 0, U ′(0), U(0), F T (u), U(1), 0, 0, 0, 0)T , (4.33)
where F (u) is defined by (4.25).
4.3.6 Non-singularity of a matrix
The non-singularity of coefficient matrix A, which is defined in (4.32), can be
checked by different methods:
95
Since the determinant of matrix A is non-zero for N ≤ 2000 then the non linear
system of equations give solution for N ≤ 2000.
It is also observed that all the eigenvalues are non-zero for N ≤ 2000. Hence by
Strang (2011) matrix A is non-singular. For large N > 2000 the matrix may or
may not be non-singular.
The coefficient matrix A is neither symmetric nor diagonally dominant. Though
it can be prove that A is non-singular/invertible. The matrix is almost symmet-
ric except the first and last few rows and columns due to its boundary treatment
for large value of N . Therefore, we first consider the square band matrix B of
order N + 1 defined as
B = (−1)×
g′′′1 g′′′2 g′′′3 g′′′4 · · · · · · 0 0 0
g′′′0 g′′′1 g′′′2 g′′′3 · · · · · · 0 0 0
g′′′−1 g′′′0 g′′′1 g′′′2 · · · · · · 0 0 0
g′′′−2 g′′′−1 g′′′0 g′′′1 · · · · · · 0 0 0
g′′′−3 g′′′−2 g′′′−1 g′′′0 · · · · · · 0 0 0
· · · · · · · · · · · · · · · · · · · · · · · · · · ·
0 0 0 0 · · · · · · g′′′2 g′′′2 g′′′3
0 0 0 0 · · · · · · g′′′0 g′′′1 g′′′2
0 0 0 0 · · · · · · g′′′−1 g′′′0 g′′′1
.
Kilic and Stanica (2013) presented a method to find the inverse of banded matrix
of order n by using the LU factorization of the banded matrix. As B is a band
matrix of order (N + 1) so by LU factorization method its inverse exist.
96
4.3.7 Iterative algorithm and its convergence
In this section, we propose an iterative algorithm and discuss its convergence.
4.3.8 Iterative algorithm based on basis function
The iterative algorithm is based on basis function of the subdivision scheme
(4.7) as defined in the following three steps.
First step: Initial approximation
Initial approximation is important because numerical solution depends on the
initial approximation. We define the process for finding the initial approxima-
tion as follows:
Let initial approximate solution U0 be the solution of the following linear sys-
tem
AU0 = G0, (4.34)
where
G0 = (0, 0, 0, 0, 0, y′(a), y(a), p0, p1, p2, · · · , pN , y(b), 0, 0, 0, 0)T ,
pi = h3p(xi,mi, S), i = 0, 1, 2, · · ·N, a ≤ x ≤ b,
mi = y(a) + ih(
y(b)−y(a)b−a
),
S = y(b)− y(a).
G0 is the initial linear approximation of the nonlinear vector G(u). By solving
the linear system of equations (4.34) we get initial approximate solution.
97
Second step: Numerical solution
The numerical solutions U∗ of the nonlinear system are obtained by using sim-
ple iterative scheme
AU (m+1) = G(Um), m = 0, 1, 2, 3, · · · (4.35)
Third step: Stopping condition
The above iterative processes will terminate when the following condition is
satisfied
||Um − Um−1|| ≤ tol (4.36)
where tol is supposed value i.e. tol = 10−6. The convergence of the above it-
erative algorithm is guaranteed by the following propositions. The solutions of
linear system of equations (4.34) and (4.35) are obtained by Gaussian elimina-
tions method.
Proposition 4.3.3. The successive solutions {Um} for the nonlinear system (4.31) gen-
erated by the iterative algorithm (4.35) linearly converge to the solution U∗ provided
that the M0 and M1 are Lipschitz constants and step size h is small.
i.e.
∥∥A−1∥∥(M0h
3 +867307
212370M1h
2
)≤ 1. (4.37)
Proof. Let U∗ and U (m) be the solutions of the nonlinear system (4.31) then by
definition, for small h we have
AU∗ = G(U∗),
AU (m+1) = G(U (m)).
98
Let the error vector be defined as e(m) = Um−U∗ at mth iteration which satisfies
AU (m+1) − AU∗ = G(U (m))−G(U∗),
G(U (m+1) − U∗) = G(U (m))−G(U∗),
Ae(m+1) = G(Um)−G(U∗).
For i = 0, 1, 2, · · · , N ,
D3e(m+1)i = (F (U (m))− F (U∗))i,
by using the mean value theorem, which is stated as “If a function f(x, y, z) is
continuously differentiable in an open set of R3 containing points (x1, y1, z1) and
(x2, y2, z2) and the line segment connecting them, then an equation
f(x2, y2, z2)− f(x1, y1, z1) = f′
x(r, s, t)(x2 − x1) + f′
y(r, s, t)(y2 − y1)
+f′
z(r, s, t)(z2 − z1),
is valid for the interior point (a, b, c) of the segment.”, we have
D3e(m+1)i = f(xi, U
(m)i , U ′(m))− f(xi, U
(∗)i , U ′(∗)).
The above equation can be written as (by using mean value theorem)
D3e(m+1)i = f ∗
x(xi − xi) + f ∗y (U
(m)i − U
(∗)i ) + f ∗
y′(U′(m) − U ′(∗)),
by using the definition of error vector, we have
D3e(m+1)i = f ∗
y e(m)i + f ∗
y′e′(m)i ,
D3e(m+1)i = f ∗
y e(m)i + f ∗
y′D1e(m)i ,
where D1 and D3 are the derivative difference operators defined as
D1fi =1
2973180h[−5(fi+6 − fi−6) + 1024(fi+5 − fi−5) + 13225(fi+4 − fi−4)
−199680(fi+3 − fi−3) + 1141695(fi+2 − fi−2)− 4715520(fi+1 − fi−1)] ,
99
D3fi =1
1879920h3[−225(fi+6 − fi−6) + 11520(fi+5 − fi−5)− 10952(fi+4 − fi−4)
−476928(fi+3 − fi−3) + 3047987(fi+2 − fi−2)− 4677632(fi+1 − fi−1)] .
This implies
D3e(m+1)i = h3f ∗
y e(m)i + h2f ∗
y′D1e(m)i .
Since ei = eN−i = 0, i = 0,−1,−2, · · · ,−6, therefore we have
Ae(m+1)i = h3f ∗
y e(m)i + h2f ∗
y′D1e(m)i .
This can be written as
e(m+1)i = A−1(h3f ∗
y e(m)i + h2f ∗
y′D1e(m)i ).
By taking norm on both sides, we get
∥e(m+1)i ∥ = ∥A−1∥∥h3f ∗
y e(m)i + h2f ∗
y′D1e(m)i ∥.
By using the definition of Lipschitz condition, we get
∥e(m+1)i ∥ ≤ ∥A−1∥(h3M0∥e(m)
i ∥+ h2M1∥D1∥∥e(m)i ∥). (4.38)
This implies
∥e(m+1)i ∥
∥e(m)i ∥
≤ ∥A−1∥(h3M0 + h2M1∥D1∥
),
which is equivalent to
∥e(m+1)i ∥
∥e(m)i ∥
≈ h2M1∥A−1∥∥D1∥,
The results follows immediately from this inequality and the following fact
∥D1∥ =867307
212370.
A simple approximation of condition by omitting the cubic term is
h ≤(212370
867307M−1
1
∥∥A−1∥∥−1
) 12
.
This complete the proof.
100
Proposition 4.3.4. If f satisfies the Lipschitz condition (4.3) and Lipschitz constants
M0, M1 and M2 and mesh size h are small enough then nonlinear system (4.23) has
a unique solution U∗. A sufficient condition for the existence of a solution is given by
(4.37).
Proof. From the proof of previous proposition, we observe that if (4.37) holds,
than an inequality similar to that of (4.38) holds, which implies that the sequence
{Um} is contracting and hence converges. The limit U∗ of (4.35) also satisfies
(4.31) due to the continuity of the right hand side function f(x, y, y′, y′′).
Remark 4.3.1. The numerical complexity for the solution of the linear system
(4.35), where the matrix A is almost band matrix with half band 7, using Gaus-
sian elimination method is about 49(N + 13) multiplications. The number of
complexity depends upon the efficient boundary treatment. If more efficient
boundary treatment is constructed than number of complexity will be reduced.
4.4 Error analysis
From the approximation properties of the basis function g(x), it is shown that
the collocation method (4.11) with nonic precision treatments at end points has
at least the power of approximation O(h3). Here we present our main results for
error estimation. Proof of these results are similar to the proof of proposition in
Mustafa and Ejaz (2014).
Proposition 4.4.1. Suppose the exact solution y(x) ∈ C3[0, 1] and {ui} are obtained
by (4.38) then absolute error by interpolating collocation algorithm is
||err(x)||∞ = ||U (l)(x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.
where l denotes the order of derivative.
101
Proof. Since the order of approximation of subdivision scheme (4.2) is ten so
by direct calculation (third left eigenvector), we can find derivative of smooth
function y(x) as
y′′′(xj) =1
1879920h3 [−225{y(xj + 6h)− y(xj − 6h)}+ 11520{y(xj + 5h)− y(xj − 5h)}
−10952{y(xj + 4h) + y(xj − 4h)} − 476928{y(xj + 3h) + y(xj − 3h)}
+3047987{y(xj + 2h)− y(xj − 2h)} − 4677632{y(xj + h)− y(xj − h)}]
+O(h8).
This can be written as
y′′′(xj) =1
1879920h3[−225(yj+6 − yj−6) + 11520(yj+5 − yj−5)− 10952(yj+4 − yj−4)
−476928(yj+3 − yj−3) + 3047987(yj+2 − yj−2)− 4677632(yj+1 − yj−1)]
+O(h8). (4.39)
Similarly, we have
U ′′′(xj) =1
1879920h3[−225(uj+6 − uj−6) + 11520(uj+5 − uj−5)− 10952(uj+4 − uj−4)
−476928(uj+3 − uj−3) + 3047987(uj+2 − uj−2)− 4677632(uj+1 − uj−1)]
+O(h8). (4.40)
If we define error function e(x) = U(x)− y(x) and error vectors at the nodes by
e(xj) = U(xj)− y(xj + jh), −6 ≤ j ≤ N + 6,
or equivalently ej = Zj − yj, −6 ≤ j ≤ N + 6, then this impliese′j = U ′
j − y′j,
e′′j = U ′′j − y′′j ,
e′′′j = U ′′′j − y′′′j .
By subtracting (4.40) from (4.39), we get
102
y′′′
j − U′′′
j =1
1879920h3[−225{(yj+6 − yj−6)− (uj+6 − uj−6)}+ 11520{(yj+5
−yj−5)− (uj+5 − uj−5)} − 10952{(yj+4 − yj−4)− (uj+4 − uj−4)}
−476928{(yj+3 − yj−3)− (uj+3 − uj−3)}+ 3047987{(yj+2 − yj−2)
−(uj+2 − uj−2)} − 4677632{(yj+1 − yj−1)− (uj+1 − uj−1)}] +O(h8).
This implies
e′′′
j =1
1879920h3[−225(ej+6 − ej−6) + 11520(ej+5 − ej−5)− 10952(ej+4
−ej−4)− 476928(ej+3 − ej−3) + 3047987(ej+2 − ej−2)− 4677632(ej+1
−ej−1)] +O(h8). (4.41)
From (4.1), (4.11), (4.41) and by assuming the eighth order boundary treatments
at the end points, we have
e′′′
j = ajej + bje′
j, 0 ≤ i ≤ N (4.42)
and
ej =
max0≤k≤6
{|ek|}O(h8), −6 ≤ j ≤ 0
maxN−6≤k≤N
{|ek|}O(h8), N ≤ j ≤ N + 6(4.43)
where j = 0, 1, · · · , N ,
aj = fy(tj, y∗j , y
′∗j ), bj = fy′(tj, y
∗j , y
′∗j ),
and
y∗j = yj + θjej, y′∗j = y′j + θje
′
j, 0 ≤ θj ≤ 1.
Using the results (4.41) and
D1Ui =1
2973180h[−5(ui+6 − ui−6) + 1024(ui+5 − ui−5) + 13225(ui+4 − ui−4)
−199680(ui+3 − ui−3) + 1141695(ui+2 − ui−2)− 4715520(ui+1 − ui−1)] .
103
It can be conclude that relation (4.42) and (4.43) is equivalent to
(A+O(h6)−O(h3)−D1O(h))E = O(h8)∥E∥,
where E = (e−6, e−5, · · · , e5, e6).
Hence for small h, the coefficient matrix A+O(h2), will be invertible, thus us-
ing the standard result from algebra and effect of ∥A−1∥ , we have the following
estimate
∥E∥ ≤ ∥A−1∥1−O(h2)
O(h8) = O(h3).
This completes the result.
4.5 Examples, comparison and conclusion
In this section, we use subdivision based collocation algorithm to find the solu-
tion of some nonlinear third order boundary value problems. We present nu-
merical results in table format along with their graphical representations. We
also give comparison of the results obtained by our algorithm and the results
computed by existing algorithms. We end this section with precise conclusion.
4.5.1 Numerical examples
We find the approximate solutions of the following nonlinear problems to check
the accuracy and convergence of subdivision based iterative collocation algo-
rithm.
Example 4.5.1. The nonlinear boundary value problem
y′′′= −2e−3y + 4(1 + x)−3, (4.44)
with boundary conditions
y(0) = 0, y′(0) = 1, y(1) = ln(2).
104
Example 4.5.2. The nonlinear boundary value problem
y′′′(x) = e−xy2(x) (4.45)
subject to the boundary conditions
y(0) = 1, y′(0) = 1, y(1) = e.
Example 4.5.3. The nonlinear boundary value problem
y′′′(x) = −exy2(x) (4.46)
subject to the boundary conditions
y(0) = 1, y′(0) = −1, y(1) =1
e.
The exact solutions of the problems (4.44), (4.45) and (4.46) are y = ln(1 + x),
y = ex and y = e−x respectively.
Example 4.5.4. We consider the third order ordinary differential equation
y′′′ = y−k, k > 0, (4.47)
where k is constant. The initial conditions imposed by Tanner (1979) are
y(0) = 1, y′(0) = 0. (4.48)
The problem is closed by the boundary condition
y(r) = 0, (4.49)
and the problem become singular at y = 0. The boundary condition is imposed
Momonait (2011) as
y(r) = ϵ, (4.50)
105
where r is a constant satisfying r > 0. The analytic solution of (4.47) by Momon-
ait (2011)
y(x) =1
6(6− 7x2 + 6ϵx2 + x3). (4.51)
Case-1( Momonait (2011)): For k = 0 the problem (4.47) takes the form
y′′′ = y0 = 1, (4.52)
We solve (4.52) along with the conditions (4.48) and (4.50) by letting r = 1
then ϵ = 1112
and obtain the results which also support our algorithm. That is the
numerical results have the order of approximation O(h3). The numerical results
are tabulated in Table 4.4 and graphical representation of these results is shown
in Figure 4.4. These results are obtained after first iteration level. The maximum
absolute error is 6.1250× 10−3.
Case-2 (Momonait (2011)): For k = 12
the problem (4.47) takes the form
y′′′ = y−12 , (4.53)
with the boundary conditions y(0) = 1, y′(0) = 0 and y(r) = ϵ. The numerical
solution of (4.53) is obtained by using the proposed numerical algorithm. The
solution after third iteration level is presented in Table 4.5 and their graphical
representation is presented in Figure 4.5. The maximum absolute error for this
problem is 6.4004× 10−3.
Case-3 (Duffy and Wilson (1979) and Momonait (2011)): In this case, we nu-
merically solve the problem (4.47) for k = 2 i. e.
y′′′ = y−2, (4.54)
together with the boundary conditions y(0) = 1, y′(0) = 0 and y(r) = ϵ. Its
exact solution is given in (4.52). The numerical solution of (4.54) with given and
106
imposed condition at node points is tabulated in Table 4.6 and graphical repre-
sentation is given in Figure 4.6. The maximum absolute error for this problem
is 4.93276× 10−3.
4.5.2 Comparison and discussion
For Examples 4.5.1, 4.5.2 and 4.5.3, we use the iterative collocation algorithm
described in Section 4.3, for h = 10−1, 20−1, 50−1 (i-e. for N=10, 20, 50) and tol =
10−6 along with eighth order boundary treatment at end points, to get solutions
of nonlinear boundary value problems. The numerical results are obtained after
third iteration with condition (4.36).
• The numerical solutions of the problems (4.44), (4.45) and (4.46) are pre-
sented in Tables 4.1, 4.2 and 4.3 respectively.
• Caglar et al. (1999) solved the problem (4.44) by fourth degree B-spline
algorithm. The maximum absolute error obtained by the proposed algo-
rithm and by Caglar et al. (1999) are 4.77×10−3 and 5.80×10−2 respective-
ly. The graphical comparison between exact and approximate solutions is
shown in Figure 4.1. We observe that numerical results obtained by pro-
posed algorithm are better than the results of caglar et al. (1999).
• Hasan (2012) solved the problem (4.45) by modified Adomian decompo-
sition method (MADM). We also solve this problem by subdivision based
collocation algorithm. Here we observe that the order of approximation
by proposed and MADM algorithms is same (i.e.O(h3)). The graphical
comparison between exact and approximate solutions is shown in Figure
4.2.
• The comparison between exact and approximate solutions of problem (4.46)
is given in Figure 4.3.
107
In Example 4.5.4, we consider the problem related to thin film flows. We solve
(4.47) by assuming different values of k and we observer from the numerical
results tabulated in Table 4.4, 4.5 and 4.6 accuracy of the approximate solution
is O(h3). The numerical results are obtained after first and third iteration for
k = 0 and k = 12, 1 with condition (4.36) respectively.
The numerical solutions of Examples 1, 2 and 3 at different step sizes are shown
in Figure 4.7. From this figure, we see that step sizes have small effect on the
numerical solutions of BVPs.
4.5.3 Conclusion
In this chapter, we have presented subdivision based iterative collocation algo-
rithm for the solution of nonlinear third order boundary value problems. The
proposed algorithm has been applied on different nonlinear third order bound-
ary value problems. Numerical results show that the accuracy of approximate
solution is O(h3). We have also observed that the accuracy of the solution can
be improved by choosing different subdivision schemes with the proper adjust-
ment of boundary conditions. Our proposed algorithm gives better results com-
parative to the solution obtained by fourth degree B-spline Caglar et al. (1999).
The order of approximation by the proposed algorithm and modified Adomian
decomposition method Hasan (2012) is the same.
108
Table 4.1: Numerical results of Example 4.5.1: h = 10−1
xi Analytic Approximate Error by prop- Error by Cag-
solution yi solution Zi osed method lar et al. (1999)
0.0 0 0 0 0
0.1 0.0953101798 0.0957100706 0.0003998908 0.009
0.2 0.1823215568 0.1836519852 0.0013304284 0.013
0.3 0.2623642645 0.2648256813 0.0024614168 0.031
0.4 0.3364722366 0.3400108156 0.0035385790 0.045
0.5 0.4054651081 0.4098275640 0.0043624559 0.054
0.6 0.4700036292 0.4747771017 0.0047734725 0.058
0.7 0.5306282511 0.5352702927 0.0046420416 0.058
0.8 0.5877866649 0.5916474874 0.0038608225 0.053
0.9 0.6418538862 0.6441934825 0.0023395963 0.044
1.0 0.6931471806 0.6931471806 0 0.000
109
Table 4.2: Numerical results of Example 4.5.2: h = 10−1
xi Analytic Approximate Error by prop-
solution yi solution Zi osed method
0.0 1 1 0
0.1 1.105170918 1.1056664532 0.000495535
0.2 1.221402758 1.2232054699 0.001802712
0.3 1.349858808 1.3534912400 0.003632432
0.4 1.491824698 1.4974920669 0.005667369
0.5 1.648721271 1.6562805667 0.007559296
0.6 1.822118800 1.8310443828 0.008925583
0.7 2.013752707 2.0230974700 0.009344763
0.8 2.225540928 2.2338920118 0.008351084
0.9 2.459603111 2.4650310392 0.005427928
1.0 2.718281828 2.7182818285 0
110
Table 4.3: Numerical results of Example 4.5.3 : h = 10−1
xi Analytic Approximate Error by prop-
solution yi solution Zi osed method
0.0 1 1 0
0.1 0.9048374180 0.9045439989 0.0002934191
0.2 0.8187307531 0.8177120092 0.0010187439
0.3 0.7408182207 0.7388586762 0.0019595445
0.4 0.6703200460 0.6674013997 0.0029186463
0.5 0.6065306597 0.6028146540 0.0037160057
0.6 0.5488116361 0.5446245817 0.0041870544
0.7 0.4965853038 0.4924038346 0.0041814692
0.8 0.4493289641 0.4457666384 0.0035623257
0.9 0.4065696597 0.4043640577 0.0022056020
1.0 0.3678794412 0.3678794412 0
111
Table 4.4: Numerical results of Example 4.5.4 : k = 0 and h = 10−1
xi Analytic Approximate Error by prop-
solution yi solution Zi osed method
0.0 1 1 0
0.1 0.9976666667 0.9980416667 0.0003750000
0.2 0.9913333333 0.9926666667 0.0013333334
0.3 0.9820000000 0.9846250000 0.0026250000
0.4 0.9706666667 0.9746666667 0.0040000000
0.5 0.9583333333 0.9635416667 0.0052083334
0.6 0.9460000000 0.9520000000 0.0060000000
0.7 0.9346666667 0.9407916667 0.0061250000
0.8 0.9253333333 0.9306666667 0.0053333334
0.9 0.9190000000 0.9223750000 0.0033750000
1.0 0.9166666667 0.9166666667 0
112
Table 4.5: Numerical results of Example 4.5.4 : k = 12
and h = 10−1
xi Analytic Approximate Error by prop-
solution yi solution Zi osed method
0.0 1 1 0
0.1 0.9976666667 0.9980508018 0.0003841351
0.2 0.9913333333 0.9927028673 0.0013695340
0.3 0.9820000000 0.9847044600 0.0013695340
0.4 0.9706666667 0.9748013302 0.0041346635
0.5 0.9583333333 0.9637358441 0.0054025108
0.6 0.9460000000 0.9522463844 0.0062463844
0.7 0.9346666667 0.9410670255 0.0064003588
0.8 0.9253333333 0.9309274865 0.0055941532
0.9 0.9190000000 0.9225533666 0.0035533666
1.0 0.9166666667 0.9166666692 0
113
Table 4.6: Numerical results of Example 4.5.4 : k = 2 and h = 10−1
xi Analytic Approximate Error by prop-
solution yi solution Zi osed method
0.0 1 1 0
0.1 0.9976666667 0.9980024045 0.0003357378
0.2 0.9913333333 0.9925110306 0.0011776973
0.3 0.9820000000 0.9842830603 0.0022830603
0.4 0.9706666667 0.9740862436 0.0034195769
0.5 0.9583333333 0.9627028681 0.0043695348
0.6 0.9460000000 0.9509327630 0.0049327630
0.7 0.9346666667 0.9395952385 0.0049285718
0.8 0.9253333333 0.9295297470 0.0041964137
0.9 0.9190000000 0.9215950588 0.0025950588
1.0 0.9166666667 0.9166666667 0
114
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
Sol
utio
n
Figure 4.1: Comparison of the analytic and approximate solution of Example 4.5.1
by proposed algorithm and Caglar et al. (1999). In this figure solid line shows exact
solution, dotted lines show approximate solution by proposed algorithm and dashed
lines show the solution obtained by Caglar et al. (1999).
115
0 0.2 0.4 0.6 0.8 11
1.5
2
2.5
3
X
Sol
utio
n
Figure 4.2: Comparison of the analytic and approximate solution of Example 4.5.2 by
proposed algorithm and Hasan (2012). In this figure solid line shows exact solution,
dotted lines show approximate solution by proposed algorithm and dashed lines show
the solution obtained by Hasan (2012).
116
0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Sol
utio
n
Figure 4.3: Comparison of the analytic and approximate solution of Example 4.5.3 by
proposed algorithm. In this figure solid line shows exact solution and dashed lines show
approximate solution by proposed algorithm.
117
0 0.2 0.4 0.6 0.8 10.9
0.92
0.94
0.96
0.98
1
X
Sol
utio
n
Figure 4.4: Comparison between exact and approximate solutions of Example 4.5.4
for k = 0. Solid line represents exact solution and dash line represents approximate
solution.
118
0 0.2 0.4 0.6 0.8 10.9
0.92
0.94
0.96
0.98
1
X
Sol
utio
n
Figure 4.5: Comparison between exact and approximate solutions of Example 4.5.4
for k = 12. Solid line represents exact solution and dash line represents approximate
solution.
119
0 0.2 0.4 0.6 0.8 10.9
0.92
0.94
0.96
0.98
1
X
Sol
utio
n
Figure 4.6: Comparison between exact and approximate solutions of Example 4.5.4
for k = 2. Solid line represents exact solution and dash line represents approximate
solution.
120
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
Sol
utio
n
h=1/10h=1/20h=1/50
0 0.2 0.4 0.6 0.8 11
1.5
2
2.5
3
X
Sol
utio
n
h=1/10h=1/20h=1/50
0 0.2 0.4 0.6 0.8 1
0.4
0.5
0.6
0.7
0.8
0.9
1
X
Sol
utio
n
h=1/10h=1/20h=1/50
Figure 4.7: Approximate solutions of Examples 4.5.1, 4.5.2 and 4.5.3 at different step
sizes. 121
Chapter 5
A Numerical Approach Based on
Subdivision Schemes for Solving
Nonlinear Fourth Order Boundary
Value Problems
In this chapter, we present an iterative collocation algorithm based on interpo-
lating subdivision schemes for the solution of nonlinear fourth order boundary
value problems. Main purpose of this chapter is to explore and seek the ap-
plications of subdivision schemes in the field of physics and engineering. We
consider the following type of nonlinear boundary value problem
y(iv) = f(x, y, y′), (5.1)
with the boundary condition y(a) = α1, y′(a) = α2,
y(b) = α3, y′(b) = α4,(5.2)
where αi, i = 1, 2, 3, 4 are constants. We assume that the problem is well-posed.
In Section 5.1, some results about subdivision algorithms and basis function are
122
given. In Section 5.2, a numerical method to solve (5.1) using the refinable ba-
sis functions is formulated and its convergence properties are discussed. Error
properties are given in Section 5.3. Numerical examples illustrating the feasibil-
ity of our proposed algorithm are given in Section 5.4.
5.1 Basis functions and their derivatives
Some useful results for the solution of nonlinear boundary value problem are
discussed in this section. Introduction to the basis functions of subdivision
schemes that are used to construct the approximate solutions of (5.1) is also
part of this section.
5.1.1 Interpolating subdivision scheme
A mathematical formulation of binary subdivision scheme is defined asP k+12i =
∑j∈Z
a−2jPki+j
P k+12i+1 =
∑j∈Z
ai−2jPki+j
(5.3)
thus the scheme is a stepwise interpolatory scheme iff the coefficient ai satisfy
a2i = δi ∀ i ∈ Z. We consider the following binary interpolating subdivision
scheme by Deslauriers and Dubuc (1989), Qu (1994) and Deng and Ma (2013)
pk+12i = pki ,
pk+12i+1 =
3565536
(pki−4 + pki+5
)− 405
65536
(pki−3 + pki+4
)+ 567
16384
(pki−2 + pki+3
)− 2205
16384
(pki−3 + pki+4
)+ 19845
32768
(pki + pki+1
).
(5.4)
The scheme (5.4) is C4-continuous, having support length (−9, 9) and approxi-
mation order is ten.
123
5.1.2 Basis functions
The basis functions is the limit function resulting from cardinal data, where all
vertices of the polygon have value zero except for one. Let ϕ(x), x ∈ R be the
fundamental solution of (5.4) satisfies the two scale equation
ϕ(x) = ϕ(2x) +1
65536[39690{ϕ(2x− 1) + ϕ(2x+ 1)} − 8820{ϕ(2x− 3)
+ϕ(2x+ 3)}+ 2268{ϕ(2x− 5) + ϕ(2x+ 5)} − 405{ϕ(2x− 7)
+ϕ(2x+ 7)}+ 35{ϕ(2x− 9) + ϕ(2x+ 9)}] , x ∈ R (5.5)
and
ϕ(x) ∈ C4, ϕ(x) = 0, x ∈]− 8, 8[, ϕ(i) = δ0, i ∈ Z. (5.6)
Furthermore, it has the following derivatives, first derivatives of ϕ(i) at i ∈
[−8, 8] are
ϕ(i)(0) = 0, ϕ(i)(±1) = ∓19146219521159104017
,
ϕ(i)(±2) = ± 5304527961159104017
, ϕ(i)(±3) = ∓ 147046413780629
,
ϕ(i)(±4) = ± 172970691159104017
, ϕ(i)(±5) = ∓ 27729925795520085
,
ϕ(i)(±6) = ∓ 112763610431936153
, ϕ(i)(±7) = ∓ 40968113728119
,
ϕ(i)(±8) = ∓ 59272832136
,
(5.7)
second derivatives of ϕ(i) at i ∈ [−8, 8] are
ϕ(ii)(0) = −2370618501415309077185968
, ϕ(ii)(±1) = 3265310153216676106344305
,
ϕ(ii)(±2) = −878265102572676106344305
, ϕ(ii)(±3) = 7340630594562028319032915
,
ϕ(ii)(±4) = − 808839012771352212688610
, ϕ(ii)(±5) = 214899200135221268861
,
ϕ(ii)(±6) = 297875188405663806583
, ϕ(ii)(±7) = 6400019317324123
,
ϕ(ii)(±8) = 4375618154371936
,
(5.8)
124
third derivatives of ϕ(i) at i ∈ [−8, 8] are
ϕ(iii)(0) = 0, ϕ(iii)(±1) = ±4331751500815295995855
,
ϕ(iii)(±2) = ∓12153051235761183983420
, ϕ(iii)(±3) = ±240606976566518365
,
ϕ(iii)(±4) = ∓ 5285889107244735933680
, ϕ(iii)(±5) = ∓ 374141443059199171
,
ϕ(iii)(±6) = ± 1090169453214692
, ϕ(iii)(±7) = ∓ 21760437028453
,
ϕ(iii)(±8) = ∓ 297513984910496
,
(5.9)
fourth derivatives of ϕ(i) at i ∈ [−8, 8] are
ϕ(iv)(0) = 33869667457408
, ϕ(iv)(±1) = −529505475289730585
,
ϕ(iv)(±2) = 10404741119358922340
, ϕ(iv)(±3) = −748795849970065
,
ϕ(iv)(±4) = 2950208692871378720
, ϕ(iv)(±5) = 923862417946117
,
ϕ(iv)(±6) = − 9001877976052
, ϕ(iv)(±7) = 7184017946117
,
ϕ(iv)(±8) = 11225328157568
.
(5.10)
The above derivative values are found by using the left eigenvectors of the sub-
division process (5.4). The detailed description about these left eigenvectors
and derivatives can be found in Qu (1996) and Mustafa and Ejaz (2014). The
graphical representations of above derivatives are given in Figure 5.1
5.2 Description of iterative numerical algorithm
This section describes the method for the numerical solution of nonlinear bound-
ary value problem (5.1). The detail of the method is given below:
5.2.1 The collocation algorithm
In this subsection, the collocation method is constructed based on the interpo-
lating subdivision scheme (5.4). Our numerical approach for nonlinear fourth
order boundary value problem using collocation method based on subdivision
125
Figure 5.1: Graphical representation of basis functions is shown in figure (a), and first,
second, third and fourth derivatives of basis function are shown in figure (b), (c), (d)
and (e). 126
scheme is to seek an approximate solution as
Z(x) =N+8∑i=−8
ziϕ
(x− xi
h
), 0 6 x 6 1, (5.11)
where N is the positive integer N > 8, h = 1/N and xi = i/N = ih, and {zi}
are the unknown to be determined for the solution of (5.1). In order to solve
the problem, a collocation method Z(x) is considered to be the solution of the
above differential equation at x = xj and we substitute equation (5.11) into the
equation (5.1). This leads to
Z(iv)(xj) = f(xj, Z(xj), Z′(xj)), j = 0, 1, 2, · · · , N, (5.12)
and boundary conditions
Z(0) = α1, Z′(0) = α2, Z(N) = α3, Z
′(N) = α4, (5.13)
From (5.11), we get
Z(iv)(x) =1
h4
N+8∑i=−8
ziϕ(iv)
(x− xi
h
), 0 6 x 6 1, (5.14)
substituting (5.14) into (5.12), we obtain
N+8∑i=−8
ziϕ(iv)
(xj − xi
h
)= h4f(xj, Z(xj), Z
′(xj)), j = 0, 1, 2, · · · , N.
This can be written asN+8∑i=−8
ziϕ(iv)j−i = h4f(xj, Z(xj), Z
′(xj)), j = 0, 1, 2, · · · , N.
Since ϕ(iv)i = ϕ
(iv)−i , the above system of equations become
N+8∑i=−8
ziϕ(iv)i−j = h4f(xj, Z(xj), Z
′(xj)), j = 0, 1, 2, · · · , N. (5.15)
The nonlinear system of equations (5.15) can be simply in following Theorems
5.2.1 and 5.2.2.
127
Theorem 5.2.1. The nonlinear system of equations (5.15) for j = 0 becomes
8∑i=−8
ziϕ(iv)i = h4f(x0, Z(x0), Z
′(x0)). (5.16)
Proof. By expanding (5.15) for j = 0, we get
N+8∑i=−8
ziϕ(iv)i = h4f(x0, Z(x0), Z
′(x0)),
z−8ϕiv−8 + z−7ϕ
iv−7 + z−6ϕ
iv−6 + · · ·+ z7ϕ
iv7 + z8ϕ
iv8 + z9ϕ
iv9 + · · ·+ zN+7ϕ
ivN+7
+zN+8ϕivN+8 = h4f(x0, Z(x0), Z
′(x0)).
Since ϕiv(i) exist only for the interval for i ∈ [−8, 8] and outside the interval it
will be zero. Then above equation can be written as
z−8ϕiv−8 + z−7ϕ
iv−7 + z−6ϕ
iv−6 + · · ·+ z7ϕ
iv7 + z8ϕ
iv8 = h4f(x0, Z(x0), Z
′(x0)).
Theorem 5.2.2. For j = 1, 2, · · · , N , the nonlinear system of equations (5.15) becomes
8+j∑i=−8+j
ziϕ(iv)i−j = h4f(xj, Z(xj), Z
′(xj)). (5.17)
Proof. By expanding (5.15), for j = 1, 2, 3, · · · , N, we get
z−8ϕiv−8−j + z−7ϕ
iv−7−j + z−6ϕ
iv−6−j + · · ·+ z7ϕ
iv7−j + z8ϕ
iv8−j + z9ϕ
iv9−j + z10ϕ
iv10−j
+ · · ·+ zN+6ϕivN+6−j + zN+7ϕ
ivN+7−j + zN+8ϕ
ivN+8−j = h4f(xj, Z(xj), Z
′(xj)).
(5.18)
Substituting j = 1 in (5.18), it becomes
z−8ϕiv−8−1 + z−7ϕ
iv−7−1 + z−6ϕ
iv−6−1 + · · ·+ z7ϕ
iv7−1 + z8ϕ
iv8−1 + z9ϕ
iv9−1 + z10ϕ
iv10−1
+ · · ·+ zN+6ϕivN+6−1 + zN+7ϕ
ivN+7−1 + zN+8ϕ
ivN+8−1 = h4f(x1, Z(x1), Z
′(x1)).
128
This implies
z−8ϕiv−9 + z−7ϕ
iv−8 + z−6ϕ
iv−7 + · · ·+ z7ϕ
iv6 + z8ϕ
iv7 + z9ϕ
iv8 + z10ϕ
iv9 + · · ·
+zN+6ϕivN+5 + zN+7ϕ
ivN+6 + zN+8ϕ
ivN+7 = h4f(x1, Z(x1), Z
′(x1)).
Since ϕiv(i) is non-zero only for the interval for i ∈ [−8, 8] and outside the inter-
val it will be zero. Then above equation becomes
z−7ϕiv−8 + z−6ϕ
iv−7 + · · ·+ z7ϕ
iv6 + z8ϕ
iv7 + z9ϕ
iv8 = h4f(x1, Z(x1), Z
′(x1)). (5.19)
For j=2, (5.18) becomes
z−8ϕiv−8−2 + z−7ϕ
iv−7−2 + z−6ϕ
iv−6−2 + · · ·+ z7ϕ
iv7−2 + z8ϕ
iv8−2 + z9ϕ
iv9−2 + z10ϕ
iv10−2
+ · · ·+ zN+6ϕivN+6−2 + zN+7ϕ
ivN+7−2 + zN+8ϕ
ivN+8−2 = h4f(x2, Z(x2), Z
′(x2)).
This implies
z−8ϕiv−10 + z−7ϕ
iv−9 + z−6ϕ
iv−8 + · · ·+ z7ϕ
iv5 + z8ϕ
iv6 + z9ϕ
iv7 + z10ϕ
iv8
+ · · ·+ zN+6ϕivN+4 + zN+7ϕ
ivN+5 + zN+8ϕ
ivN+6 = h4f(x2, Z(x2), Z
′(x2)).
By using the definition of ϕivi given in (5.10), above equation yields
z−6ϕiv−8 + z−5ϕ
iv−7 + · · ·+ z7ϕ
iv5 + z8ϕ
iv6 + z9ϕ
iv7 + z10ϕ
iv8 = h4f(x2, Z(x2), Z
′(x2)).
(5.20)
By using the similar pattern for j = 1, 2, we can find the expression for j =
3, 4, · · ·N ,
z−8+jϕiv−8−j + z−7+jϕ
iv−7−j + z−6+jϕ
iv−6−j + · · ·+ z7+jϕ
iv7−j + z8+jϕ
iv8−j + z9+jϕ
iv9−j
+ · · ·+ zN+6+jϕivN+6−j + zN+7+jϕ
ivN+7−j + zN+8+jϕ
ivN+8−j = h4f(xj, Z(xj), Z
′(xj)).
(5.21)
129
The nonlinear system of equation (5.15) is equivalent to the following non-
linear system of N + 1 equations with (N+17) unknowns {zi}
AZ = F (z) (5.22)
where A is banded matrix of order (N + 1)× (N + 17), Z is the unknown vector
of order N + 17 and F (z) is the vector of order N + 1 depends on z. The matrix
A, vectors Z and F (z) are given explicitly by
A = [ϕivpq(q − p− 8)](N+1)×(N+17), (5.23)
where p = 1, 2, 3 · · · , N + 1 and q = 1, 2, 3, · · · , N + 17 represent the row and
column respectively.
F (z) =(h4f(x0, Z(x0), Z
′(x0)), · · · , h4f(xN , Z(xN), Z
′(xN))
)T
, (5.24)
Z = (z−8, z−7, z−6, · · · , zN+6, zN+7, zN+8)T , (5.25)
Z′(xj) =
N+8∑i=−8
zjϕ′(xj − xi
h
),
where ϕ′(i) is already defined in (5.7) with ϕ(i) = ϕi.
5.2.2 Boundary conditions at end points
For unique solution of nonlinear systems (5.15), we need sixteen more condi-
tions. Four conditions can be attained from given boundary conditions for non-
linear systems of equations and remaining conditions are attained by setting
some extrapolation method. The detail of the given boundary conditions and
extrapolation method are given below:
5.2.3 Boundary conditions
The given boundary conditions are :
Z(0) = α1, Z′(0) = α2, Z(N) = α3, Z
′(N) = α4.
130
The approximation of derivative conditions at ends point is defined as
Z ′(0) =
(N
2520
){−7381z0 + 25200z1 − 56700z2 + 100800z3 − 132300z4 + 127008z5
−88200z6 + 43200z7 − 14175z8 + 28800z9 − 252z10}+O(h10), (5.26)
Z ′(N) =
(N
2520
){7381zN − 25200zN−1 + 56700zN−2 − 100800zN−3 + 132300zN−4
−127008zN−5 + 88200zN−6 − 43200zN−7 + 14175zN−8 − 28800zN−9
+252zN−10}+O(h10). (5.27)
5.2.4 Extrapolation method
The remaining twelve conditions for the nonlinear systems (5.15) to obtain sta-
ble systems for the solution of (5.1) are obtained by setting the following extrap-
olation method.
We define six conditions at left end points and six conditions at the right end
points. Since subdivision scheme (5.4) reproduces nine degree (i.e. tenth or-
der) polynomials, so we define boundary conditions of order ten for solution of
(5.15). For simplicity only left end points z−7, z−6, z−5, z−4, z−3, z−2 are discussed
and the values of right end points zN+2, zN+3, zN+4, zN+5, zN+6, zN+7 can be treat-
ed similarly.
The values z−7, z−6, z−5, z−4, z−3, z−2 can be determined by the polynomial q(x)
interpolating (xi, zi), 2 ≤ i ≤ 7. Precisely, we have
z−i = q(−xi), i = 2, 3, 4, 5, 6, 7,
where
q(xi) =10∑j=1
10
j
(−1)j+1Z(xi−j).
From (5.13), Z1(xi) = zi for i = 2, 3, 4, 5, 6, 7 and replacing xi by −xi, we have
q(−xi) =10∑j=1
10
j
(−1)j+1z−i+j.
131
Hence the following boundary conditions can be employed at the left end
10∑j=0
10
j
(−1)jz−i+j = 0, i = 7, 6, 5, 4, 3, 2. (5.28)
Similarly for the right end, we can define zi = q(−xi), i = N + 2, N + 3, N + 4,
N + 5, N + 6, N + 7 and
q(xi) =10∑j=1
10
j
(−1)j+1zi−j.
So we have the following boundary conditions at the right end
10∑j=0
10
j
(−1)jzi−j = 0, i = N + 2, N + 3, N + 4, N + 5, N + 6, N + 7.
(5.29)
Finally, we obtain a new system of (N + 17) linear equations with (N + 17)
unknowns {zi}. The N + 1 equations are obtained from (5.15), four equations
from boundary conditions (5.13) and twelve from boundary conditions (5.28)
and (5.29).
Hence the stable nonlinear system of equations is define as
BZ = R(z), (5.30)
where the matrix B is given by
B = (CT0 , A
T , CT1 )
T , (5.31)
A is defined in (5.23), C0, C1 and the vector R(z) is defined as
132
C0 =
0 1 −10 45 −120 210 −252 210 −120 45 −10 1
0 0 1 −10 45 −120 210 −252 210 −120 45 −10
0 0 0 1 −10 45 −120 210 −252 210 −120 45
0 0 0 0 1 −10 45 −120 210 −252 210 −120
0 0 0 0 0 1 −10 45 −120 210 −252 210
0 0 0 0 0 0 1 −10 45 −120 210 −252
0 0 0 0 0 0 0 0 7381N2520
25200N2520 −56700N
2520100800N
2520
0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 · · · 0 0
1 0 0 0 0 0 0 · · · 0 0
−10 1 0 0 0 0 0 · · · 0 0
45 −10 1 0 0 0 0 · · · 0 0
−120 45 −10 1 0 0 0 · · · 0 0
210 −120 45 −10 1 0 0 · · · 0 0
− 132300N2520
127008N2520 − 88200N
252043200N2520 −14175N
25202800N2520 −252N
2520 · · · 0 0
0 0 0 0 0 0 0 · · · 0 0
,(5.32)
the first six rows of C0 are obtained from (5.28), second last row is obtained
from (5.26) and last row is taken from given boundary conditions Z1(0) which
is defined in (5.13) and
C1 =
0 0 · · · 0 0 0 0 0 0 0 0 0 0
0 0 · · · N10 −10N
945N8 − 120N
7 35N − 252N5
105N2 −40N 45N
2 −10N
0 0 · · · 0 0 1 −10 45 −120 210 −252 210 −120
0 0 · · · 0 0 0 1 −10 45 −120 210 −252 210
0 0 · · · 0 0 0 0 1 −10 45 −120 210 −252
0 0 · · · 0 0 0 0 0 1 −10 45 −120 210
0 0 · · · 0 0 0 0 0 0 1 −10 45 −120
0 0 · · · 0 0 0 0 0 0 0 1 −10 45
133
1 0 0 0 0 0 0 0 0
7381N2520 0 0 0 0 0 0 0 0
45 −10 1 0 0 0 0 0 0
−120 45 −10 1 0 0 0 0 0
210 −120 45 −10 1 0 0 0 0
−252 210 −120 45 −10 1 0 0 0
−120 210 −252 210 −120 45 −10 1 0
, (5.33)
first row of C1 is obtained from Z1(N) which is defined in (5.13), second row is
obtained from (5.27) and the last six rows are obtained from (5.29), Z which is
defined in (5.25) and R(z) is defined as
R(z) = (0, 0, 0, 0, 0, 0, Z ′(0), Z(1), F T (z), Z(1), Z ′(1), 0, 0, 0, 0, 0, 0)T , (5.34)
where F (z) is defined by (5.24).
5.2.5 Non-singularity of a matrix
We can check the non-singularity of coefficient matrix B defined in (5.31) by
different methods. We observe that the determinant of matrix B is non-zero for
N ≤ 500. Hence the nonlinear system of equations have a solution for N ≤
500. We also check the non singularity of matrix by finding eigenvalues up to
N ≤ 500 and we observe that all the eigenvalues are non-zero. Hence by Strang
(2011) we conclude that the B is non-singular. For large N > 500 the matrix may
or may not be singular.
5.2.6 Iterative algorithm and its convergence
An iterative algorithm and its convergence are described in this section.
134
5.2.7 Iterative algorithm based on basis function
The iterative algorithm based on basis function of the subdivision scheme (5.4)
are as defined in the following three steps.
First step: Initial approximation
The initial approximation is important because numerical solution depends on
the initial approximation. We define the process for finding the initial approxi-
mation as follows:
Let initial approximate solution Z0 be the solution of the following linear system
BZ0 = F 0, (5.35)
where
F 0 = (0, 0, 0, 0, 0, 0, y′(a), y(a), f0, f1, f2, · · · , fN , y(b), y′(b), 0, 0, 0, 0, 0, 0)T ,
fi = h4f(xi, Li, D), i = 0, 1, 2, · · ·N
Li = y(0) + ih(
y(b)−y(a)b−a
),
D = y(b)− y(a).
(5.36)
F 0 is the initial linear approximation of the nonlinear vector R(z).
Second step: Numerical solution
The numerical solutions Z∗ of the nonlinear system are obtained by using sim-
ple iterative scheme
BZ(m+1) = R(Zm), m = 0, 1, 2, 3, · · · (5.37)
135
Third step: Stopping condition
The above iterative processes will terminate when the following condition sat-
isfied
||z(m) − z(m−1)|| ≤ tol. (5.38)
where tolerance is supposed value i.e. tol = 10−6. The convergence of the above
iterative algorithm is guaranteed by the following proposition.
Proposition 5.2.3. The successive solutions {Z(m)} generated by the iterative algorith-
m (5.37) linearly converges to the solution Z∗ of the nonlinear solution of the system
(5.30) provided that the M0 and M1 are Lipschitz constants and step size h is small.
i.e.
∥∥B−1∥∥ ≤
(M0h
4 +4994220330463
1460471061420M1h
3
). (5.39)
Proof. Let Z∗ and Z(m) be the solutions of the nonlinear system (5.30). Then by
definition, for small h we have
BZ∗ = R(Z∗), (5.40)
BZm+1 = R(Zm). (5.41)
Let the error vector is defined as e(m) = Z(m)−Z∗ at mth iteration which satisfies
BZ(m+1) −BZ∗ = R(Z(m))−R(Z∗),
B(Z(m+1) − Z∗) = R(Z(m))−R(Z∗),
Be(m+1) = R(Z(m))−R(Z∗). (5.42)
For i = 0, 1, 2, · · · , N ,
D4e(m+1)i = (F (Z(m))− F (Z∗))i.
136
By mean value theorem, which is stated as “If a function f(x, y, z) is continuous-
ly differentiable in an open set of R3 containing points (x1, y1, z1) and (x2, y2, z2)
and the line segment connecting them, then an equation
f(x2, y2, z2)− f(x1, y1, z1) =
f′
x(r, s, t)(x2 − x1) + f′
y(r, s, t)(y2 − y1) + f′
z(r, s, t)(z2 − z1),
is valid for the interior point (a, b, c) of the segment.”, we have
D4e(m+1)i = f(xi, Z
(m)i , Z ′(m))− f(xi, Z
(∗)i , Z ′(∗)).
The above equation can be written as (by using mean value theorem)
D4e(m+1)i = f ∗
x(xi − xi) + f ∗y (Z
(m)i − Z
(∗)i ) + f ∗
y′(Z′(m) − Z ′(∗)),
by using the definition of error vector, we have
D4e(m+1)i = f ∗
y e(m) + f ∗
y′e′(m),
D4e(m+1)i = f ∗
y e(m) + f ∗
y′D1e(m),
where D1 and D4 are the derivative difference operators defined as
D1fi =1
2920942122840h[1575(fi−8 − fi+8) + 1474560(fi−7 − fi+7)
+315738080(fi−6 − fi+6) + 1397587968(fi−5 − fi+5)
−43588613880(fi−4 − fi+4) + 311679549440(fi−3 − fi+3)
−1336741045920(fi−2 − fi+2) + 4824847319040(fi−1 − fi+1)] ,
D4fi =1
183768238080h4[392875(fi+8 − fi−8) + 45977600(fi+7 − fi−7)
−1296269280(fi+6 − fi−6) + 5912719360(fi+5 − fi−5)
+1180083476(fi+4 − fi−4)− 86261280768(fi+3 − fi−3)
+332951715808(fi+2 − fi−2)− 677767008256(fi+1 − fi−1)
+850467338370fi] .
137
This implies
D4e(m+1)i = h4f ∗
y e(m) + h3f ∗
y′D1e(m).
Since ei = eN−i = 0, i = 0,−1,−2, · · · ,−8, we have
Be(m+1)i = h4f ∗
y e(m) + h3f ∗
y′D1e(m).
This can be written as
e(m+1)i = B−1(h4f ∗
y e(m) + h3f ∗
y′D1e(m)).
By taking norm on both sides, we get
∥e(m+1)i ∥ = ∥B−1(h4f ∗
y e(m) + h3f ∗
y′D1e(m))∥.
This implies
∥e(m+1)i ∥ = ∥B−1∥∥(h4f ∗
y e(m) + h3f ∗
y′D1e(m))∥.
By using the definition of Lipschitz condition, we get
∥e(m+1)∥ ≤ h4M0(b− a)∥B−1∥∥e(m)∥+ h3M1∥D1∥∥e(m)∥.
This implies
∥e(m+1)i ∥
∥e(m)∥≤ ∥B−1∥
(h4M0(b− a) + h3M1∥D1∥
),
which is equivalent to
∥e(m+1)i ∥
∥e(m)∥≈ h3M1∥B−1∥∥D1∥ ≤ hM1∥B−1∥∥D1∥,
i-e
∥e(m+1)i ∥
∥e(m)∥≈ hM1∥B−1∥∥D1∥.
The results follows immediately from this inequality and the following fact
∥D1∥ =4994220330463
1460471061420.
A simple approximation of condition by omitting the quatric term is
h ≤ 1460471061420
4994220330463M−1
1
∥∥B−1∥∥−1
.
This complete the proof.
138
5.3 Error estimation
From the approximation properties of the basis function ϕ(x), it is shown that
the collocation method (5.11) with nonic precision treatments at end points has
at least the power of approximation O(h3). Here we present our main results for
error estimation. Proof of these results are similar to the proof of proposition by
Qu and Agarwal (1996) and Mustafa and Ejaz (2014).
Proposition 5.3.1. Suppose the exact solution y(x) ∈ C4[0, 1] and {zi} are obtained
by (5.30) then absolute error by interpolating collocation algorithm is
||err(x)||∞ = ||Z(l)(x)− y(l)(x)||∞ = O(h3−l), l = 0, 1, 2, 3.
where l denotes the order of derivative.
Proof. Since the order of approximation of subdivision scheme (5.4) is ten so
by direct calculation (fourth left eigenvector), we can find derivative of smooth
function y(x) as
yiv(xj) =24
183768238080h4{392875y(xj − 8h) + 45977600y(xj − 7h)
−1296269280y(xj − 6h) + 5912719360y(xj − 5h) + 1180083476y(xj − 4h)
−86261280786y(xj − 3h) + 332951715808y(xj − 2h)− 677767008256y(xj − h)
+850467338370y(xj)− 677767008256y(xj + h) + 332951715808y(xj + 2h)
−86261280786y(xj + 3h) + 1180083476y(xj + 4h) + 5912719360y(xj + 5h)
−1296269280y(xj + 6h) + 45977600y(xj + 7h) + 392875y(xj + 8h)}+O(h10).
This can be written as
yivj =24
183768238080h4{392875yj−8 + 45977600yj−7 − 1296269280yj−6
+5912719360yj−5 + 1180083476yj−4 − 86261280786yj−3 + 332951715808yj−2
−677767008256yj−1 + 850467338370yj − 677767008256yj+1 + 332951715808yj+2
−86261280786yj+3 + 1180083476yj+4 + 5912719360yj+5 − 1296269280yj+6
+45977600yj+7 + 392875yj+8}+O(h10). (5.43)
139
Similarly, we have
Zivj =
24
183768238080h4{392875zj−8 + 45977600zj−7 − 1296269280zj−6
+5912719360zj−5 + 1180083476zj−4 − 86261280786zj−3 + 332951715808zj−2
−677767008256zj−1 + 850467338370zj − 677767008256zj+1 + 332951715808zj+2
−86261280786zj+3 + 1180083476zj+4 + 5912719360zj+5 − 1296269280zj+6
+45977600zj+7 + 392875zj+8}+O(h10). (5.44)
If we define error function e(x) = Z(x)− y(x) and error vectors at the nodes by
e(xj) = Z(xj)− y(xj + jh), −8 ≤ j ≤ N + 8,
or equivalently ej = Zj − yj, −8 ≤ j ≤ N + 8, then this implies
e′j = Z ′j − y′j,
e′′j = Z ′′j − y′′j ,
e′′′j = Z ′′′j − y′′′j
eivj = Zivj − yivj .
(5.45)
By subtracting (5.44) from (5.43), we get
yivj − Zivj = 24
183768238080h4 {392875(yj−8 − zj−8) + 45977600(yj−7 − zj−7)
−1296269280(yj−6 − zj−6) + 5912719360(yj−5 − zj−5) + 1180083476(yj−4 − zj−4)
−86261280786(yj−3 − zj−3) + 332951715808(yj−2 − zj−2)− 677767008256(yj−1 − zj−1)
+850467338370(yj − zj)− 677767008256(yj+1 − zj+1) + 332951715808(yj+2 − zj+2)
−86261280786(yj+3 − zj+3) + 1180083476(yj+4 − zj+4) + 5912719360(yj+5 − zj+5)
−1296269280(yj+6 − zj+6) + 45977600(yj+7 − zj+7) + 392875(yj+8 − zj+8)}+O(h10).
This implies
eivj =24
183768238080h4{392875ej−8 + 45977600ej−7 − 1296269280ej−6
+5912719360ej−5 + 1180083476ej−4 − 86261280786ej−3 + 332951715808ej−2
−677767008256ej−1 + 850467338370ej − 677767008256ej+1 + 332951715808ej+2
−86261280786ej+3 + 1180083476ej+4 + 5912719360ej+5 − 1296269280ej+6
+45977600ej+7 + 392875ej+8}+O(h10). (5.46)
140
From (5.1), (5.11), (5.45) and by assuming the tenth order boundary treatments
at the end points, we have
eivj = ajej + bje′
j, 0 ≤ i ≤ N (5.47)
and
ej =
max0≤k≤7
{|ek|}O(h10), −8 ≤ i ≤ 0
maxN−3≤k≤N
{|ek|}O(h10), N ≤ i ≤ N + 8(5.48)
where j = 0, 1, · · ·N
aj = fy(tj, y∗j , y
′∗j ), bj = fy′(tj, y
∗j , y
′∗j ),
and
y∗j = yj + θjej, y′∗j = y′j + θje
′
j, 0 ≤ θj ≤ 1.
Using the results (5.46) and
[1575(zi−8 − zi+8) + 1474560(zi−7 − zi+7) + 315738080(zi−6 − zi+6) + 1397587968
(zi−5 − zi+5)− 43588613880(zi−4 − zi+4) + 311679549440(zi−3 − zi+3)− 1336741045920
(zi−2 − zi+2) + 4824847319040(zi−1 − zi+1)] = 2920942122840hZ′+O(h10),
it can be conclude that relation (5.47) and (5.48) is equivalent to
(B +O(h8)−O(h4)−D1O(h3))E = O(h10)∥E∥,
where E = (e−8, e−7, · · · , e7, e8).
Hence for small h, the coefficient matrix B +O(h), will be invertible, thus us-
ing the standard result from algebra and effect of ∥B−1∥ , we have the following
estimate
∥E∥ ≤ ∥B−1∥1−O(h)
O(h10) = O(h3).
This completes the result.
141
5.4 Results and discussions
In this section, we test the proposed method on some nonlinear problems. Nu-
merical results for each problems are presented in the tables. These values are
very close to the corresponding true solutions and the values of the correspond-
ing errors are also given in the table.
Example 5.4.1. ( Agarwal (1986)) Consider the following nonlinear BVP
yiv − 6e(−4y) = −12(1 + x)−4, (5.49)
with boundary conditions
y(0) = 0, y′(0) = 1, y(1) = ln(2) = y′(1) = 0.5.
The exact solution of the problem (5.49) is y = ln(1 + x). Using the collocation
method described in Section 5.2 for N = 10, h = 10−1 and tol = 10−6 with tenth
order boundary treatment at end points. The numerical results are obtained
after third iteration with the condition (5.38). The obtained numerical results for
this problem are presented in Table 5.1. The maximum absolute error obtained
by the proposed method is 1.78×10−3. The graphical comparison between exact
and approximate solutions is shown in Figure 5.2.
Example 5.4.2. (Agarwal (1986)) Consider the nonlinear BVP
y(iv) = y2 − x10 + 4x9 − 4x8 − 4x7 + 8x6 − 4x4 + 120x− 48 (5.50)
subject to the boundary conditions
y(0) = y′(0) = 0, y(1) = y′(1) = 1.
Using the collocation method described in Section 5.2 for N = 10, h = 10−1 and
tol = 10−6 with tenth order boundary treatment at end points. The numerical
results are obtained after third iteration with the condition (5.38). The obtained
142
Table 5.1: Numerical results of Example 5.4.1
xi Analytic Approximate Error
solution Yi solution Zi = ||Yi − Zi||∞
0.0 0 0 0
0.1 0.0953101798 0.0950147533 0.0002954265
0.2 0.1823215568 0.1814496227 0.0008719341
0.3 0.2623642645 0.2609546573 0.0014096072
0.4 0.3364722366 0.3347370220 0.0017352146
0.5 0.4054651081 0.4036840381 0.0017810699
0.6 0.4700036292 0.4684459279 0.0015577013
0.7 0.5306282511 0.5294932609 0.0011349902
0.8 0.5877866649 0.5871580370 0.0006286279
0.9 0.6418538862 0.6416636708 0.0001902154
1.0 0.6931471806 0.6931471806 0
143
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
Sol
utio
n
Analytic solution YApproximate solution Z
Figure 5.2: Comparison of the analytic and approximate solution of Example 5.4.1.
numerical results for this problem are presented in Table 5.2. The maximum
absolute error obtained by the proposed method is 1.73 × 10−2. The graphical
comparison between exact and approximate solutions is shown in Figure 5.3.
144
Table 5.2: Numerical results of Example 5.4.2
xi Analytic Approximate Error
solution Yi solution Zi = ||Yi − Zi||∞
0.0 0 0 0
0.1 0.01981 0.0202195 0.0004095
0.2 0.07712 0.0796952 0.0025752
0.3 0.16623 0.1728732 0.0066432
0.4 0.27904 0.2905995 0.0115595
0.5 0.40625 0.4219208 0.0156708
0.6 0.53856 0.5558846 0.0173246
0.7 0.66787 0.6833406 0.0154706
0.8 0.78848 0.7987412 0.0102612
0.9 0.89829 0.9019417 0.0036517
1.0 1.00000 1.0000000 0
145
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
X
Sol
utio
n
Analytic solution YApproximate solution Z
Figure 5.3: Comparison of the analytic and approximate solution of Example 5.4.2.
5.5 Conclusion
This study has presented a numerical approach based on subdivision based col-
location algorithm for solving the numerical solution of nonlinear fourth order
boundary value problems. The proposed iterative algorithm has been applied
on different nonlinear fourth order boundary value problems. Numerical re-
sults show that the accuracy of approximate solution is O(h3). We have also
observed that the accuracy of the solution can be improved by choosing differ-
ent subdivision schemes with the proper adjustment of boundary conditions.
146
Chapter 6
Subdivision Based Collocation
Algorithm for Singularly Perturbed
Boundary Value Problems
This chapter investigates the approximate solutions of singularly perturbed sec-
ond order boundary value problems using binary interpolating subdivision sch-
eme based collocation algorithm. These type of problems frequently occur in
various field of science and engineering. We consider a second order singularly
perturbed boundary value problem
εy′′(x) = A(x)y
′(x) +B(x)y(x) + F (x), (6.1)
y(a) = α0, y(b) = α1 , a 6 x 6 b (6.2)
where A(x), B(x), F (x) are smooth bounded real functions and ε is a parameter
such that 0 < ε ≪ 1. Generally, the inhomogeneities F and α0, α1 may depends
on ε as well. It is well known that boundary value problem (6.1) with condition
(6.2) shows boundary layers at one or both ends of the interval depending on
the choice of the function A(x) detail is given in Ascher et al. (1988).
147
In this chapter, we are using subdivision based technique for the solution of
second order singularly perturbed boundary value problems. We develop sub-
division collocation methods based on following 6-point binary interpolating
C2 scheme by Lee et al. (2006)pk+12i = pki ,
pk+12i+1 = ω
(pki−2 + pki+3
)−
(3ω + 1
16
)(pki−1 + pki+2) +
(2ω + 9
16
)(pki + pki+1),
(6.3)
where 0 < ω < 0.042, for the solution of (6.1). The scheme (6.3) has support
length (−5, 5), approximation order four and satisfies following two scale rela-
tion
Ψ(x) = Ψ(2x) +
[ω {Ψ(2x− 1) + Ψ(2x+ 1)} −
(3ω +
1
16
){Ψ(2x− 3) + Ψ(2x
+3)}+(2ω +
9
16
){Ψ(2x− 5) + Ψ(2x+ 5)}
], x ∈ R (6.4)
where
Ψ(x) =
1 for x = 0,
0 for x = 0.(6.5)
This chapter is arranged in following way: In Section 6.1, we discuss derivatives
of the subdivision scheme and construct the subdivision based numerical algo-
rithm for the approximate solution of the singularly perturbed boundary value
problems. In Section 6.2, convergence of the method has been discussed. Sec-
tion 6.3, contains three numerical examples for the illustration of the algorithm
and their results are compared with other methods to show the efficiency of the
method. Conclusion about the numerical results is presented in Section 6.4.
6.1 Derivation of numerical algorithm
In this section, first we discuss the derivatives of (6.4), then we formulate the
subdivision based collocation algorithm for the solution of second order singu-
148
larly perturbed boundary value problems.
6.1.1 First and second derivatives of Ψ(x)
Since the function Ψ(x) ∈ C2, then the first and second derivatives can be
obtained by using the left and right eigenvectors of the subdivision matrix of
scheme (6.3). The first and second derivatives of (6.4) for the parametric value
ω = 0.04 are given in Table 6.1. Similar approach of Mustafa and Ejaz (2014) and
Ejaz et al. (2015) has been used to find derivatives.
Table 6.1: First and second derivatives of Ψ
i 0 ±1 ±2 ±3 ±4
Ψ(i) 1 0 0 0 0
Ψ′(i) 0 ±91004313
∓1967325878
± 160012939
± 12812939
Ψ′′(i) −144314224
43252112
−15752816
25132
133
6.1.2 The subdivision based collocation algorithm
Let N be a positive integer (N ≥ 4), h = 1/N and xi = i/N = ih, i = 0, 1, 2, · · ·N ,
and
U(x) =N+4∑i=−4
uiΨ
(x− xi
h
), 0 ≤ x ≤ 1, (6.6)
be the approximate solution of (6.1) where {ui} are the unknowns to be deter-
mined then
εU ′′(xj) = A(xj)U′(xj) +B(xj)U(xj) + F (xj), j = 0, 1, 2, · · · , N, (6.7)
with given boundary conditions
U(0) = α0, U(1) = α1.
149
From (6.6), we have
U ′(xj) =1h
N+4∑i=−4
uiΨ′ (xj−xi
h
),
U ′′(xj) =1h2
N+4∑i=−4
uiΨ′′ (xj−xi
h
).
(6.8)
Using (6.6) and (6.8) in (6.7), we get following N + 1 system of equations
εN+4∑i=−4
uiΨ′′(xj − xi
h
)− hAj
N+4∑i=−4
uiΨ′(xj − xi
h
)− h2Bj
N+4∑i=−4
uiΨ
(xj − xi
h
)= h2Fj,
where Aj = A(xj), Bj = B(xj) and Fj = F (xj). This implies
N+4∑i=−4
ui
{εΨ′′
(xj − xi
h
)− hAjΨ
′(xj − xi
h
)− h2BjΨ
(xj − xi
h
)}= h2Fj.
Further implies
N+4∑i=−4
ui
{εΨ′′(j − i)− hAjΨ
′(j − i)− h2BjΨ(j − i)}= h2Fj, (6.9)
where j = 0, 1, 2, · · · , N and xi = ih or xj = jh . By using Ψ(i) = Ψi, (6.9) can be
written as
N+4∑i=−4
ui
{εΨ′′
j−i − hAjΨ′j−i − h2BjΨj−i
}= h2Fj, j = 0, 1, 2, · · · , N. (6.10)
As we observe from Table 6.1, Ψ′−i = −Ψ′
i and Ψ′′−i = Ψ′′
i so (6.10) becomes
N+4∑i=−4
ui
{εΨ′′
i−j + hAjΨ′i−j − h2BjΨi−j
}= h2Fj, j = 0, 1, 2, · · · , N. (6.11)
The following proposition is the simplified form of the above system of equa-
tions.
Proposition 6.1.1. The system (6.11) is equivalent to
4∑i=−4
uj+iPji = h2Fj, j = 0, 1, 2, · · · , N, (6.12)
150
where
P ji =
εΨ′′0 − h2Bj, for i = 0,
εΨ′′i − hAjΨ
′i, for i = 0.
(6.13)
Proof. Substituting j = 0 in (6.11), we get
N+4∑i=−4
ui
{εΨ′′
i + hA0Ψ′i − h2B0Ψi
}= h2F0, j = 0, 1, 2, · · · , N.
By expanding above equation, we get
u−4{εΨ′′−4 + hA0Ψ
′−4 − h2B0Ψ−4}+ u−3{εΨ′′
−3 + hA0Ψ′−3 − h2B0Ψ−3}+ · · ·
+u0{εΨ′′0 + hA0Ψ
′0 − h2B0Ψ0}+ · · ·+ uN+3{εΨ′′
N+3 + hA0Ψ′N+3 − h2B0ΨN+3}
+uN+4{εΨ′′N+4 + hA0Ψ
′N+4 − h2B0ΨN+4} = h2F0.
Since Ψ′i and Ψ′′
i are non-zero in the interval [−4, 4], outside the interval these
are zero, then above equation becomes
u−4{εΨ′′−4 + hA0Ψ
′−4 − h2B0Ψ−4}+ u−3{εΨ′′
−3 + hA0Ψ′−3 − h2B0Ψ−3}+ u−2{εΨ′′
−2
+hA0Ψ′−2 − h2B0Ψ−2}+ u−1{εΨ′′
−1 + hA0Ψ′−1 − h2B0Ψ−1}+ u0{εΨ′′
0 + hA0Ψ′0
−h2B0Ψ0}+ u1{εΨ′′1 + hA0Ψ
′1 − h2B0Ψ1}+ u2{εΨ′′
2 + hA0Ψ′2 − h2B0Ψ2}+ u3{εΨ′′
3
+hA0Ψ′3 − h2B0Ψ3}+ u4{εΨ′′
4 + hA0Ψ′4 − h2B0Ψ4} = h2F0.
By using the definition of Ψi given in (6.5) and Ψ′0 = 0, given in Table 6.1, above
expression becomes
u−4{εΨ′′−4 + hA0Ψ
′−4}+ u−3{εΨ′′
−3 + hA0Ψ′−3}+ u−2{εΨ′′
−2 + hA0Ψ′−2}
+u−1{εΨ′′−1 + hA0Ψ
′−1}+ u0{εΨ′′
0 − h2B0Ψ0}+ u1{εΨ′′1 + hA0Ψ
′1}
+u2{εΨ′′2 + hA0Ψ
′2}+ u3{εΨ′′
3 + hA0Ψ′3}+ u4{εΨ′′
4 + hA0Ψ′4} = h2F0.
If
P 0±4 = εΨ′′
±4 + hA0Ψ′±4, P 0
±3 = εΨ′′±3 + hA0Ψ
′±3, P 0
±2 = εΨ′′±2 + hA0Ψ
′±2
P 0±1 = εΨ′′
±1 + hA0Ψ′±1, P 0
0 = εΨ′′0 − h2B0,
151
so above equation becomes
4∑i=−4
uiP0i = h2F0.
Similarly for j = 1, 2, 3, · · · , N , we get
4∑i=−4
ui+jPji = h2Fj.
where for i = −4,−3, · · · , 3, 4 and j = 1, 2, 3, · · ·N, we have
P ji =
εΨ′′0 − h2Bj for i = 0,
εΨ′′i − hAjΨ
′i for i = 0.
This completes the proof.
6.1.3 Singularly perturbed linear system of equations
The system of equations (6.12) are the singularly perturbed linear equations.
These equations can be written in matrix form as
AU = F1, (6.14)
In (6.14) matrix A is (N + 1)× (N + 9) defined as
A = (pr−1s )(N+1)×(N+9), (6.15)
where “r” represents rows i.e. r = 1, 2, · · ·N + 2 , “s” represents columns i.e.
s = −4,−3, · · · , N + 3, N + 4 and
pr−1s =
P r−1i , for − 4 6 i 6 4,
0, for otherwise,
where P r−1i defined in (6.13). The vectors U and F1 are defined as
U = (u−4, u−3, · · · , uN+3, uN+4)T , (6.16)
152
and
F1 = h2 × (F0, F1, · · · , FN−1, FN)T . (6.17)
To find the unique solution of the system (6.14), we need eight more conditions.
Two conditions are given in (6.2) i.e. U(0) and U(1), we construct remaining
conditions as given in next section.
6.1.4 Compelled conditions
By adding two given conditions in the system (6.14), the system remains un-
stable. We need six more conditions to get stable system. Since approximation
order of subdivision scheme (6.3) is four, so we define compelled conditions of
order four for solution of (6.14). In this section, we discuss two methods for
the construction of compelled condition. For simplicity only the left end points
u−1, u−2, u−3 are discussed and right end points uN+1, uN+2, uN+3 can be treated
similarly.
153
C-1: Conditions by using cubic polynomial:
The values u−3, u−2, u−1 can be determined by the fourth order polynomial S1(x)
interpolating (xi, ui), 0 ≤ i ≤ 3. Precisely, we have
u−i = S1(−xi), i = 1, 2, 3,
where
S1(xi) =4∑
j=1
4
j
(−1)j+1U(xi−j).
Since by (6.6), U(xi) = ui for i = 1, 2, 3 then by replacing xi by −xi, we have
S1(−xi) =4∑
j=1
4
j
(−1)j+1u−i+j.
Hence the following compelled conditions can be employed at the left end
4∑j=0
4
j
(−1)ju−i+j = 0, i = 1, 2, 3. (6.18)
Similarly for the right end, we can define ui = S1(xi), i = N + 1, N + 2, N + 3
and
S1(xi) =4∑
j=1
4
j
(−1)j+1ui−j.
So we have the following boundary conditions at the right end
4∑j=0
4
j
(−1)jui−j = 0, i = N + 1, N + 2, N + 3. (6.19)
C-2: Conditions by using cardinal basis functions:
The values u−3, u−2, u−1 are determined by fourth order polynomial S2(x), i.e.
u−i = S2(−xi), i = 1, 2, 3.
154
The cubic polynomial S2(x) is defined as
S2(x) = u0L0
(x− x0
h
)+ u1L1
(x− x0
h
)+ u′′
0L∗0
(x− x0
h
)+u′′
1L∗1
(x− x0
h
), (6.20)
where the basis functions are defined as
L0
(x− x0
h
)= 1−
(x− x0
h
),
L1
(x− x0
h
)=
(x− x0
h
),
L∗0
(x− x0
h
)= −1
6
(x− x0
h
)(x− x0
h− 1
)(x− x0
h− 2
),
L∗1
(x− x0
h
)=
1
6
(x− x0
h
)(x− x0
h− 1
)(x− x0
h+ 1
),
and for t = 0, 1
u′′t = A
(xt − x0
h
)ut +B
(xt − x0
h
)ut + F
(xt − x0
h
).
Similarly for the right end, we can define ui = S2(−xi), i = N + 1, N + 2, N + 3.
The cubic polynomial in this case is
S2(x) = uNLN
(x− xN
h
)+ uN+1LN+1
(x− x0
h
)+ u′′
NL∗N
(x− x0
h
)+u′′
N+1L∗N+1
(x− x0
h
), (6.21)
where the basis functions are given below
LN
(x− xN
h
)= 1−
(x− xN
h
),
LN+1
(x− xN
h
)=
(x− xN
h
),
L∗N
(x− xN
h
)= −1
6
(x− xN
h
)(x− xN
h− 1
)(x− xN
h− 2
),
L∗N+1
(x− xN
h
)=
1
6
(x− xN
h
)(x− xN
h− 1
)(x− xN
h+ 1
),
155
and for t = N,N + 1
u′′t = A
(xt − xN
h
)ut +B
(xt − xN
h
)ut + F
(xt − xN
h
).
6.1.5 Stable singularly perturbed linear system of equations
By using any of above compelled conditions, we get a following new system
of (N + 9) singularly perturbed linear equations with (N + 9) unknowns {ui},
in which N + 1 equations are obtained from (6.12) and two equations obtained
from given boundary conditions (6.2). Further six equations are obtained from
compelled conditions either from (6.18) and (6.19) or (6.20) and (6.21).
By using C-1 compelled conditions:
If we use compelled conditions defined in (6.18) and (6.19) then stable singularly
perturbed linear system of equations is define as
B1U = F, (6.22)
where the coefficients matrix B1 = (LT1 ,AT ,RT
1 )T , A is defined by (6.15). The
matrix L1 of order 4× (N + 9) for left end boundary conditions is defined as
L1 =
0 1 −4 6 −4 1 1 0 0 0 · · · 0 0
0 0 1 −4 6 −4 1 0 0 0 · · · 0 0
0 0 0 1 −4 6 −4 1 0 0 · · · 0 0
0 0 0 0 1 0 0 0 0 0 · · · 0 0
,
where the first three rows are obtained from (6.18) and last row is obtained by
using left end compelled condition (6.2) i.e. U(0).
The matrix R1 of order 4 × (N + 9) for right end boundary conditions is con-
structed as
R1 =
0 0 · · · 0 0 0 0 0 0 1 0 0 0 0
0 0 · · · 0 0 0 1 −4 6 −4 1 0 0 0
0 0 · · · 0 0 0 0 1 −4 6 −4 1 0 0
0 0 · · · 0 0 0 0 0 1 −4 6 −4 1 0
,
156
where the first row of above matrix is obtained from given condition (6.2) i.e.
U(1) and the remaining three conditions are obtained from the right end com-
pelled conditions (6.19). The vector U is defined in (6.16) and F is define as
F = (0, 0, 0, α0,FT1 , α1, 0, 0, 0)
T , (6.23)
where F1 is defined in (6.17) and α0, α1 given in (6.2).
By using C-2 compelled conditions:
If we use compelled conditions defined in (6.20) and (6.21) then stable singularly
perturbed linear system takes the form as
B2U = F, (6.24)
where the coefficients matrix B2 = (LT2 ,AT ,RT
2 )T , A is defined by (6.15). The
matrix L2 of order 4× (N +9) for left end boundary conditions is defined as: the
first three rows are obtained from (6.20) and last row is obtained by using left
end compelled condition (6.2) i.e. U(0). The matrix R2 of order 4 × (N + 9) for
right end boundary conditions is constructed as: the first row of above matrix is
obtained from given condition (6.2) i.e. U(1) and the remaining three conditions
are obtained from the right end compelled conditions (6.21). The vectors U and
F are defined in (6.16) and (6.23).
Hence for different compelled conditions we have two different singularly
perturbed linear system of equations (6.22) and (6.24). The non-singularity of
the coefficient matrices B1 and B2 has been discussed in next section.
6.1.6 Existence of the solution
In this section, we discuss the non-singularity of the coefficient matrix. Non-
singularity of the matrices B1 and B2 by finding the eigenvalues of both the
matrices. We observe that all the eigenvalues up to N ≤ 500 are non-zero. Hence
157
by Strang (2011), we conclude that the B1 and B2 are non-singular. For large
N > 500, the matrices may or may not be singular.
6.2 Convergence of the method
In this section, we discuss convergence of the method described in Section 6.1.
Let Y be the analytic solution of the problem (6.1) with (6.2) then
εY′′(x) = A(x)Y
′(x) +B(x)Y (x) + F (x).
The above result can be written for node points for j = 0, 1, · · ·N, as
εY′′(xj) = A(xj)Y
′(xj) +B(xj)Y (xj) + F (xj). (6.25)
Let the vector Y (x) be defined as
Y (x) = (y(x0), y(x1), · · · , y(xN))T .
By Taylor’s series
Y′(xj) =
1
25878h[−256y(xj − 4h)− 3200y(xj − 3h) + 19673y(xj − 2h)
−54600y(xj − h) + 54600y(xj + h)− 19673y(xj + 2h)
+3200y(xj + 3h) + 256y(xj + 4h)] + o(h4),
and
Y′′(xj) =
1
8448h2[256y(xj − 4h) + 1600y(xj − 3h)− 4725y(xj − 2h)
+17300y(xj − h)− 28862y(xj) + 17300y(xj + h)− 4725y(xj + 2h)
+1600y(xj + 3h) + 256y(xj + 4h)] + o(h4).
The system of equations (6.22) and (6.24) provide the required subdivision based
approximate solution U(x) for (6.1) then by (6.7), for j = 0, 1, · · · , N,
εU ′′(xj) = A(xj)U′(xj) +B(xj)U(xj) + F (xj), (6.26)
158
where U ′(xj) and U ′′(xj) are defined as
U ′(xj) =1
25878h[−256u(xj − 4h)− 3200u(xj − 3h) + 19673u(xj − 2h)
−54600u(xj − h) + 54600u(xj + h)− 19673u(xj + 2h)
+3200u(xj + 3h) + 256u(xj + 4h)] + o(h4),
and
U ′′(xj) =1
8448h2[256u(xj − 4h) + 1600u(xj − 3h)− 4725u(xj − 2h)
+17300u(xj − h)− 28862u(xj) + 17300u(xj + h)− 4725u(xj + 2h)
+1600u(xj + 3h) + 256u(xj + 4h)] + o(h4).
Let the error function E is defined as E(x) = Y (x)− U(x) and
E = (E−4, E−3, · · · , EN+3, EN+4).
Then error vector at the node points is
E(xj) = Y (xj)− U(xj), −4 6 j 6 N + 4.
This implies
E′(xj) = Y
′(xj)− U ′(xj), −4 6 j 6 N + 4,
E′′(xj) = Y
′′(xj)− U ′′(xj), −4 6 j 6 N + 4.
By subtracting (6.26) from (6.25), we get
ε[Y
′′(xj)− U ′′(xj)
]= A(xj)
[Y
′(xj)− U ′(xj)
]+B(xj)
[Y (xj)− U(xj)
].
By definition of error vector
εE′′(xj) = A(xj)E
′(xj) + B(xj)E(xj), 0 6 j 6 N.
This implies
εE′′(xj)− A(xj)E
′(xj)−B(xj)E(xj) = 0, 0 6 j 6 N, (6.27)
159
where for 0 6 j 6 N
E′(xj) =
1
25878h[−256E(xj − 4h)− 3200E(xj − 3h) + 19673E(xj − 2h)
−54600E(xj − h) + 54600E(xj + h)− 19673E(xj + 2h)
+3200E(xj + 3h) + 256E(xj + 4h)] + o(h4),
and for 0 6 j 6 N
E′′(xj) =
1
8448h2[256E(xj − 4h) + 1600E(xj − 3h)− 4725E(xj − 2h)
+17300E(xj − h)− 28862E(xj) + 17300E(xj + h)− 4725E(xj + 2h)
+1600E(xj + 3h) + 256E(xj + 4h)] + o(h4).
As 0 ≤ x ≤ 1 and xj = jh, j = 0, 1, 2, · · · , N so E0, E1, · · · , EN are non zero while
E−4, · · · , E−1 and EN+1, · · · , EN+4 are zero because they lie outside the interval
[0, 1]. Let us define error values at the end points as
Ej =
max0≤k≤4
{|Ek|}O(h4), −4 ≤ j < 0,
maxN−4≤k≤N
{|Ek|}O(h4), N < j ≤ N + 4.
(6.28)
By expanding (6.27) similar to Proposition 6.1.1 and using the algorithm defined
in Section 6.1, we get
(B1 +O(h4))E = 0.
Similarly
(B2 +O(h4))E = 0.
These are equivalent to
(B1 +O(h4))E = O(h4) ∥ E ∥= O(h4),
and
(B2 +O(h4))E = O(h4) ∥ E ∥= O(h4).
160
Since for small h, the coefficient matrix Bi + O(h4), i = 1, 2 will be invertible
and thus using the standard result from linear algebra and effect of ∥B−1i ∥, we
have
||E|| ≤(
||B−1i ||
1−O(h4)O(h4)
)= O(h2), i = 1, 2.
Hence ∥ E ∥= O(h2). The result is summarized in the following theorem.
Theorem 6.2.1. Let Y be the exact solution of the system (6.1) and Uj , j = 0, 1, · · · , N
be the approximate solution of (6.7) then ∥ E ∥= O(h2).
6.3 Numerical examples and discussions
In this section, we have implemented our method on three examples which sup-
port the theoretical analysis of our findings about order of convergence. We
solve these examples by solving two singularly perturbed linear systems of e-
quations (6.22) and (6.24). The maximum absolute errors in the exact and ap-
proximate solutions are also calculated at the different step sizes. For the sake
of comparisons, we have also tabulated the numerical results. The physical be-
havior of analytic and approximate solutions is also presented in this section.
We consider the following three examples of second order singularly perturbed
boundary value problems:
Example 6.3.1.
εy′′(x) = y + cos2(πx) + 2επ2 cos(2πx), 0 < x < 1,
with boundary conditions
y(0) = 0 = y(1).
The analytic solution is given by
y(x) =
[e
(−(1−x)√
ε
)+ e
(−x√
ε
)][1 + e
(−1√ε
)] − cos2(πx).
161
Example 6.3.2.
εy′′(x)− (1 + x)y(x) = 40[x(x2 − 1)− 2ε], 0 < x < 1,
with boundary conditions
y(0) = 0 = y(1).
The analytic solution of this problem is
y(x) = 40x(1− x).
Example 6.3.3.
εy′′(x)− {1 + x(1− x)}y(x) = − [1 + x(1− x) + {2√ε− x2(1− x)}
e
{(1−x)√
ε
}+ {2
√ε− x(1− x)2}e
{− x√
ε
}],
where 0 6 x 6 1 with boundary conditions
y(0) = y(1) = 0.
The analytic solution for the above problem is
y(x) = 1 + (x− 1)e
{− x√
ε
}− xe
{− (1−x)√
ε
}.
6.3.1 Results and discussion
All above examples are solved by subdivision based algorithm at different val-
ues of N and small values of ε. We have observed the following facts:
• The numerical results of Example 6.3.1 are shown in Tables 6.2 - 6.6 and
in Figures 6.1 - 6.3. Tables 6.2 and 6.3 show the maximum absolute errors
at different values of N and small values of ε. Tables 6.4 - 6.6 provide a
comparison of maximum absolute error with the existing methods. i.e.
comparison with spline based method by Aziz and Khan (2002) and Bawa
162
and Natesan (2005), tension spline method by Khan and Aziz (2005), fit-
ted mesh B-spline method by Kadalbajoo and Aggarwal (2005) and Haar
wavelet based method by Pandit and Kumar (2014). It is concluded that
our method gives better results than the methods of Aziz and Khan (2002),
Bawa and Natesan (2005), Khan and Aziz (2005), Kadalbajoo and Aggar-
wal (2005) and Pandit and Kumar (2014). Figure 6.1 compares the analytic
and numerical solutions while Figures 6.2 and 6.3 delineate the physical
behavior of the Example 6.3.1 at different values of N and ε.
• We present the numerical results of Example 6.3.2 in Tables 6.7 - 6.10 and
graphical representation in Figures 6.4 - 6.6. Tables 6.7 and 6.10 show the
maximum absolute errors at different value of ε and N while Tables 6.9
and 6.10 present the comparison of maximum absolute errors with exist-
ing methods Miller (1979) and Niijima (1980a, 1980b). From these com-
parison, it is concluded that our method gives better results than that of
existing methods. Figure 6.4 compare the analytic and numerical solution-
s while Figures 6.5 and 6.6 depicts the physical behavior of the Example
6.3.2 at different values of N and ε.
• The numerical results of Example 6.3.3 are reported in Tables 6.11 - 6.13
and in Figures 6.7 - 6.9. Tables 6.11 and 6.12 shows the maximum absolute
errors at different values of N and ε. We compare numerical results of Ex-
ample 6.3.3 with wavelet and finite difference methods Kumar and Mehra
(2009) and Lubuma and Patidar (2006) and it is concluded that our method
gives better approximation comparative to other methods. The graphical
representation of the solution of Example 6.3.3 for different values of N
and ε is given in Figures 6.8 and 6.9.
• From the tabulated results of these examples, we observe that the condi-
tion C-2 gives less maximum absolute errors comparative to the condition
163
C-2.
• It is also observed that for fixed N maximum absolute errors increase with
the decrease of ε while for fixed value of ε maximum absolute errors de-
crease with the increase of N .
Table 6.2: Maximum absolute errors of Example 6.3.1
N = 10 Our method Our method
for ε with C-1 with C-2
0.1× 10−3 2.3445E-02 1.8459E-02
0.1× 10−4 2.4427E-03 1.9197E-03
0.1× 10−5 2.4525E-04 1.9270E-04
0.1× 10−6 2.4535E-05 1.9278E-05
0.1× 10−7 2.4535E-06 1.9279E-06
0.1× 10−8 2.4536E-07 1.9279E-07
0.1× 10−9 2.4536E-08 1.9279E-08
164
Tabl
e6.
3:M
axim
umab
solu
teer
rors
ofEx
ampl
e6.
3.1
Nε=
10−5
ε=
10−8
ε=
10−10
ε=
10−5
ε=
10−8
ε=
10−10
wit
hC
-1w
ith
C-2
102.
4427
E-03
2.45
35E-
062.
4536
E-08
1.91
97E-
031.
9279
E-06
1.92
78E-
08
100
1.20
05E-
012.
3115
E-04
2.31
26E-
069.
3578
E-02
1.88
87E-
041.
8895
E-06
150
1.51
52E-
015.
1955
E-04
5.20
08E-
061.
1209
E-01
4.24
58E-
044.
2498
E-06
200
1.59
50E-
019.
2276
E-04
9.24
42E-
061.
1327
E-01
7.54
13E-
047.
5541
E-06
250
1.56
92E-
011.
4402
E-03
1.44
43E-
051.
0808
E-01
1.17
71E-
031.
1802
E-05
165
Tabl
e6.
4:M
axim
umab
solu
teer
rors
ofEx
ampl
e6.
3.1
λ1
andλ2
ε=
10−5
ε=
10−5
ε=
10−8
ε=
10−10
Azi
zan
dK
han
(200
2)N
=10
N=
100
N=
200
N=
250
&K
han
and
Azi
z(2
005)
1/18
,4/9
···
1.44
463E
-03
6.22
342E
-02
6.27
380E
-02
1/14
,3/7
···
1.52
823E
-02
8.33
647E
-02
8.39
115E
-02
1/24
,11/24
···
1.00
616E
-02
4.50
702E
-02
4.55
413E
-02
1/30
,14/30
···
1.67
078E
-02
3.52
995E
-02
3.57
527E
-02
1/6,
1/3
···
1.19
71E-
012.
6683
E-01
2.67
93E-
01
Our
met
hod
2.44
27E-
031.
2005
E-01
9.22
76E-
041.
4443
E-05
wit
hC
-1
Our
met
hod
1.91
97E-
039.
3579
E-02
7.54
13E-
041.
1802
E-05
wit
hC
-2
166
Table 6.5: Maximum absolute errors of Example 6.3.1
for N = 32 Bawa and Pandit and Our method Our method
and ε = (2−r)2 Natesan (2005) Kumar (2014) with C-1 with C-2
r=10 5.022E-02 1.23E-02 2.2646E-03 1.8478E-03
r=20 3.125E-02 1.23E-08 2.1692E-09 1.7695E-09
r=25 3.125E-02 1.20E-11 2.1200E-12 1.7283E-12
Table 6.6: Maximum absolute errors of Example 6.3.1
for N = 32 Kadalbajoo and Pandit and Our method Our method
and ε = 2−r Aggarwal (2005) Kumar (2014) with C-1 with C-2
r=10 5.022E-02 1.80E-03 1.5551E-01 1.0545E-01
r=20 3.125E-02 1.23E-03 2.2645E-03 1.8478E-03
r=25 3.125E-02 4.04E-4 7.1071E-05 5.7977E-05
167
Table 6.7: Maximum absolute errors of Example 6.3.2
N = 10 Our method Our method
for ε with C-1 with C-2
0.1× 10−3 1.4049E-02 2.4080E-02
0.1× 10−4 1.4331E-03 2.4866E-03
0.1× 10−5 1.4360E-04 2.4948E-04
0.1× 10−6 1.4363E-05 2.4957E-05
0.1× 10−7 1.4364E-06 2.4958E-06
0.1× 10−8 1.4364E-07 2.4958E-07
0.1× 10−9 1.4364E-08 2.4958E-08
6.4 Conclusions
The subdivision techniques appear to be an ideal tool to attain the numerical so-
lution of singularly perturbed boundary value problem. The subdivision based
method is developed for the approximate solution of the singularly perturbed
boundary value problems. The proposed method is computationally efficient
and the algorithm can be easily implemented on computer. In addition, we in-
troduced some compelled conditions for the existence of approximate solution.
The method has been proved to be second order convergent. It is concluded
that our method for the solutions of singularly perturbed boundary value prob-
lems is batter than spline based methods by Aziz and Khan (2002), Bawa and
Natesan (2005), Khan and Aziz (2005), Kadalbajoo and Aggarwal (2005) , finite
difference methods by Lubuma (2006), Miller (1979), Niijima (1980a), Niijima
(1980b) and Haar wavelet methods by Kumar and Mehra (2002), Pandit and
Kumar (2014).
168
Tabl
e6.
8:M
axim
umab
solu
teer
rors
ofEx
ampl
e6.
3.2
Nε=
10−5
ε=
10−8
ε=
10−10
ε=
10−5
ε=
10−8
ε=
10−10
wit
hC
-1w
ith
C-2
101.
4331
E-03
1.43
64E-
061.
4364
E-08
2.48
66E-
032.
4958
E-06
2.49
58E-
08
100
1.50
99E-
031.
5640
E-06
1.56
44E-
081.
1720
E-02
1.72
89E-
051.
7298
E-07
150
1.52
17E-
031.
5687
E-06
1.56
95E-
081.
3907
E-02
2.68
51E-
052.
6881
E-07
200
1.52
62E-
031.
5706
E-06
1.57
21E-
081.
4855
E-02
3.63
98E-
053.
6471
E-07
250
1.52
8E-0
31.
5714
E-06
1.57
36E-
081.
5259
E-02
4.59
19E-
054.
6062
E-07
169
Table 6.9: Maximum absolute errors of Example 6.3.2
for N = 16 Miller Niijima Niijima Our Our
and ε (1979) (1980a) (1980b) method method
with C-1 with C-2
0.1× 10−3 0.25E-01 0.26E-01 0.65E-04 0.1408E-01 0.3079E-01
0.1× 10−4 0.21E-01 0.24E-01 0.36E-04 0.1478E-02 0.3374E-02
0.1× 10−5 0.70E-02 0.17E-01 0.33E-04 0.14862E-03 0.3407E-03
0.1× 10−6 0.75E-03 0.69E-02 0.26E-04 0.14870E-04 0.3410E-04
0.1× 10−7 0.74E-04 0.23E-02 0.20E-04 0.14871E-05 0.3411E-05
0.1× 10−8 0.67E-05 0.76E-03 0.11E-04 0.14872E-06 0.3411E-06
Table 6.10: Maximum absolute errors of Example 6.3.2
for N = 32 Miller Niijima Niijima Our Our
and ε (1979) (1980a) (1980b) method method
with C-1 with C-2
0.1× 10−3 0.64E-02 0.65E-02 0.59E-04 0.1414E-01 0.3861E-01
0.1× 10−4 0.61E-02 0.64E-02 0.21E-04 0.1497E-02 0.5501E-02
0.1× 10−5 0.41E-02 0.56E-02 0.35E-04 0.1529E-03 0.5756E-03
0.1× 10−6 0.77E-03 0.31E-02 0.39E-04 0.1531E-04 0.5783E-04
0.1× 10−7 0.76E-04 0.12E-02 0.21E-04 0.1532E-05 0.5786E-05
0.1× 10−8 0.67E-05 0.38E-03 0.21E-04 0.1532E-06 0.5786E-06
170
Table 6.11: Maximum absolute errors of Example 6.3.3
N = 10 Our method Our method
ε with C-1 with C-2
0.1× 10−3 2.0312E-02 7.4455E-03
0.1× 10−4 2.1110E-03 7.6536E-04
0.1× 10−5 2.1189E-04 7.6679E-05
0.1× 10−6 2.1197E-05 7.6682E-06
0.1× 10−7 2.1198E-06 7.6679E-07
0.1× 10−8 2.1198E-07 7.6678E-08
0.1× 10−9 2.1198E-08 7.6679E-09
171
Tabl
e6.
12:M
axim
umab
solu
teer
rors
ofEx
ampl
e6.
3.3
wit
hC
-1w
ith
C-2
Nε=
10−5
ε=
10−8
ε=
10−10
ε=
10−5
ε=
10−8
ε=
10−10
102.
1110
E-03
2.11
98E-
062.
1198
E-08
7.65
36E-
047.
6679
E-07
7.66
77E-
09
165.
5268
E-03
5.58
76E-
065.
5877
E-08
2.00
82E-
032.
0212
E-06
2.02
12E-
08
322.
1930
E-02
2.29
64E-
052.
2965
E-07
8.02
32E-
038.
3069
E-06
8.30
69E-
08
100
1.19
19E-
012.
2869
E-04
2.28
79E-
062.
3666
E-02
8.27
35E-
058.
2759
E-07
200
1.59
31E-
019.
1801
E-04
9.19
65E-
063.
7244
E-02
3.32
26E-
043.
3265
E-06
250
1.56
91E-
011.
4343
E-03
1.43
84E-
054.
6497
E-02
5.19
30E-
045.
2028
E-06
172
Tabl
e6.
13:M
axim
umab
solu
teer
rors
ofEx
ampl
e6.
3.3
forN
=16
Kum
aran
dM
ehra
Lubu
ma
and
Pati
dar
Our
met
hod
Our
met
hod
andε=
10−r
(200
9)(2
006)
wit
hC
-1w
ith
C-2
r=3
0.77
E-01
0.28
E-01
0.15
24E-
000.
2703
E-01
r=5
0.46
E-02
0.53
E-02
0.55
27E-
020.
2008
E-02
r=7
0.46
E-04
0.53
E-02
0.55
87E-
040.
2021
E-04
r=8
0.46
E-05
0.53
E-02
0.55
89E-
052.
0212
E-05
173
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(c)
Figure 6.1: Physical behavior of analytic and approximate solutions of Example 6.3.1
for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c) respectively.
174
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
Figure 6.2: Physical behavior of analytic and approximate solutions of Example 6.3.1
for N = 32 and ε = 2−25.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
Figure 6.3: Physical behavior of analytic and approximate solutions of Example 6.3.1
for N = 32 and ε = (2−20)2.
175
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
(c)
Figure 6.4: Physical behavior of analytic and approximate solutions of Example 6.3.2
for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c) respectively.
176
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
Figure 6.5: Physical behavior of analytic and approximate solutions of Example 6.3.2
for N = 16 and ε = 10−8.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
8
9
10
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1 Approximate solution with C−2
Figure 6.6: Physical behavior of analytic and approximate solutions of Example 6.3.2
for N = 32 and ε = 10−9.
177
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1Approximate solution with C−2
(a)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1Approximate solution with C−2
(b)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1Approximate solution with C−2
(c)
Figure 6.7: Physical behavior of analytic and approximate solutions of Example 6.3.3
for N = 10 with ε = 10−4, 10−7, 10−10 shown in (a), (b) and (c) respectively.
178
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1Approximate solution with C−2
Figure 6.8: Physical behavior of analytic and approximate solutions of Example 6.3.3
for N = 16 and ε = 10−5.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
App
roxi
mat
e so
lutio
n
Analytic solutionApproximate solution with C−1Approximate solution with C−2
Figure 6.9: Physical behavior of analytic and approximate solutions of Example 6.3.3
for N = 16 and ε = 10−8.
179
Chapter 7
A Subdivision Collocation
Algorithm for Solving Two point
Boundary value Problems of Order
Three
In this chapter, we propose an algorithm for the numerical solution of self ad-
joint singularly perturbed third order boundary value problems in which the
highest order derivative is multiplied by a small parameter ε. This chapter is
divided into five sections. Self adjoint singularly perturbed BVP, subdivision
scheme and their derivative are presented in Section 7.1. Numerical algorithm
for the solution of third order singularly perturbed boundary value problem is
discussed in Section 7.2. Convergence analysis of the numerical algorithm is
given in Section 7.3. Numerical examples and their discussion for the illustra-
tion of algorithm is given in Section 7.4. Conclusion about the proposed algo-
rithm is given in Section 7.5.
180
7.1 Preliminaries
We present third order singularly perturbed boundary value problem, binary
subdivision scheme and their derivatives in this section.
7.1.1 Third order singularly perturbed BVP
We consider the general expression for third order singulary perturbed problem
as
−εy′′′(x) + p(x)y(x) = f(x), p(x) > 0,
y(0) = β0, y(1) = β1, y′(0) = β2,(7.1)
or
−εy′′′(x) + p(x)y(x) = f(x), p(x) > 0,
y(0) = β0, y(1) = β1, y′′(0) = β3,(7.2)
where β0, β1, β2 and β3 are constant, p(x), f(x) are smooth functions and ε is a
small positive parameter with ε ≪ 1. These type of problems usually occur in
quantum mechanics, fluid mechanics, optical control, chemical reactions etc.
7.1.2 Subdivision scheme and derivatives
The subdivision schemes have been considered esteemed in many arenas of
computational sciences. Such as computer animation, computer graphics and
computer aided geometric design due to its efficient and simple characteristics.
The subdivision scheme defines a curve out of an initial control polygon by
subdividing them according to some refining rules recursively. Consider eight
181
point binary interpolating scheme presented by Lee et al. (2006) as
pk+12i = pki ,
pk+12i+1 = −ω
(pki−3 + pki+4
)+(5ω + 3
256
)(pki−2 + pki+3)−
(9ω + 25
256
)(pki−1
+pki+2) +(5ω + 75
128
)(pki + pki+1).
(7.3)
The scheme (7.3) is C3 derivable continuous for 0.0016 < ω < 0.0084, the support
width for the mask of the scheme is [−6, 6], the approximation order is six and
satisfies following two scale relation
Ψ(x) = Ψ(2x) +[−ω {Ψ(2x− 1) + Ψ(2x+ 1)}+
(5ω + 3
256
){Ψ(2x− 3)
+Ψ(2x+ 3)} −(9ω + 25
256
){Ψ(2x− 5) + Ψ(2x+ 5)}+
(5ω + 75
128
){Ψ(2x− 7) + Ψ(2x+ 7)}] , x ∈ R
(7.4)
where
Ψ(x) =
1 for x = 0,
0 for x = 0.(7.5)
As the function Ψ(x) ∈ C3, then the first, second and third derivatives can be
obtained by using the similar approach as in Mustafa and Ejaz (2014). The third
derivatives of (7.4) for the parametric value ω = 0.0032 are given below:
Ψ′′′(0) = 0, Ψ′′′(±1) = ±1122400355000418234124847
,
Ψ′′′(±2) = ∓1502922273911836468249694
, Ψ′′′(±3) = ±1166667500038021284077
,
Ψ′′′(±4) = ± 159804307271672936499388
, Ψ′′′(±5) = ∓ 4221440000418234124847
,
Ψ′′′(±6) = ± 108068864418234124847
.
(7.6)
7.2 Subdivision collocation algorithm
This section describes an algorithm for the numerical solutions of singularly
perturbed linear third order boundary value problems with non-homogeneous
boundary conditions. An algorithm is described as follows:
182
Let N be a positive integer (N > 6), h = 1/N and xi = i/N = ih, i =
0, 1, 2, · · ·N , and
U(x) =N+6∑i=−6
uiΨ
(x− xi
h
), 0 ≤ x ≤ 1, (7.7)
be the approximate solution to (7.1) or (7.2) where {ui} are the unknown to be
determined. Then
−εU ′′′(xj) + p(xj)U(xj) = f(xj), j = 0, 1, 2, · · · , N,
with the following given boundary conditions
U(0) = β0, U(1) = β1 U ′(0) = β2,
or
U(0) = β0, U(1) = β1 U ′′(0) = β3.
(7.8)
Let we define p(xj) = pj , and f(xj) = fj , then above equation can be written as
−εU ′′′(xj) + pjU(xj) = fj, j = 0, 1, 2, · · · , N, (7.9)
From (7.7) we have
U ′′′(xj) =1
h3
N+6∑i=−6
uiΨ′′′(xj − xi
h
). (7.10)
Using (7.7) and (7.10) in (7.9), we get following N + 1 system of equations
−ε
N+6∑i=−6
uiΨ′′′(xj − xi
h
)+ h3pj
N+6∑i=−6
uiΨ
(xj − xi
h
)= h3fj.
This implies
N+6∑i=−6
ui
{−εΨ′′′
(xj − xi
h
)+ h3pjΨ
(xj − xi
h
)}= h3fj.
Further implies
N+6∑i=−6
ui
{−εΨ′′′(j − i) + h3pjΨ(j − i)
}= h3fj, (7.11)
183
where j = 0, 1, 2, · · · , N and xi = ih or xj = jh. By using Ψ(i) = Ψi, (7.11) can
be written asN+6∑i=−6
ui
{−εΨ′′′
j−i + h3pjΨj−i
}= h3fj, j = 0, 1, 2, · · · , N. (7.12)
As we observe from (7.6), Ψ′′′−i = −Ψ′′′
i , then (7.12) becomes
N+6∑i=−6
ui
{εΨ′′′
i−j + h3pjΨi−j
}= h3fj, j = 0, 1, 2, · · · , N. (7.13)
Remark 7.2.1. The system (7.13) is equivalent to
6∑i=−6
uj+iQji = h3fj, j = 0, 1, 2, · · · , N, (7.14)
where
Qji =
εΨ′′′0 + h3pj, i = 0,
εΨ′′′i , i = 0.
(7.15)
7.2.1 Singularly perturbed system
The system of equations (7.14) are the singularly perturbed linear equations.
These equations can be written in matrix form as
AU = F1, (7.16)
where
A = (qr−1s )(N+1)×(N+13), (7.17)
U = (us)(N+13)×1, (7.18)
F1 = h3 × (uℓ)(N+13)×1, (7.19)
“r” and “s” represent rows and columns respectively. Where
qr−1s =
Qr−1i , for − 6 6 i 6 6,
0, for otherwise,
184
ℓ = 0, 1, · · · , N, r = 1, 2, · · · , N + 2, s = −6,−5, · · · , N + 5, N + 6.
and Qr−1i defined in (7.15).
To find the unique solution of the system (7.16), we need twelve more condi-
tions. Three conditions are given in (7.8) i.e. U(0), U(1) and U ′(0) or U ′′(0). As
in given conditions first or second derivative is involved so first we replace first
or second derivative conditions by their approximation. The approximation of
these derivatives is given as follows:
7.2.2 Approximation of derivative conditions
We approximate first and second derivatives of the function U(x) by finite dif-
ferences algorithm. Given a non-zero value of h, the lth order derivative satisfies
the following equation where the integer order of error p > 0 may be selected
as desired
U l(x) =l!
hl
imax∑i=imin
ciU(x+ ih) +O(hp). (7.20)
A forward difference approximation occurs if we set imin = 0 and imax = l +
p − 1. The vector C = (cimin, · · · , cimax) is called the convolution mask for the
approximation. In order for equation (7.20) satisfied, it is necessary that
imax∑i=imin
inci =
0, for 0 6 n 6 l + p− 1 and n = l,
1, for n = l.(7.21)
Approximation of U ′(x) with error O(h7), we have imin = 0 and imax = 7. The
convolution matrix (c0, c1, · · · , c7) is obtained by solving the linear system
7∑i=0
inci =
0, for 0 6 n 6 7 and n = 1,
1, for n = 1.(7.22)
185
After solving (7.22) substituting the values of ci in (7.20), we obtain first deriva-
tive approximation as
U ′(0) = −(N
60
)[−147u0 + 360u1 − 450u2 + 400u3 − 225u4 + 72u5 − 10u6] .
(7.23)
Similarly approximation of U ′′(x) with error O(h7), we have imin = 0 and imax =
8. The convolution matrix (c0, c1, · · · , c8) is obtained by solving the linear system
8∑i=0
inci =
0, for 0 6 n 6 8 and n = 2,
1, for n = 2.(7.24)
After solving (7.24) substituting the values of ci in (7.20), we obtain second
derivative approximation as
U ′′(0) = −(
N
360
)[938u0 − 4014u1 + 7911u2 − 9490u3 + 7380u4 − 3618u5
+1019u6 − 126u7] . (7.25)
The remaining nine conditions are discussed in the next section.
7.2.3 Necessitated conditions
To find the unique solution of (7.16) with (7.8), we require nine more conditions.
For this purpose we define these conditions, named necessitated conditions, in
this section. These conditions can be defined as follows:
The values u−5, u−4, u−3, u−2, u−1 can be determined by the sixth order polyno-
mial S1(x) interpolating (xi, ui), 0 ≤ i ≤ 5. Precisely, we have
u−i − S1(−xi) = 0, i = 1, 2, 3, 4, 5,
where
S1(xi) =6∑
j=1
6
j
(−1)j+1U(xi−j).
186
Since by (7.7), U(xi) = ui for i = 1, 2, 3, 4, 5 then by replacing xi by −xi, we have
S1(−xi) =6∑
j=1
6
j
(−1)j+1u−i+j.
Hence the following necessitated conditions can be employed at the left end
6∑j=0
6
j
(−1)ju−i+j = 0, i = 1, 2, 3, 4, 5. (7.26)
Similarly for the right end, we can define ui−S1(xi) = 0, i =N +1, N +2, N +3,
N + 4 So we have the following necessitated boundary conditions at the right
end
6∑j=0
6
j
(−1)jui−j = 0, i = N + 1, N + 2, N + 3, N + 4. (7.27)
7.2.4 Stable linear system of equations
By using above necessitated conditions, we get a following new system of (N +
13) singularly perturbed linear equations with (N+13) unknowns {ui}, in which
N+1 equations are obtained from (7.14) and three equations obtained from giv-
en boundary conditions (7.8). Further nine equations are obtained from neces-
sitated conditions defined in (7.26) and (7.27).
If we use necessitated conditions then stable singularly perturbed linear system
of equations becomes
BU = F, (7.28)
where the coefficients matrix B = (LT ,AT ,RT )T , A is defined by (7.17).
The matrix L of order 7 × (N + 13) for left end boundary conditions is defined
as:
First five rows are obtained from (7.26), second last row is obtained from (7.23)
or (7.25) either U ′(0) or U ′′(0) and last row is also obtained from (7.8) i.e. U(0).
187
The matrix R of order 6 × (N + 13) for right end boundary conditions is con-
structed as: First row of above matrix is obtained from given condition (7.8) i.e.
U(1) and the remaining five conditions are obtained from the conditions (7.27).
The vector U is defined in (7.18) and F is define as
F = (0, 0, 0, 0, 0, β2, β0,FT1 , β1, 0, 0, 0, 0, 0)
T ,
or
F = (0, 0, 0, 0, 0, β3, β0,FT1 , β1, 0, 0, 0, 0, 0)
T ,
(7.29)
where F1 is defined in (7.19) and β0, β1, β2, β3, given in (7.8). For the existences
of the unique solution, first we check the non-singularity of the coefficient B.
We observed that B remains non-singulary for N 6 500 and for large N it may
or may not be non-singular.
7.3 Convergence of collocation algorithm
In this section, we discuss convergence of collocation algorithm described in
Section 7.2.
Let y(x) be the exact solution of the problem (7.1) or (7.2) then for j = 0, 1, · · ·N,
we have
−εy′′′(xj) + p(xj)y(xj) = f(xj). (7.30)
Let the vector y(x) be defined as
y(x) = (y(x0), y(x1), · · · , y(xN))T .
188
By Taylor’s series
y′′′(xj) = − 1
1672936499388h3
[−432275456y(j−6)h + 16885760000y(j−5)h
−15980430727y(j−4)h − 513333700000y(j−3)h + 3005844547822y(j−2)h
−4489601420000y(j−1)h + 4489601420000y(j+1)h − 3005844547822y(j+2)h
+513333700000y(j+3)h + 15980430727y(j+4)h − 16885760000y(j+5)h
+432275456y(j+6)h
]+ o(h7),
where y(xj − th) = y(j−t)h for t = −6,−5, · · · , N + 6. The system of equations
(7.28) provides the required subdivision based approximate solution U(x) for
(7.1) or (7.2) then by (7.8), for j = 0, 1, · · · , N
−εU ′′′(xj) + p(xj)U(xj) = f(xj), (7.31)
where U ′′′(xj) is defined as
U ′′′(xj) = − 1
1672936499388h3
[−432275456u(j−6)h + 16885760000u(j−5)h
−15980430727u(j−4)h − 513333700000u(j−3)h + 3005844547822u(j−2)h
−4489601420000u(j−1)h + 4489601420000u(j+1)h − 3005844547822u(j+2)h
+513333700000u(j+3)h + 15980430727u(j+4)h − 16885760000u(j+5)h
+432275456u(j+6)h
]+ o(h7),
where u(xj − th) = u(j−t)h for t = −6,−5, · · · , N + 6. Let the error function E is
defined as E(x) = y(x)− U(x) and
E = (E−6, E−5, · · · , EN+5, EN+6).
Then error vector at the node points is
E(xj) = y(xj)− U(xj), −6 6 j 6 N + 6.
This implies
E′′′(xj) = y′′′(xj)− U ′′′(xj), −6 6 j 6 N + 6.
189
By subtracting (7.31) from (7.30), we get
−ε [y′′′(xj)− U ′′′(xj)] + p(xj) [y(xj)− U(xj)] = 0.
By definition of error vector
−εE′′′(xj) + p(xj)E(xj) = 0, 0 6 j 6 N.
This implies
−εE′′′(xj) + p(xj)E(xj) = 0, 0 6 j 6 N, (7.32)
where for 0 6 j 6 N
E′′′(xj) =1
1672936499388h3
[−432275456E(j−6)h + 16885760000E(j−5)h
−15980430727E(j−4)h − 513333700000E(j−3)h + 3005844547822E(j−2)h
−4489601420000E(j−1)h + 4489601420000E(j+1)h − 3005844547822E(j+2)h
+513333700000E(j+3)h + 15980430727E(j+4)h − 16885760000E(j+5)h
+432275456E(j+6)h
]+ o(h7).
As 0 ≤ x ≤ 1 and xj = jh, j = 0, 1, 2, · · · , N so E0, E1, · · · , EN are non zero while
E−6, · · · , E−1 and EN+1, · · · , EN+6 are zero because they lie outside the interval
[0, 1]. Let us define error values at the end points as
Ej =
max0≤k≤4
{|Ek|}O(h6), −6 ≤ j < 0,
maxN−4≤k≤N
{|Ek|}O(h6), N < j ≤ N + 6.
(7.33)
By expanding (7.32), we get
(B+O(h5))E = 0.
These are equivalent to
(B1 +O(h5))E = O(h5) ∥ E ∥= O(h5),
190
Since for small h, the coefficient matrix B + O(h5) will be invertible and thus
using the standard result from linear algebra and effect of ||B−1||, we have
||E|| ≤(
||B−1||1−O(h6)
O(h5))= O(h3).
Hence ∥ E ∥= O(h3). The result is summarized in the following theorem.
Theorem 7.3.1. Let y be the exact solution of the system (7.1) and U , be the approxi-
mate solution of (7.1) then ∥ E ∥∞=∥ y − U ∥∞= O(h3).
7.4 Numerical results and discussion
In this section, we have solved four examples by using subdivision based nu-
merical algorithm to show the accuracy of our algorithm. Numerical results of
these examples are calculated by using MATLAB. We observed that accuracy
between the exact and approximate solutions is good.
Example 7.4.1. Consider the following singularly perturbed boundary value prob-
lem:
−εy(3) + y(x) = f(x), x ∈ [0, 1]
y(0) = 0, y(1) = 0, y(1)(0) = 0,(7.34)
where
f(x) = 6ε(1− x)5x3 − 6ε2[6(1− x)5 − 90x(1− x)4 + 180x2(1− x)3
−60x3(1− x)2].
The analytic solution of Example (7.34) is
y(x) = 6x3ε(1− x)5.
The numerical results of Example 7.4.1 for different values of N and ε are giv-
en in Tables 7.1 and 7.2. Graphical representation of these numerical results is
shown in Figures 7.1 and 7.2.
191
Example 7.4.2. Consider the following singularly perturbed boundary value prob-
lem:
−εy(3) + y(x) = f(x), x ∈ [0, 1]
y(0) = 0, y(1) = 0, y(2)(0) = 0,(7.35)
where
f(x) = 6ε(1− x)5x3 − 6ε2[6(1− x)5 − 90x(1− x)4 + 180x2(1− x)3
−60x3(1− x)2].
The analytic solution of (7.35) is y(x) = 6x3ε(1 − x)5. Numerical results of this
example is shown in Tables 7.3 and 7.4 for different values of N and ε . Graphical
representation of these numerical results is shown in Figures 7.3 and 7.4.
Example 7.4.3. Consider the following boundary value problems
−εy3(x) + y(x) = 81ε2 cos 3x+ 3ε sin 3x, x ∈ [0, 1]
y(0) = 0, y(1) = 3ε sin 3, y(1)(0) = 9ε,(7.36)
The analytic solution of the system (7.36)
y(x) = 3ε sin 3x.
The numerical results for Example 7.4.3 is given in Table 7.4. Graphical repre-
sentation of these numerical results is shown in Figures 7.5 and 7.6.
Example 7.4.4. Consider the following boundary value problems
−εy3(x) + y(x) = 81ε2 cos 3x+ 3ε sin 3x, x ∈ [0, 1]
y(0) = 0, y(1) = 3ε sin 3, y(2)(0) = 0.(7.37)
The analytic solution of the system (7.37)
y(x) = 3ε sin 3x.
Numerical results of (7.37) is shown in Table 7.5 . Graphical representation of
these numerical results is shown in Figures 7.7 and 7.8.
192
Table 7.1: Maximum absolute errors for N = 10 of Example 7.4.1
ε Our algorithm By Akram (2012)
116
9.5454E-04 2.9E-03132
4.2571E-04 9.2E-04164
1.7964E-04 1.4E-04
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
−6
x
Sol
utio
n
Analytic solution YApproximate solution U
Figure 7.1: Comparability of analytic and approximate solutions of Example 7.4.1 for
N = 100 with ε = 10−4 .
193
Tabl
e7.
2:M
axim
umab
solu
teer
rors
ofEx
ampl
e7.
4.1
εN
=10
N=
50N
=100
N=
150
N=
200
N=
250
0.1
1.60
53E
-03
7.48
32E
-04
7.14
46E
-04
7.09
10E
-04
7.07
44E
-04
7.06
94E
-04
0.01
1.00
97E
-04
4.73
93E
-05
4.68
96E
-05
4.74
64E
-05
4.78
77E
-05
4.81
67E
-05
0.001
9.86
78E
-06
1.56
62E
-06
1.19
65E
-06
1.38
00E
-06
1.49
73E
-06
1.57
54E
-06
0.0001
1.06
34E
-04
2.39
26E
-07
1.04
01E
-07
6.18
03E
-08
4.32
21E
-08
3.34
48E
-08
194
Tabl
e7.
3:M
axim
umab
solu
teer
rors
ofEx
ampl
e7.
4.2
εN
=10
N=
50N
=100
N=
150
N=
200
N=
250
0.1
1.61
90E
-02
7.33
71E
-04
6.44
63E
-04
6.36
71E
-04
6.34
96E
-04
6.34
46E
-04
0.01
5.47
77E
-04
3.53
02E
-05
3.27
08E
-05
3.30
05E
-05
3.33
31E
-05
3.35
79E
-05
0.001
4.38
14E
-05
2.41
50E
-06
1.39
66E
-06
1.15
44E
-06
1.23
48E
-06
1.29
13E
-06
0.0001
7.56
23E
-06
2.43
29E
-07
1.12
23E
-07
7.63
23E
-08
6.15
21E
-08
5.38
11E
-08
195
Table 7.4: Maximum absolute errors for N = 10 of Example 7.4.2
ε Our algorithm By Akram (2012)
116
6.2854E-03 1.3E-02132
1.9707E-03 3.2E-03164
3.9065E-04 3.4E-04
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
−6
x
Sol
utio
n
Analytic solution YApproximate solution U
Figure 7.2: Comparability of analytic and approximate solutions of Example 7.4.1 for
N = 200 with ε = 10−4 .
196
Tabl
e7.
5:M
axim
umab
solu
teer
rors
ofEx
ampl
e7.
4.3
εN
=10
N=
50N
=100
N=
150
N=
200
N=
250
0.001
6.38
64E
-04
9.24
76E
-04
1.00
85E
-03
1.03
71E
-03
1.05
11E
-03
1.05
93E
-03
0.0001
4.74
79E
-04
3.25
67E
-05
4.14
41E
-05
4.44
05E
-05
4.58
84E
-05
4.67
75E
-05
0.00001
···
2.30
18E
-06
1.43
49E
-06
1.72
61E
-06
1.88
11E
-06
1.97
31E
-06
197
Tabl
e7.
6:M
axim
umab
solu
teer
rors
ofEx
ampl
e7.
4.4
εN
=10
N=
50N
=100
N=
150
N=
200
N=
250
0.001
2.52
76E
-03
1.75
08E
-04
7.01
00E
-05
3.67
47E
-05
2.03
55E
-05
1.06
33E
-05
0.0001
1.99
38E
-03
2.52
43E
-05
1.14
40E
-05
7.34
69E
-06
5.37
22E
-06
4.20
54E
-06
0.00001
···
2.04
15E
-05
1.33
33E
-06
8.32
94E
-07
6.09
41E
-07
4.80
45E
-07
198
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
−6
X
Sol
utio
n
Analytic solution YApproximate solution U
Figure 7.3: Comparability of analytic and approximate solutions of Example 7.4.2 for
N = 100 with ε = 10−4 .
0 0.2 0.4 0.6 0.8 1−0.5
0
0.5
1
1.5
2
2.5
3
3.5x 10
−6
X
Sol
utio
n
Analytic solution YApproximate solution U
Figure 7.4: Comparability of analytic and approximate solutions of Example 7.4.2 for
N = 200 with ε = 10−4 .
199
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3x 10
−5
x
Solu
tion
Analytic solution YApproximate solution U
Figure 7.5: Comparability of analytic and approximate solutions of Example 7.4.3 for
N = 250 with ε = 10−5 .
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3x 10
−5
x
Solu
tion
Analytic solution YApproximate solution U
Figure 7.6: Comparability of analytic and approximate solutions of Example 7.4.3 for
N = 300 with ε = 10−5 .
200
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3x 10
−5
x
Solu
tion
Analytic solution YApproximate solution U
Figure 7.7: Comparability of analytic and approximate solutions of Example 7.4.4 for
N = 250 with ε = 10−5 .
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3x 10
−5
x
Solu
tion
Analytic solution YApproximate solution U
Figure 7.8: Comparability of analytic and approximate solutions of Example 7.4.4 for
N = 300 with ε = 10−5 .
201
7.5 Concluding remark
A binary interpolating subdivision scheme is used to construct a numerical al-
gorithm for solving third order singularly perturbed boundary value problem.
The algorithm is third order convergent. The numerical illustration shows that
the developed algorithm maintains a very remarkable high accuracy that makes
it very encouraging for dealing with the solution of singularly perturbed bound-
ary value problems. We compare numerical results with Akram (2012) algorith-
m and observed that our results are better than their results.
202
Chapter 8
Comparison, Conclusions,
Limitations and Future Work
In this chapter, we present the comparison, conclusion, limitations of proposed
algorithms and future research directions.
In this dissertation, we have presented the subdivision schemes based algo-
rithms for numerical solutions of the following types of problems:
• The linear boundary value problems of order three and four.
• The nonlinear third and fourth order boundary value problems.
• The second and third order singularly perturbed boundary value prob-
lems.
8.1 Comparison and Conclusion
In this section, first we present a comparison of proposed algorithms with other
numerical methods in literature and then we give concluding remarks about
our results.
203
8.1.1 Comparison with existing methods
In this section, we present comparison of our numerical algorithms with other
existing methods.
We compared numerical results by proposed algorithms for third order lin-
ear and nonlinear BVP’s with the numerical method based on fourth degree B-
splines of Caglar et al. (1999). We observed that our algorithm is more efficient
than that of Caglar et al.’s method. The order of approximation of proposed
algorithm and modified Adomian decomposition method of Hasan (2012) is
same. We have presented two collocation algorithms based on interpolating
and approximating subdivision schemes for the solution of linear fourth order
boundary value problems. From the numerical results, we observed that ap-
proximating subdivision scheme based collocation algorithm gives better result-
s comparative to the second order finite difference method. It is also observed
that, approximating subdivision scheme based collocation algorithm and quin-
tic spline based collocation algorithm have same order of approximation. The
proposed iterative algorithms for nonlinear fourth order BVP’s have been ap-
plied on different problems. Numerical results show that the accuracy of ap-
proximate solution is O(h3). It has been observed that proposed algorithm for
the solution of second order singularly perturbed boundary value problems is
better than spline based methods by Aziz and Khan (2002), Bawa and Natesan
(2005), Khan and Aziz (2005), Kadalbajoo and Aggarwal (2005) , finite difference
methods by Lubuma (2006), Miller (1979), Niijima (1980a), Niijima (1980b) and
Haar wavelet methods by Kumar and Mehra (2002), Pandit and Kumar (2014).
Numerical results of singulary third order boundary value problems by pro-
posed algorithm showed that our results are better than the results of Akram
(2012).
204
8.1.2 Conclusion
It is concluded that the subdivision based collocation algorithm is an ideal tool
to attain the numerical solutions of boundary value problems. The proposed al-
gorithms are computationally efficient and can be easily implemented on com-
puter. The numerical results showed that the adjustment of boundary condi-
tions at the end points influence the accuracy of the approximate solution. That
is, the accuracy of the solution can be improved by the proper adjustment of
boundary conditions. So our algorithm has flexibility to improve the results
by adjusting boundary conditions. The automatic selection and adjustment of
the boundary conditions to improve the approximation order of the solution is
possible future research direction.
8.2 Limitations of algorithms
In this section, we presented limitations of our work. These are the followings
• The proposed algorithms are applicable only for two point boundary val-
ue problems.
• We used only primal symmetric binary subdivision schemes, either they
are interpolating or approximating, to construct a subdivision based nu-
merical collocation algorithms.
• Our algorithms are applicable for all types of boundary value problems
with either the coefficients are polynomial or trigonometric functions.
• Our proposed algorithms are not applicable when the boundary condi-
tions are defined other than the interval 0 6 x 6 1.
205
8.3 Future Work
The proposed subdivision based collocation algorithms are limited to two point
boundary value problems. So the extension of these algorithms for the numer-
ical solutions of n-point nth order boundary value problems is possible future
research direction.
206
Bibliography
Abdullah, A. S., Majid, Z. A. and Senu, N. (2013) Solving third order
boundary value problem using fourth order block method. Applied Mathe-
matical Sciences. 7(53): 2629-2645.
Ablasiny, E. L. and Hoskins, W.D. (1969) Cubic spline solutions to
two-point boundary value problems. Computer Journal . 12: 151-153.
Agarwal, R. P. ( 1973) Non-linear two point boundary value prob-
lems. F. C. Auluck, FNA. 4(7 and 8).
Agarwal, R. P. (1986) Boundary value problems for higher order dif-
ferential equations. World Scientific publisher, Singapore.
Akram, G. (2012) Quartic spline solution of a third order singularly
perturbed boundary value problems. The ANZIAM Journal. 53(E): E44-E58.
Ascher, U. M., Mattheij, R. M. M. and Russell, R. D. (1988) Numerical
solution of boundary value problems for ordinary differential equations.
Prentice-Hall, Englewood cliffs, NJ.
Aslam, M., Mustafa, G. and Ghaffar, A. (2011) (2n − 1)-point ternary
207
aproximating and interpolating subdivision schemes. Journal of Applied
Mathematics. 2011: Article ID 832630, 12 pages.
Aziz, T. and Khan, A. (2002) A spline method for second order sin-
gularly pertubed boundary value problems. Journal of Computational and
Applied Mathematics. 147: 445-452.
Baccou, J. and Liandrat, J. (2013) Kriging-based interpolatory subdi-
vision schemes. Applied and Computational Harmonic Analysis. 35: 228-250.
Bawa, R. K. and Natesan, S. (2005) A computational method for self
adjoint singular perturbation problem using quintic spline. Computers and
Mathematics with Applications. 50: 1371-1382.
Bhrawy, A. H. and Abd-Elhameed, W. M. (2011) New algorithm for
the numerical solutions of nonlinear third-order differential equations us-
ing jacobi-gauss collocation method. Mathematical Problems in Engineering.
2011: Article ID 837218, 14 pages.
Bickley, W. G. (1968) Piecewise cubic interpolation and two-point
boundary value problems. Computer Journal. 11: 206-208.
Caglar, H. N., Caglar, S. H. and Twizell, E. H. (1999) The numerical
solution of third-order boundary value problems with fourth-degree B-
spline functions. International Journal of Computer Mathematics. 71: 373-381.
Conti, C. and Hormann, K. (2011) Polynomial reproduction for uni-
variate subdivision scheme of any arity. Approximation Theory. 163: 413-437.
208
Craster, R. V. and Matar, O. K. (2009) Dynamics and stability of thin
liquid films. Review of Modern Physics. 81(3): 1131-1198.
Cui, M. and Geng, F. (2008) A computational method for solving third
order singularly perturbed boundary value problems. Applied Mathematics
and Computation. 198: 896-903.
de Rham, G. (1956) Sur une courbe plane, Journal de Mathmatiques
Pures et Appliques. 35: 25-42.
Deng, C., Lia, Y. and Xu, H. (2016) Repeated local operations for m-
ary 2N -point Dubuc-Deslauriers subdivision schemes. Computer Aided
Geometric Design. 44: 10-14.
Deng, C. and Ma, W. (2013) A unified interpolatory subdivision schemes
for quadrilateral meshes. ACM Transactions on Graphics. 32(3): Article No.
23, 11 pages.
Deng, C. and Ma, W. (2016) Efficient evaluation of subdivision schemes
with polynomial reproduction property. Journal of Computational and
Applied Mathematics. 294: 403-412.
Deslauriers, G. and Dubuc, S. (1989) Symmetric iterative interpola-
tion processes. Constructive Approximation. 5: 49-68.
Duan, J. S. and Rach, R. (2011) A new modification of the Adomian
decomposition method for solving boundary value problems for higher
209
order nonlinear differential equations. Applied Mathematics and Computa-
tion. 218: 4090-4118.
Duffy, B. R. and Wilson, S. K. (1997) A third order differential equation
arising in thin-film flows and relvent to tanner’s law. Applied Mathematics
Letters. 10(3): 63-68.
Dyn, N. (2002) Tutorial on multiresolution in geometric modelling
summer school lecture notes series. Mathematics and Visualization, (Iske,
Armin; Quak, Ewald; Floater, Michael S. (Eds.)) Springer. ISBN: 3-540-43639-1.
Dyn, N., Heard, A., Hormann, K. and Sharon, N. (2015) Univariate
subdivision schemes for noisy data with geometric applications. Computer
Aided Geometric Design. 37: 85-104.
Dyn, N., Levin, D. and Gregory, J. (1987) A 4-point interpolatory
subdivision scheme for curve design. Computer Aided Geometric Design.
4(4): 257-268.
Ejaz, S. T. and Mustafa, G. (2016) A subdivision based iterative collo-
cation algorithm for nonlinear third order boundary value problems.
Advances in Mathematical Physics. 2016: Article ID 5026504, 15 pages.
Ejaz, S. T., Mustafa, G. and Khan, F. (2015) Subdivision schemes based col-
location algorithms for solution of fourth order boundary value problems.
Mathematical Problems in Engineering. 2015: Article ID 240138, 18 pages.
El-Danaf, T. S. (2008) Quartic non polynomial spline solutions for
210
third order two-point boundary value problem. International Journal of
Mathematical, Computational, Physical, Electrical and Computer Engineering.
2(9): 453-456.
El-Salam, F. A. Abd and Zaki, Z.A. (2010) The numerical solution of
linear fourth order boundary value problems using non polynomial spline
technique. Journal of American Science. 6(12): 310-316.
Ford, W. F. (1992) A third order differential equation. SIAM Review.
34: 121-122.
Fyfe, D. J. (1969) The use of cubic splines in the solution of two-point
boundary value problems. Computer Journal. 12: 188-192.
Fyfe, D. J. (1970) The use of cubic splines in the solution of certain
fourth order boundary value problems. Computer Journal. 13: 204-205.
Gupta Y. and Srivastava P. K. (2011) A computational method for
solving two point boundary value problems of order four. International
Journal of Computer Technology and Applications. 2(5): 1426-1431.
Hamid, N. N. A., Majid, A. A. and Ismail, A. I. Md., (2012) Quartic
B-spline interpolation method for linear two-point boundary value prob-
lem. World Applied Sciences Journal. 17 (Special Issue of Applied Math):
39-43.
Haq, F-i., Hussain, I. and Ali, A. (2011) A haar wavelets based nu-
merical method for third-order boundary and initial value problems.
211
World Applied Sciences Journal. 13(10): 2244-2251.
Hasan, Y. Q. (2012) The numerical solution of third-order boundary
value problems by the modified decomposition method. Advances in
Intelligent Transportation Systems. 1(3): 71-74.
Hassan, M. F. and Dodgson, N. A. (2003) Ternary and three-point u-
nivariate subdivision schemes. in: A. Cohen, J. L. Marrien, L. L. Schumaker
(Eds.), Curve and Surface Fitting: Sant-Malo 2002. Nashboro Press, Brent-
wood pp. 199-208.
Hassan, M. F., Ivrissimitzis, I. P., Dodgson, N. A. and Sabin, M. A.
(2002) An interpolating 4-point C2 ternary stationary subdivision scheme.
Computer Aided Geometric Design. 19: 1-18.
Islam, Md. S. and Shirin, A. (2011) Numerical solutions of a class of
second order boundary value problems on using bernoulli polynomials.
Applied Mathematics. 2: 1059-1067.
Kadalbajoo, M. K. and Aggarwal, V. K. (2005) Fitted mesh B-spline
collocation method for solving self-adjoint singularly perturbed boundary
value problems. Applied Mathematics and Computation. 161: 973-987.
Kadalbajoo, M. K. and Patidar, K. C. (2002) Numerical solution of
singularly perturbed two point boundary value problems by spline in
tension. Applied Mathematics and Computations. 131: 299-320.
Kalyani, P. and Rama Chandra Rao, P. S. (2013) Solution of boundary
212
value problems by approaching spline techniques. International Journal of
Engineering Mathematics. 2013: Article ID 482050, 9 pages.
Khan, I. and Aziz, T. (2005) Tension spline method for second order
singularly perturbed boundary value problems. International Journal of
Computing Science and Mathematics. 82: 1547-1553.
Kilic, E. and Stanica, P. (2013) The inverse of banded matrix. Journal
of Computational and Applied Mathematics. 237: 126-135.
Ko, K. P., Lee, B. G. and Yoon, D. G. J. (2007) A study on the mask
of interpolatory symmetric subdivision schemes. Applied Mathematics and
Computation. 187: 609-621.
Kumar, M. (2002) A fourth order finite difference method for a class
of singular two point boundary value problems. Applied Mathematics and
Computation. 133: 539-545.
Kumar, V. and Mehra, M. (2009) Wavelet optimized finite difference
method using interpolating wavelets for self-adjoint singularly perturbed
problems. Journal of Computational and Applied Mathematics. 230: 803-812.
Lee, B. G., Lee, Y. J. and Yoon, J. (2006) Stationary binary subdivision
schemes using radial basis function interpolation. Advances in Computa-
tional Mathematics. 25: 57-72.
Lian, J.-ao (2008) On a-ary subdivision for curve design: I. 3-point
and 5-point interpolatory schemes. Applications and Applied Mathematics:
213
An International Journal. 3(2): 176-187.
Lian, J.-ao (2009) On a-ary subdivision for curve design: I. 2m-point
and (2m + 1)-point interpolatory schemes. Applications and Applied Mathe-
matics: An International Journal. 4(1): 434-444.
Lubuma, J. M. S. and Patidar, K. C. (2006) Uniformly convergent non-
standard finite difference methods for self-adjoint singularly perturbed
problems. Journal of Computational and Applied Mathematics. 191: 228-238.
Miller, J. J. H. (1979) On the convergence, uniformly in ε, of differ-
ence schemes for a two point singular pertubation problem. Proceedings
Conference on Numerical Analysis of Singular Perturbation Problems. (Aca-
demic press, New york). 467-474.
Momonait, E. (2011) Numerical investigation of a third order ODE
from thin film flow. Maccanica. 46: 313-323.
Mustafa, G., Abbas, M., Ejaz, S. T., Ismail, A. I. M. and Khan, F. (2017) A
numerical approach based on subdivision schemes for solving non-linear
fourth order boundary value problems. Journal of Computational Analysis
and Applications. 23(4): 607-623.
Mustafa, G., Ashraf, P. and Aslam, M. (2014a) Binary univariate dual
and primal subdivision schemes. SeMA Journal. 65(1): 23-35.
Mustafa, G., Ashraf, P. and Deng, J. (2014b) Generalized and unified
families of interpolating subdivision schemes. Numerical Mathematics
214
Theory Methods and Applications. 7(2): 193-213.
Mustafa, G., Deng, J., Ashraf, P. and Rehman, N. A. (2012) The mask
of odd point n-ary interpolating subdivision scheme. Journal of Applied
Mathematics. 2012: Article ID 205863, 20 pages.
Mustafa, G. and Ejaz, S. T. (2014) Numerical solution of two point
boundary value problems by interpolating subdivision schemes. Abstract
and Applied Analysis. 2014: Article ID: 721314, 13 pages.
Mustafa, G. and Ejaz, S. T. (2017) A Subdivision collocation method
for solving two point boundary value problems of order three. Journal of
Applied Analysis and Computation. 2017: Accepted.
Mustafa, G. and Khan, F. (2009) A new 4-point C3 quaternary ap-
proximating subdivision scheme. Abstract and Applied Analysis. 2009
Article ID: 301967, 14 pages.
Mustafa, G., Li, H., Zhang, J. and Deng, J. (2015) l1-Regression based
subdivision schemes for noisy data. Computer-Aided Design. 58: 189-199.
Mustafa, G. and Rehman, N. A. (2010) The mask of (2b + 4)-point n-
ary subdivision scheme. Computing: Archieves for Scientific Computing. 90:
1-14.
Niijima, K. (1980a) On a finite difference scheme for a singular per-
turbation problem without a first derivative term I. Memoirs of Numerical
Mathematics. 7: 1-10.
215
Niijima, K. (1980b) On a finite difference scheme for a singular per-
turbation problem without a first derivative term II. Memoirs of Numerical
Mathematics. 7: 11-27.
Oron, A. S., Davis, H. and Bankoff, S. G. (1997) Long-scale evolution
of thin liquid films. Review of Modern Physics. 69(3): 931-980.
Pandey, P. K. (2016) Solving third order boundary value problems
with quartic splines. SpringerPlus. 5(1): 326, 10 pages.
Pandit, S. and Kumar, M. (2014) Haar wavelet approch for numerical
solution of two parameters singularly perturbed boundary value prob-
lems. Applied Mathematics & Information Sciences. 8: 2965-2974.
Qu, R. (1994) Curve and surface interpolation by subdivision algo-
rithms. Computer Aided Drafting Design and Manufacturing. 4(2): 28-39.
Qu, R. (1996) A new approach to numerical differentiation and inte-
gration. Mathematical and Computer Modelling. 24: 55-68.
Qu, R. and Agarwal, R. P. (1995) A cross difference approach to the
analysis of subdivision algorithms. Neural, Parallel and Scientific Computa-
tions. 3(3): 393-416.
Qu, R. and Agarwal, R. P. (1996) Solving two point boundary value
problems by interpolatory subdivision algorithms. International Journal of
Computer Mathematics. 60: 279-294.
216
Qu, R. and Agarwal, R. P. (1997a) A collocation method for solving a
class of singular nonlinear two-point boundary value problems. Journal of
Computational and Applied Mathematics. 83: 147-163.
Qu, R. and Agarwal, R. P. (1997b) An iterative scheme for solving
nonlinear two point boundary value problems. International Journal of
Computer Mathematics. 64(3-4): 285-302.
Qu, R. and Agarwal, R. P. (1998) A subdivision approach to the con-
struction of approximate solutions of boundary-value problems with
deviating arguments. Journal of Computers and Mathematics with Applica-
tions. 35(11): 121-135.
Rahman, M. M., Hossen, M. A., Islam, M. N. and Ali, Md. S. (2012)
Numerical solutions of second order boundary value problems by
Galerkin method with Hermite polynomials. Annals of Pure and Applied
Mathematics. 1(2): 138-148.
Rehan, K. and Siddiqi, S. S. (2015) A combined binary 6-points sub-
division scheme. Applied Mathematics and Computation. 270: 130-135.
Russell, R. D. and Shampine, L. F. (1972) A collocation method for
boundary value problems. Numerische Mathematik. 19(1): 1-28.
Sakai, M. (1971) Piecewise cubic interpolation and two-point bound-
ary value problems. Publications of the Research Institute for Mathematical
Sciences. 7: 345-362.
217
Sakai, M. and Usmani, R. (1983) Spline solutions for nonlinear fourth-order
two-point boundary value problems. Publication of the Research Institute for
Mathematical Sciences. 19: 135-144.
Si, X., Baccou, J. and Liandrat, J. (2016) On four-point penalized La-
grange subdivision schemes. Applied Mathematics and Computation. 281:
278-299.
Siddiqi S. S. and Akram, G. (2008) Quintic spline solutions of fourth
order boundary-value problems. International Journal of Numerical Analysis
and Modeling. 5(1): 101-111.
Strang, G. (2011) Linear algebra and its applications, fourth edition.
Cengage Learning India Private Limited. ISBN-10: 81-315-0172-8.
Su-rang, L., Gen-bao, T. and Zong-chi, L. (2001) Singularly perturba-
tion of boundary value problem for quasilinear third order ordinary
differential equations involing two small parameters. Applied Mathematics
and Mechanics. 22(2): 229-236.
Tanner, L. H. (1979) The spreading of silicone oil drops on horizontal
surfaces. Journal of Physics D: Applied Physics. 12(9): 1473-1485.
Tirmizi, I. A., i-Haq, F. and ul-Islam, S. (2008) Non-polynomial spline
solution of singularly perturbed boundary value problems. Applied Mathe-
matics and Computation. 196: 6-16.
218
Tuck, E. O. and Schwartz, L. W. (1990) A numerical and asymptotic
study of some third order ordinary differential equations relevent to
draining and coating flows. SIAM Review. 32(3): 453-469.
Usmani, R. A. (1983) Finite difference methods for a certain two point
boundary value problems. Indian Journal of Pure and Applied Mathematics.
14(3): 398-411.
Valarmatht, S. and Ramanujam, N. (2002) Boundary value technique
to boundary value problems for third order singularly perturbed ordinary
differential equations. International Journal of Computer Mathematics, 79(6):
747-763.
Wang, J. and Zhang, Z. (1998) A boundary value problem from draining
and coating flows involving a third-order ordinary differential equation.
Zeitschrift fur Angewandte Mathematik und Physik. 49: 506-513.
Wazwaz, A. M. (2000) The modified Adomian decomposition method
for solving linear and nonlinear boundary value prblems of tenth-order
and 12th-order. International Journal of Nonlinear Sciences and Numerical
Simulation. 1: 17-24.
Wazwaz, A. M. (2001a) A new algorithm for solving boundary for
higher-order integro-differential equations. Applied Mathematics and Com-
putation. 118: 327-342.
Wazwaz, A. M. (2001b) A reliable algorithm for obtaining positive
solutions for nonlinear boundary value problems. Computers & Mathemat-
219
ics with Applications. 41: 1237-1246.
Wazwaz, A. M. (2001c) Blow-up for solution of some linear wave e-
quations with mixed nonlinear boundary conditions. Applied Mathematics
and Computation. 123: 133-140.
Wazwaz, A. M. (2001d) The numerical solution of fifth order BVP by
the decomposition method. Journal of Computational and Applied Mathemat-
ics. 136: 259-270.
Yalcinbas, S., Bicer, K. E. and Tastekin, D. (2016) Fermat collocation
method for the solutions of nonlinear system second order boundary
value problems. New Trends in Mathematical Sciences. 4(1): 87-96.
Zheng, Hong-C., Huang, Shu-P., Guo, F. and Peng, Guo-H. (2014)
Designing multi-parameter curve subdivision schemes with high continu-
ity. Applied Mathematics and Computation. 243: 197-208.
Zheng, H., Hu, M. and Peng, G. (2009a) Constructing (2n − 1)-point
ternary interpolatory subdivision schemes by using variation of constants.
International Conference on Computational Intelligence and Software Engineer-
ing. 2009: 1-4.
Zheng, H., Hu, M. and Peng, G. (2009b) Ternary even symmetric
2n-point subdivision. In 2009 International Conference on Computational
Intelligence and Software Engineering. 2009: 1-4.
Zheng, H., Hu, M. and Peng, G. (2009c) p-ary subdivision generaliz-
220
ing B-splines. In 2009 Second International Conference on Computer and
Electrical Engineering. 2009: 214-218.
221
Publications of Syeda Tehmina Ejaz
1. Numerical solution of two-point boundary value problems by interpolat-
ing subdivision schemes, Abstract and Applied Analysis, Vol. 2014, Article
ID 721314, 13 pages, (2014).
2. Subdivision schemes based collocation algorithms for solution of fourth
order boundary value problems, Mathematical Problems in Engineering, Vol.
2015, Article ID 240138, 18 pages, (2015). Impact factor = 0.644
3. A subdivision based iterative collocation algorithm for nonlinear third or-
der boundary value problems, Advances in Mathematical Physics, Vol. 2016,
Article ID 5026504, 15 pages, (2016). Impact factor = 1.12
4. A numerical approach based on subdivision schemes for solving nonlinear
fourth order boundary value problems, Journal of Computational Analysis
and Applications, Vol. 23(4), page no. 607-623, (2017). Impact factor = 0.481
5. Subdivision based collocation method for singularly perturbed boundary
value problems. Article submitted.
6. A subdivision collocation method for solving two point boundary value
problems of order three. Accepted in Journal of Applied Analysis and Com-
putation, 2017. Impact factor = 0.844
222