akramIMF41-44-2012

International Mathematical Forum, Vol. 7, 2012, no. 44, 2179 - 2190

Solution of a Fourth Order Singularly Perturbed

Boundary Value Problem Using Quintic Spline

Ghazala Akram and Nadia Amin

Department of MathematicsUniversity of the PunjabLahore 54590, Pakistan

[email protected]@hotmail.com

Abstract

Singularly perturbed boundary value problem can be solved using var-ious techniques. The solution of the following fourth order self adjointsingularly perturbed boundary value problem is approximated usingquintic spline

Ly = −εy(4) + p(x)y = f(x), p(x) ≥ p > 0,

y(a) = α0, y(b) = α1, y(1)(a) = α2, y(1)(b) = α3,

}

or

Ly = −εy(4) + p(x)y = f(x), p(x) ≥ p > 0,

y(a) = α0, y(b) = α1, y(2)(a) = α4, y(2)(b) = α5,

}

Convergence analysis of the method confirms second order convergence.The numerical description of the method is shown by two examples.

Keywords: Singularly perturbed boundary value problems; Quintic spline;Self adjoint; Transition Layer; Uniform convergence

1 Introduction

The solution of singularly perturbed boundary value problem exhibits a multiscale character. Since there is a thin transition layer, where the solution variesrapidly, while away from the layer the solution behaves regularly and variesgradually, therefore many complications may be faced in solving singularlyperturbed boundary value problems using standard numerical methods. Inrecent years, a large number of special purpose methods have been establishedto provide accurate results.

2180 Ghazala Akram and Nadia Amin

Consider the self adjoint singularly perturbed boundary value problem ofthe form:

Ly = −εy(4) + p(x)y = f(x), p(x) ≥ p > 0,y(a) = α0, y(b) = α1, y(1)(a) = α2, y(1)(b) = α3,

}(1.1)

or

Ly = −εy(4) + p(x)y = f(x), p(x) ≥ p > 0,y(a) = α0, y(b) = α1, y(2)(a) = α4, y(2)(b) = α5,

}(1.2)

where α0, α1, α2, α3, α4 and α5 are constants and ε is a small positive param-eter (0 < ε ≤ 1), also f(x) and p(x) are smooth functions. In this problemp(x) = p = constant.Singularly perturbed problems are very famous in the field of science and engi-neering e.g., fluid dynamics, quantum mechanics, optimal control, convection-diffusion processes and chemical reactor theory etc. Singularly perturbedboundary value problems have been solved numerically using three basic tech-niques named as finite difference method, finite element method and splineapproximation method.

Second order singularly perturbed boundary value problem with boundarylayer at one end is replaced by three first order initial value problems(IVPs)and these problems have been solved using numerical patching method in [11].Second order singularly perturbed boundary value problem has been convertedinto IVP for system of two first order ODEs which are solved using two nu-merical schemes namely, classical and exponentially fitted difference schemeand the adaptive single-step exponential fitting scheme in [15]. Various typesof uniformly convergent mesh and finite difference schemes for the solutionof second order singularly perturbed boundary value problem with numericalexperiments have been discussed in [5]. A new approach has been defined tosolve second order singularly perturbed boundary value problem in which theinner and outer region of the problem is considered as two point boundary layercorrection and initial value problem respectively in [7]. A numerical methodhas been proposed to define second order semi linear singularly perturbedboundary value problem with violating stability condition described uniformconvergence in [16]. Second order singularly perturbed boundary value prob-lem has been solved using spline approach where singular and non singularcases are discussed in [10]. Second order singularly perturbed boundary valueproblem using difference scheme based on quintic spline has been considered,in which the domain of definition is converted into three non overlapping subdomains and this scheme shows fourth order accuracy in [4]. Second order sin-gularly perturbed boundary value problem has been solved using fourth ordermethod based on quintic spline in [2]. Sextic spline has been used to solve thesecond order singularly perturbed boundary value problem and the method

Solution of a fourth order singularly perturbed BVP 2181

is proved to be fifth order accurate in [6]. A summary of linear, non linearsecond order two point singular boundary value problems, two point singu-larly perturbed BVPs in ordinary differential equations and singularly ellipticBVPs in partial differential equation has been discussed using various compu-tational techniques in [8]. Third order singularly perturbed boundary valueproblem using quartic spline has been solved and the method is proved to besecond order accurate [1]. Solution of third order singularly perturbed bound-ary value problem has been developed using sextic spline and the convergenceanalysis is shown to have third order convergence in [14]. Second and fourthorder method have been developed using quintic non polynomial spline for thesolution of fourth order two point boundary value problem in [9]. Third andfourth order singularly perturbed boundary value problem having discontinu-ous source term has been solved using asymptotic finite element method, wherethese problems are transformed into weakly coupled system according to thetype of boundary conditions in [3]. Fourth order singularly perturbed bound-ary value problem has been converted into two linear and nonlinear ODEsin [12] with the suitable boundary conditions and the domain of definition isdivided into two non overlapping sub intervals where the non linear equationhas been solved using Newton’s method of quasilinearization. Fourth ordersingularly perturbed two point boundary value problem has been reduced intotwo ODEs in [13] w.r.t suitable boundary conditions where the domain of def-inition is transformed into three non overlapping subintervals and Newton’smethod of quasilinearization is used for the solution of nonlinear equation.

The paper is organized in four sections. In section 2, the consistency rela-tion and end conditions required for the solution of BVP (1.1) and (1.2) aredetermined. In section 3, the convergence analysis of the quintic spline methodis discussed. Finally in fourth section, numerical results are illustrated.

2 Consistency Relations

To develop the consistency relations the following fifth degree spline is consid-ered:

Si(x) = ai(x−xi)5+bi(x−xi)

4 +ci(x−xi)3+di(x−xi)

2 +ei(x−xi)+fi (2.1)

defined on [a, b], where x ∈ [xi, xi+1] with equally spaced knots,

xi = a + ih, i = 0, 1, ..., N, and h = (b−a)N

.Using the following notations

Si(xi) = yi, Si(xi+1) = yi+1,

S(1)i (xi) = mi, S

(2)i (xi) = ki,

S(4)i (xi) = Ni, S

(4)i (xi+1) = Ni+1.

⎫⎪⎪⎬⎪⎪⎭


the coefficients in (2.1) can be determined, as

ai = −(Ni − Ni+1)

120h,

bi =Ni

24,

ci = −(60h2ki + 120hmi + 4h4Ni + h4Ni+1 + 120yi − 120yi+1)

120h3,

di =ki

2,

ei = mi,

fi = yi.

Applying the first, second and third derivative continuities at knots i.e S(µ)i−1(xi) =

S(µ)i (xi), for μ = 1, 2 and 3, the following relations can be obtained as

−2ki−1 − ki +(−360hmi−1 + 8h4Ni−1 + 7h4Ni − 360yi−1 + 360yi)

60h2= 0,

1

20h3(−60h2ki−1 + 60h2ki − 120hmi−1 + 120hmi + 6h4Ni−1 + 13h4Ni

+h4Ni+1 − 120yi−1 + 240yi − 120yi+1) = 0,

−1

2hki−1 − 2mi−1 − mi +

1

40h3Ni−1 +

h3Ni

60− 3yi−1

h+

3yi

h= 0,

which leads the following consistency relation in terms of Ni and yi

h4Ni−2 + 26h4Ni−1 + 66h4Ni + 26h4Ni+1 + h4Ni+2

= 120yi−2 − 480yi−1 + 720yi − 480yi+1 + 120yi+2, i = 2, 3, ..., N − 2.(2.2)

Using Eq. (1.1), the Eq. (2.2) can be written as

(ph4 − 120ε)yi−2 + (26ph4 + 480ε)yi−1 + (66ph4 − 720ε)yi + (26ph4

+480ε)yi+1 + (ph4 − 120ε)yi+2 = h4(fi−2 + 26fi−1 + 66fi + 26fi+1 + fi+2),

i = 2, ..., N − 2.(2.3)

Since above system consists of (N − 3) equations with (N − 1) unknowns, sotwo more equations are required. The following relations describing truncationerrors are used in this regard

T1 = h4(a0N0 + a1N1 + a2N2 + a3N3 + a4N4 + a5N5) − (a6y0 + a7y1

+a8y2 + a9y3 + a10hy(1)0 ), (2.4)

TN−1 = h4(b0NN + b1NN−1 + b2NN−2 + b3NN−3 + b4NN−4)

+b5NN−5 − (b6yN + b7yN−1 + b8yN−2 + b9yN−3 + b10hy(1)N ). (2.5)


Using Taylor’s series for the right hand side of Eq. (2.4) along with the coef-

ficients of h−4y0, h−3y

(1)0 , h−2y

(2)0 , h−1y

(3)0 , y

(4)0 , hy

(5)0 , h2y

(6)0 , h3y

(7)0 , h4y

(8)0 , h5y

(8)0 ,

the value of ais can be calculated, as

a0 = 1, a1 =18240

937, a2 =

5990

937,

a3 =140

937, a4 = −135

937, a5 =

28

937,

a6 = −184800

937, a7 =

302400

937, a8 = −151200

937,

a9 =33600

937, a10 = −100800

937.

Using the values of ais in Eq. (2.4), the required end conditions for i = 1 andi = N − 1 have been determined, as

(18240ph4 − 302400ε)y1 + (5990ph4 + 1512000ε)y2 + (140ph4

−33600ε)y3 − 135ph4y4 + 28ph4y5 − h4(937f0 + 18240f1

+5990f2 + 140f3 − 135f4 + 28f5) + (937ph4 + 184800ε)α0

+100800hεα2 + O(h6) = 0.

(2.6)

(18240ph4 − 302400ε)yN−1 + (5990ph4 + 1512000ε)yN−2 + (140ph4

−33600ε)yN−3 − 135ph4yN−4 + 28ph4yN−5 − h4(937fN + 18240fN−1

+5990fN−2 + 140fN−3 − 135fN−4 + 28fN−5) + (937ph4 + 184800ε)α1

+100800hεα3 + O(h6) = 0.

(2.7)

Similarly, the end conditions for the system (1.2) can be derived as

(43274ph4 − 302400ε)y1 + (5662ph4 + 241920ε)y2 + (3432ph4 − 60480ε)y3

−1391ph4y4 + 230ph4y5 − h4(4233f0 + 43274f1 + 5662f2 + 3432f3

−1391f4 + 230f5) + (4233ph4 + 120960ε)α0 − 60480h2εα4 + O(h6) = 0,

(2.8)

(43274ph4 − 302400ε)yN−1 + (5662ph4 + 241920ε)yN−2 + (3432ph4

−60480ε)yN−3 − 1391ph4yN−4 + 230ph4yN−5 − h4(4233fN + 43274fN−1

+5662fN−2 + 3432fN−3 − 1391fN−4 + 230fN−5) + (4233ph4 + 120960ε)α1

−60480h2εα5 + O(h6) = 0.

(2.9)


3 Convergence of the Method

The system of Eqns. (2.6), (2.3) and (2.7) provides the required quintic splinesolution of BVP (1.1), which can be written in the following matrix form

AY − h4DF = C, (3.1)

where

A = [A1 A2]

A1 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

18240ph4 − 302400ε 5990ph4 + 151200ε 140ph4 − 33600ε26ph4 + 480ε 66ph4 − 720ε 26ph4 + 480εph4 − 120ε 26ph4 + 480ε 66ph4 − 720ε

. . .. . .

. . .

ph4 − 120ε 26ph4 + 480εph4 − 120ε

28ph4 −135ph4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

A2 =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

−135ph4 28ph4

ph4 − 120ε26ph4 + 480ε ph4 − 120ε

. . .. . .

. . .

66ph4 − 720ε 26ph4 + 480ε ph4 − 120ε26ph4 + 480ε 66ph4 − 720ε 26ph4 + 480ε

140ph4 − 33600ε 5990ph4 + 151200ε 18240ph4 − 302400ε

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

D =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

18240 5990 140 −135 2826 66 26 11 26 66 26 1

. . .. . .

. . .. . .

. . .

1 26 66 26 11 26 66 26

28 −135 140 5990 18240

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

C = (c1, c2, ..., cN−2, cN−1)T , Y = (y1, y2, ..., yN−2, yN−1)

T and F = (f1, f2, ..., fN−2, fN−1)T .

Also

c1 = −(937ph4 + 184800ε)α0 + 937h4f0 − 100800hεα2,

c2 = −(ph4 − 120ε)α0 + h4f0,

ci = 0, i = 3, 4, ..., N − 3,

cN−2 = −(ph4 − 120ε)α1 + h4fN ,

and

cN−1 = −(937ph4 + 184800ε)α1 + 937h4fN − 100800hεα3.


If Y = [y(x1), y(x2), . . ., y(xN−1)]T denotes the exact solution then from Eq.

(3.1), it can be written as

AY − h4DF = T + C, (3.2)

where T = [t1, t2, . . ., tN−1]T denotes the truncation error and calculated, as

t1 = 35937

ε h6 y(10) (ζ1), x0 < ζ1 < x2,ti = 10 ε h6 y(6) (ζi), xi−1 < ζi < xi+1, i = 2, 3, ..., N − 2,

andtN−1 = 334

7055ε h6 y(10) (ζN−1), xN−2 < ζN−1 < xN .

⎫⎪⎪⎪⎬⎪⎪⎪⎭ (3.3)

Moreover,

A(Y − Y ) = AE = T, (3.4)

E = Y − Y = (e1, e2, ..., eN−1)T . (3.5)

To determine the error bound the row sums S1, S2, ..., SN−1 of matrix A arecalculated, as

S1 =∑

j a1,j = 24263ph4 − 184800ε,S2 =

∑j a2,j = 119ph4 + 120ε,

Si =∑

j ai,j = 120ph4, i = 3, 4, ..., N − 3,SN−2 =

∑j aN−2,j = 119ph4 + 120ε,

andSN−1 =

∑j aN−1,j = 24263ph4 − 184800ε.

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎭

(3.6)

Since matrix A has been observed to be irreducible and monotone, thereforeA−1 exists and its elements are non negative. Hence following results can beobtained from Eq. (3.4)

E = A−1T. (3.7)

Also, from the theory of matrices it can be written as

A−1A = I(N−1)×(N−1), (3.8)

where

A =

⎡⎢⎢⎢⎢⎣

a1,1 a1,2 · · · a1,n−1

a2,1 a2,2 · · · a2,n−1...

......

...an−1,1 an−1,2 · · · an−1,n−1

⎤⎥⎥⎥⎥⎦ , (3.9)

A−1 =

⎡⎢⎢⎢⎢⎣

a−11,1 a−1

1,2 · · · a−11,n−1

a−12,1 a−1

2,2 · · · a−12,n−1

......

......

a−1n−1,1 a−1

n−1,2 · · · a−1n−1,n−1

⎤⎥⎥⎥⎥⎦ (3.10)


and

I =

⎡⎢⎢⎢⎢⎣

1 0 · · · 00 1 · · · 0...

......

...0 0 · · · 1

⎤⎥⎥⎥⎥⎦ . (3.11)

Since each row sum of matrix I(N−1)×(N−1) = 1 and A−1A = I(N−1)×(N−1),therefore each row sum of A−1A equals to 1. i.e.

a−11,1(a1,1 + a1,2 + · · ·+ a1,n−1) + a−1

1,2(a2,1 + a2,2 + · · · + a2,n−1)

+ · · ·+ a−11,n−1(an−1,1 + an−1,2 + · · ·+ an−1,n−1) = 1

⇒ a−11,1S1 + a−1

1,2S2 + · · ·+ a−11,n−1SN−1 = 1

which can be written in compact form as

N−1∑i=1

a−1k,i Si = 1, k = 1, 2, ..., N − 1. (3.12)

If Sj = minSi, then from Eq. (3.12), it can be written as

1 ≥ Sj(a−1k,1 + a−1

k,2 + .... + a−1k,N−1).

It follows that

N−1∑i=1

a−1k,i ≤

1

min Si=

1

(h4Bio), (3.13)

where,

Bio = (1

h4)min Si > 0, 1 ≤ i0 ≤ N − 1.

From Eq. (3.4), it can be written as

ej =N−1∑i=1

a−1j,i Ti, j = 1, 2, ..., N − 1. (3.14)

From Eq. (3.3) and Eq. (3.14), it can be proved that

|ej | ≤ Kh2

Bio, j = 1, 2, ..., N − 1,

where K is constant and independent of h. It follows that,

‖ E ‖= O(h2).


Similarly, the method developed for the system (2.3), (2.8) and (2.9) is alsosecond order convergent. The result can be summarized in the following the-orem

TheoremLet Y (x) be the exact solution of the system (1.1) or (1.2) and let yi, i =0, 1, ......, N be the exact solution of (3.1) then

‖ E ‖= O(h2).

4 Numerical Results

Example 1:The following boundary value problem is considered, as

−εy(4) + py = εx4(32ε2x(−6(7 − 55x4 + 70x8) + ε2(x2 − 3x6 + 2x10))

cosεx + (x4(x4 − 1)2 − ε5x4(x4 − 1)2 x ∈ [0, 1]

+48ε3x2(7 − 33x4 + 30x8) − 240ε(7 − 99x4 + 182x8))sinεx),

y(1) = 0, y(1)(1) = 0, y(−1) = 0, y(1)(−1) = 0.

(4.1)

The analytical solution of the problem (4.1) is,y(x) = εx8(x4 − 1)2 sin εx.The observed maximum errors associate with yis for Example 1, correspondingto different values of ε are summarized in Table 1. It is noted from the Table1 that if h is reduced by factor 1

2, then ‖E‖ is reduced by factor 1

4, which

indicates that the present method gives second order results.

Table 1.ε h = 1

16h = 1

32h = 1

64h = 1

128116

1.7094e − 004 4.7425e − 005 1.2094e − 005 3.0303e − 006132

4.4022e − 005 1.2203e − 005 3.1120e − 006 7.7974e − 007164

1.1706e − 005 3.2459e − 006 8.2662e − 007 2.0714e − 007

Example 2:The following boundary value problem is considered, as

−εy(4) + py = ε(2x4 + cosx − ε(48 + cosx)), x ∈ [−1, 1],

y(−1) = ε(2 + cos1), y(2)(−1) = ε(24 − cos1),

y(1) = ε(2 + cos1), y(2)(1) = ε(24 − cos1).

(4.2)


The analytical solution of the problem (4.2) is,y(x) = ε(2x4 + cosx).The observed maximum errors associate with yis for Example 2, correspondingto different values of ε are summarized in Table 2.

Table 2.ε h = 1

16h = 1

32h = 1

64h = 1

128116

2.3722e − 006 5.9529e − 007 1.4896e − 007 3.7214e − 008132

4.5647e − 007 1.1462e − 007 2.8684e − 008 7.1730e − 009164

1.0356e − 007 2.6027e − 008 6.5148e − 009 1.6304e − 009

It is confirmed from the Table 2 that if h is reduced by factor 12, then ‖E‖ is

reduced by factor 14, which indicates that the present method gives second order

results.

5 Conclusion

Quintic spline method is developed for the approximate solution of fourth or-der singularly perturbed boundary value problem. In addition to the boundaryconditions corresponding to the 1st derivatives, the boundary conditions cor-responding to the 2nd derivatives are also considered. The method has beenproved to be second order convergent. Two examples are considered for nu-merical illustration of the method. It is also observed that the results of theseexamples preserve O(h2).

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Received: April, 2012

akramIMF41-44-2012

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