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On the convergence of second-orderfinite difference method for weaklyregular singular boundary valueproblemsArvind K. Singh aa Department of Mathematics, IIT, Kharagpur, 721302, IndiaVersion of record first published: 10 Oct 2008.

To cite this article: Arvind K. Singh (2008): On the convergence of second-order finite differencemethod for weakly regular singular boundary value problems, International Journal of ComputerMathematics, 85:12, 1807-1814

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International Journal of Computer MathematicsVol. 85, No. 12, December 2008, 1807–1814

On the convergence of second-order finite differencemethod for weakly regular singular boundary

value problems

Arvind K. Singh*

Department of Mathematics, IIT, Kharagpur 721302, India

(Received 16 June 2006; revised version received 02 July 2007; accepted 15 July 2007)

The second-order finite difference method developed by Chawla and Katti in 1985 based on a uniformmesh for the singular two-point boundary value problems

(p(x)y′)′ = f (x, y), 0 < x ≤ 1

with p(x) = xb0 , 0 ≤ b0 < 1 and boundary conditions y(0) = A, y(1) = B (A, B are finite constants) hasbeen extended for general class of non-negative functions p(x) = xb0 g(x), 0 ≤ b0 < 1 and the boundaryconditions

y(0) = A, αy(1) + βy′(1) = γ.

Second-order convergence of the method has been established for general non-negative function p(x) andunder quite general conditions on f (x, y). Our method is based on one evaluation of f and for p(x) = 1,it reduces to the classical second-order method for y′′ = f (x, y). In the case of p(x) = xb0 , 0 ≤ b0 < 1,this method provides better results than some existing second-order method which is corroborated by oneexample, and the order of method is also corroborated for general non-negative functions p(x).

Keywords: finite difference; two-point singular B. V. problems; uniform mesh

2000 AMS Subject Classification: 65L10

CCS Category: G1.7

1. Introduction

Consider the class of weakly regular singular two-point boundary value problems

(p(x)y ′)′ = f (x, y), 0 < x ≤ 1 (1)

y(0) = A, αy(1) + βy ′(1) = γ, (2)

*Email: arvindksingh9@yahoo.co.in

ISSN 0020-7160 print/ISSN 1029-0265 online© 2008 Taylor & FrancisDOI: 10.1080/00207160701577044http://www.informaworld.com

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1808 A.K. Singh

where α > 0, β ≥ 0 and A, γ finite constants. We assume that p(x) satisfies the followingconditions:(A) (i) p(x) > 0 on (0, 1],

(ii) p(x) ∈ C1(0, 1],(iii) p(x) = xb0g(x) on [0, 1], 0 ≤ b0 < 1 and G(x) = 1/g(x) is analytic in {z : |z| ≤ 1}.

Further we assume that(B) for (x, y) ∈ {[0, 1] × R}, f (x, y) is continuous, ∂f/∂y exists, it is continuous and ∂f/∂y ≥ 0.

Existence uniqueness of the problem (1) with boundary conditions y(0) = A, y(1) = B hasbeen established in [7] with non-linear forcing term as p(x)f (x, y), where p(x) satisfiesA(i)–(iii).

There is a considerable literature on numerical methods for such problems [1–3, 5, 6, 8]. Most ofthe authors have considered the problem (1) for p(x) = xb0 , 0 ≤ b0 < 1, with boundary conditionsy(0) = A, y(1) = B. For p(x) = xb0 , 0 ≤ b0 < 1, and for linear f , Jamet [5] proposed theusual three- point finite difference scheme for a uniform mesh h for the problem (1), and heshowed in the maximum norm that his scheme provides O(h1−b0)-convergent. In [3] Jamet’sresult is improved with an error estimate of order h2−b0 . Schreiber [8] considered the applicationof splines to self-adjoint equation with linear f . Chawla and Katti [2] have developed second-orderfinite difference method based on a uniform mesh for the problem (1) with boundary conditionsy(0) = A, y(1) = B (for g(x) = 1, 0 ≤ b0 < 1).

In this work, the second-order finite difference method described in Chawla and Katti [2] isfurther extended to a class of non-negative functions p(x) satisfying the conditions given inA(i)–(iii) with boundary conditions y(0) = A, αy(1) + βy ′(1) = γ . The finite difference methodbased on just one evaluation of f is described in Section 2, and the convergence of the methodbased on a uniform mesh is established in Section 3 under quite general conditions on f andp(x). For p(x) = 1, our method reduces to classical second-order method for y ′′ = f (x, y) andthe boundary conditions y(0) = A, y(1) = B. In Section 4, the method given in this work hasbeen compared with the method given in Kumar and Aziz [6] by one example and results aredisplayed in Table 1, which shows that our method provides better results. The order of accuracyhas also been corroborated for general non-negative functions p(x) and the maximum absoluteerrors are displayed in Tables 2 and 3.

Table 1. Maximum absolute errors for Example 1.

b0 = 0.5, λ = 4.0 b0 = 0.75, λ = 3.75N Present study Method in [6] Present study Method in [6]

16 1.0(−2)a 2.1(−2) 9.6(−3) 6.8(−2)32 2.7(−3) 5.2(−3) 2.4(−3) 1.7(−2)64 6.6(−4) 1.3(−3) 6.1(−4) 4.2(−3)

128 1.7(−4) 3.3(−4) 1.5(−4) 1.0(−3)

Note: a1.0(−2) = 1.0 × 10−2.

Table 2. Maximum absolute errors for Example 2.

N/b0 0.25 Order 0.50 Order 0.75 Order

16 1.67(−3) 3.09(−3) 5.55(−3)32 4.16(−4) 2.00 7.71(−4) 2.00 1.39(−3) 1.9964 1.04(−4) 2.00 1.93(−4) 2.00 3.47(−4) 1.99

128 2.59(−5) 2.00 4.81(−5) 2.00 8.68(−5) 2.00256 6.48(−6) 2.00 1.20(−5) 2.00 2.17(−5) 2.00512 1.62(−6) 2.00 3.01(−6) 2.00 5.43(−6) 2.00

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Table 3. Maximum absolute errors for Example 3.

N/b0 0.25 Order 0.50 Order 0.75 Order

16 8.33(−2) 8.81(−2) 9.30(−2)32 2.18(−2) 1.93 2.31(−2) 1.93 2.44(−2) 1.9364 5.52(−3) 1.98 5.84(−3) 1.98 6.18(−3) 1.98

128 1.38(−3) 2.00 1.47(−3) 2.00 1.55(−3) 2.00256 3.46(−4) 2.00 3.67(−4) 2.00 3.88(−4) 2.00512 8.66(−5) 2.00 9.16(−5) 2.00 9.69(−5) 2.00

2. Description of the method

For a positive integer N ≥ 2, consider a uniform mesh over [0, 1]:0 = x0 < x1 < x2 < · · · <

xN = 1. Denote yk = y(xk), fk = f (xk, yk) etc. We set z(x) = p(x)y ′, and integrating (1) fromxk to τ then dividing by p(τ) and again integrating from xk to xk+1, we get

yk+1 − yk = zkJk +∫ xk+1

xk

(p(τ))−1∫ τ

xk

f (t)dtdτ (3)

where we have set

Jk =∫ xk+1

xk

(p(τ))−1dτ (4)

Now changing the order of integration of Equation (3) we get

yk+1 − yk = zkJk +∫ xk+1

xk

(∫ xk+1

t

(p(τ ))−1dτ

)f (t)dt. (5)

In an analogous way, we obtain

yk − yk−1 = zkJk−1 −∫ xk

xk−1

(∫ t

xk−1

(p(τ))−1dτ

)f (t)dt. (6)

Eliminating zk from Equations (5) and (6), we obtain the identity:

(yk+1 − yk)/Jk − (yk − yk−1)/Jk−1 = I+k /Jk + I−

k /Jk−1, k = 1 (1) (N − 1) (7)

where

I±k =

∫ xk±1

xk

(∫ xk±1

t

(p(τ ))−1dτ

)f (t)dt (8)

Here, we are interested in a second-order method based on one evaluation of f . Here, p(x) =xb0g(x), and let (g(x))−1 = G(x) with Taylor series expansion about xk as

G(x) = Gk + (x − xk)G′k + (x − xk)

2

2! G′′(ζ )

where ζ lies between x and xk . Further, we assume that G′′(x) exists on (0, 1) and Gk = G(xk),G′

k = G′(xk). Then

I±k =

1∑i=0

f(i)k

⎡⎣ 1∑

j=0

1

j !A±ij,kG

(j)

k + 1

2A±

i2,kG′′(ζ±

k )

⎤⎦ + 1

2f ′′(ξ±

k )

⎡⎣ 1∑

j=0

1

j !A±2j,kG

(j)

k

+1

2A±

22,kG′′(ζ±

k )

], xk−1 < ξ−

k , ζ−k < xk < ξ+

k , ζ+k < xk+1 (9)

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1810 A.K. Singh

where

A±ij,k = 1

(i + 1)

i+j+1∑m=0

(−1)m

(i + j + 2 − b0 − m)

(i + j + 1

m

)xm

k

(x

i+j+2−b0−m

k±1

−xi+j+2−b0−m

k

), i = 0(1)2, j = 0(1)2 (10)

Now from Equations (7) and (9), we get

−yk−1/Jk−1 + (1/Jk + 1/Jk−1)yk − yk+1/Jk + B00,kGkfk + tk = 0, k = 1(1)(N − 1) (11)

where

tk =1∑

j=0

1

j !G(j)

k

[1

2B2j,kf

′′(ξk) + B1j,kf′k

]+ B01,kG

′kfk

+1

4G′′(ζk)

[2

1∑i=0

Bi2,kf(i)k + B22,kf

′′(ξk)

], xk−1 < ξk, ζk < xk+1 (12)

and Bij,k = (A+ij,k/Jk + A−

ij,k/Jk−1), i = 0(1)2, j = 0(1)2.Now from Equation (6) for k = N and the boundary condition at x = 1, we get

−yN−1/JN−1 + (1/JN−1 + α/βGN)yN + (A−00,N/JN−1)GNfN + tN = γ /βGN (13)

where

tN = 1

JN−1

⎡⎣ 1∑

j=0

1

j !G(j)

N

{1

2A−

2j,Nf ′′(ξ−N ) + A−

1j,Nf ′N

}+ A−

01,NG′NfN

+1

4G′′(ζ−

N )

{2

1∑i=0

A−i2,Nf

(i)N + A−

22,Nf ′′(ξ−N )

}], xN−1 < ξ−

N , ζ−N < xN (14)

3. Convergence of the method

In this section, we establish the second-order accuracy of the method developed in Section 2. LetF(Y ) = (f1, . . . , fN)T, Y = (y1, . . . , yN)T, T = (t1, . . . , tN )T, and Q = (q1, 0, . . . , 0, qN)T; thediscretizations (11) and (13) can be expressed in matrix form as

DY + PF(Y ) + T = Q (15)

where D = (dij ) and P = (pij ) are (N × N) tridiagonal matrix and diagonal matrix, respec-tively, with dk,k−1 = −1/Jk−1, k = 2(1)N , dk,k+1 = −1/Jk, k = 1(1)(N − 1), dk,k = (1/Jk +1/Jk−1), k = 1(1)(N − 1), dN,N = (1/JN−1 + α/βGN), pk,k = B00,kGk , k = 1(1)(N − 1),pN,N = A−

00,NGN/JN−1 and q1 = A/J0, qN = γ /βGN . Thus the method to approximate the

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solution Y can be written as

DY + PF(Y ) = Q (16)

where Y is approximation to Y . Now from Equations (15) and (16), we get the error equation

(D + PM)E = T (17)

where E = Y − Y = (e1, . . . , eN)T, F(Y ) − F(Y ) = ME, M = diag {U1, . . . , UN } (we alsonote that Uk = ∂fk/∂yk ≥ 0). Since for fixed xk and h → 0, it can be shown that

A+ij,k = hi+j+2

(i + 1)x

−b0k

[1

(i + j + 2)− b0h

(i + j + 3)xk

+ O(h2)

]

A−ij,k = (−1)i+jhi+j+2

(i + 1)x

−b0k

[1

(i + j + 2)+ b0h

(i + j + 3)xk

+ O(h2)

](18)

Jk = hGkx−b0k

[1 − h

2

{b0

xk

− G′k

Gk

}+ O(h2)

]

Jk−1 = hGkx−b0k

[1 + h

2

{b0

xk

− G′k

Gk

}+ O(h2)

](19)

and hence

Bij,k =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

2hi+j+1

(i + 1)(i + j + 2)Gk

+ O(hi+j+3), i + j = 0, 2, 4

− hi+j+2

(i + 1)(i + j + 2)xkGk

[(i + j + 1)b0

(i + j + 3)+ xkG

′k

Gk

]+ O(hi+j+4), i + j = 1, 3

(20)

With the help of Equations (18)–(20) and from corollary of Theorem 7.2 and Theorem 7.4 ofref. [4] it is easy to see that D and D + PM are irreducible, monotone and in view of PM ≥ 0we get D−1, (D + PM)−1 exists, is non-negative and (D + PM)−1 ≤ D−1.

The inverse D−1 = (d−1i,j ) of matrix D is given by

d−1i,j =

⎧⎪⎨⎪⎩

P(xi)[P(xN) − P(xj ) + (βGN)/α][P(xN) + (βGN)/α] , i ≤ j

P (xj )[P(xN) − P(xi) + (βGN)/α][P(xN) + (βGN)/α] , i ≥ j

(21)

where

P(x) =∫ x

0(p(τ))−1 dτ.

Now let b0 be fixed in [0, 1) and δ be chosen such that b0 + δ < 1. We assume that

x1+δ∣∣f ′′∣∣ < N2, xδ

∣∣f ′∣∣ < N1, |f | < N0 (22)

for suitable positive constants Ni , i = 0(1)2, then from Equations (12) and (14) we get

|tk| ≤ Ch3x−1−δk , k = 1(1)(N − 1) (23)

and

|tN | ≤ Ch2 (24)

for sufficiently small h, where C, C are suitable chosen constants.

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1812 A.K. Singh

Now from Equations (21), (23) and (24) we obtain for i = 1(1)N

|ei | ≤ C∗h3

⎡⎣{1 − P(xi)

[P(xN) + (βGN)/α]} i∑

j=1

P(xj )x−1−δj

+ P(xi)

[P(xN) + (βGN)/α]

⎧⎨⎩

N−1∑j=i+1

[P(xN) − P(xj ) + (βGN)/α

]x−1−δ

j + βGN

hα

⎫⎬⎭⎤⎦ (25)

where C∗ = max{C, C

}. It is easy to see that

h

i∑j=1

P(xj )x−1−δj <

∫ xi

0P(x)x−1−δdx (26)

and

h

N−1∑j=i+1

[P(xN) − P(xj ) + βGN

α

]x−1−δ

j

<

∫ xN

xi

[P(xN) − P(x) + βGN

α

]x−1−δ dx. (27)

From Equations (25–27), we obtain

|ei | ≤ C∗h3

(1 − b0)

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

(1 − b0)x1−b0i

δ(1 − b0 − δ)

[x−δ

i − α + (1 − b0 − δ)(1 + δ)β

α + (1 − b0)β

]sup[0,1]

G(x)+(1 − b0)d1, δ �= 0

x1−b0i

[ln

(1

xi

)+ (2 − b0)β

α + (1 − b0)β

]sup[0,1]

G(x) + d2, δ = 0

(28)where

d1 = α(1 − b0 − δ)(1−b0−δ)/δ

{α + (1 − b0)β}(1 − b0)(1−b0)/δ

[(2 − b0 − δ)(1−b0−δ)/δ

(2 − b0)(1−b0)/δsup[0,1]

∣∣G′(x)∣∣

+ (3 − b0 − δ)(1−b0−δ)/δ

2(3 − b0)(1−b0)/δsup[0,1]

∣∣G′′(x)∣∣

and

d2 = α

{α + (1 − b0)β}e2

[1

(2 − b0)e1/(2−b0) sup

[0,1]

∣∣G′(x)∣∣

+ 1

2(3 − b0)e2/(3−b0) sup

[0,1]

∣∣G′′(x)∣∣]

In view of the inequalities

x1−b0

(x−δ − α + (1 − b0 − δ)(1 + δ)β

α + (1 − b0)β

)≤(

α + (1 − b0)β

α + (1 − b0 − δ)(1 + δ)β

)(1−b0−δ)/δ

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International Journal of Computer Mathematics 1813

and

x1−b0

(ln

(1

x

)+ β

α + (1 − b0)β

)≤ 1

(1 − b0)e−[α−(1−b0)

2β]/[α+(1−b0)β]

for x ∈ (0, 1], (D + PM)−1 ≤ D−1 and from Equation (28), we have ‖E‖∞ ≤ ∥∥D−1 |T |∥∥∞,hence

‖E‖∞ = O(h2).

Thus we have established the following result.

THEOREM 1 Assume that f (x, y) satisfies (B) and p(x) satisfies the conditions in (A). Further,for fixed b0 ∈ [0, 1), let δ be chosen such that b0 + δ < 1, and assume that x1+δf ′′, xδf ′ and f arebounded on [0, 1]. Then, for a uniform mesh, the finite difference scheme based on Equations (11)

and (13) for the boundary value problems (1) and (2) is of second-order accuracy for sufficientlysmall h.

4. Numerical illustrations

In this section, we compare the method described in this work with that given in [6], and we alsocorroborate the order of convergence of the method for general non-negative function p(x).

We apply the second-order method given in [6], and second-order method in this work to solveExample 1 for different values of b0, and maximum absolute errors are displayed in Table 1. It isclear from the results displayed in the table that our method provides better results compared tothat of [6]. To corroborate the order of accuracy of the method given in this work for general non-negative function p(x), we apply the method to solve Examples 2 and 3 and maximum absoluteerrors along with the order are displayed in Tables 2 and 3 respectively.

Example 1

(xb0y ′)′ = λxb0+λ−2(λxλ + (b0 + λ − 1))y,

y(0) = 1, y(1) = e

with exact solution y(x) = exp(xλ).

Example 2

(xb0exy ′)′ = 5exxb0+3(5x5ey − (b0 + 4) − x)/(4 + x5)

y(0) = − ln 4, y(1) + 5y ′(1) = − ln 5 − 5

with exact solution y(x) = − ln(4 + x5).

Example 3

(xb0(1 + x2)y ′)′ = 5(1 + x2)xb0+3(5x5 + (b0 + 4) + 2x2/(1 + x2))y

y(0) = 1, y(1) + y ′(1) = 6e

with exact solution y(x) = exp(x5).

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1814 A.K. Singh

5. Conclusion

In this paper, we discussed the construction of finite difference method and its second-orderconvergence for the singular two-point boundary value problem (1). It may be noted here thatfor α = 0, our method defined by Equations (11) and (13) reduces to the classical second-ordermethod for y ′′ = f (x, y).

We considered three examples to illustrate computationally the second-order convergence ofthe method. In Tables 1–3, we showed values of the error norms computed for these problemsby the method. From Tables 2–3, we observe that our results confirm the theoretical analysis ofthe method, and in Table 1 the results are compared with that of ref. [6], which shows that ourmethod provides better results.

Acknowledgements

This work is supported by National Board for Higher Mathematics, Department of Atomic Energy, Mumbai, India. Theauthor is also thankful to the referees for thier valuable comments and suggestions.

References

[1] M.M. Chawla and C.P. Katti, Finite difference methods and their convergence for a class of singular two pointboundary value problems, Numer. Math. 39 (1982), pp. 341–350.

[2] ———, A uniform mesh finite difference method for a class of singular two point boundary value problems, SIAMJ. Numer. Anal. 22(3) (1985), pp. 561–565.

[3] P.G. Ciarlet, F. Natterer, and R.S.Varga, Numerical methods of high-order accuracy for singular non-linear boundaryvalue problems, Numer. Math. 15 (1970), pp. 87–99.

[4] P. Henrici, Discrete Variable-methods in Ordinary Differential Equations, John Wiley & Sons, New York, 1962.[5] P. Jamet, On the convergence of finite difference approximation to one-dimensional singular boundary value

problems, Numer. Math. 14 (1970), pp. 355–378.[6] M. Kumar and T. Aziz, A non uniform mesh finite difference method and its convergence for a class of singular two

point boundary value problems, Inter. J. Comput Math. 81(2) (2004), pp. 1507–1512.[7] R.K. Pandey, On a class of weakly regular singular two point boundary value problems-I, Nonlinear Anal. Theory,

Methods Appl. 27(1) (1996), pp. 1–12.[8] R. Schreiber, Finite element methods of high order accuracy for singular two-point boundary value problems with

nonsmooth solutions, SIAM J. Numer. Anal. 17 (1980), pp. 547–566.

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