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Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method
Applied Mathematics and Computation 215 (2009) 2095–2102
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method q Fazhan Geng Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, PR China
a r t i c l e
i n f o
Keywords: Nonlinear Singular three-point boundary value problem Reproducing kernel Hilbert space method
a b s t r a c t This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we consider the following singular second order three-point boundary value problems:
aðxÞu00 ðxÞ þ bðxÞu0 ðxÞ þ cðxÞuðxÞ ¼ hðx; uÞ; uð0Þ ¼ 0;
0 6 x 6 1;
uð1Þ ¼ auðgÞ þ c;
ð1:1Þ
where g 2 ð0; 1Þ; aðxÞ 2 C 2 ½0; 1; bðxÞ 2 C 1 ½0; 1; cðxÞ 2 C½0; 1; að0Þ ¼ 0 or að1Þ ¼ 0; aðxÞ–0; x 2 ð0; 1Þ; bð0Þ–0; cð0Þ–0; bð1Þ–0; cð1Þ–0 and bð1Þ a0 ð1Þ–0. That is, the equation may be singular at x ¼ 0; 1. We consider uð0Þ ¼ 0 since the boundary conditions uð0Þ ¼ a0 can be reduced to uð0Þ ¼ 0. Singular multi-point boundary value problems arise in a variety of applied mathematics and physics. Singular two-point boundary value problems have been extensively studied in the literature, see [1–12]. Also, the existence and multiplicity of solutions of nonsingular multi-point boundary value problems have been studied by many authors, see [13–18] and the references therein. However, research for singular multi-point boundary value problems has proceeded very slowly. Recently, the existence and multiplicity of solutions of singular multi-point boundary value problems have been studied by some authors, see [19–24]. However, to the best of our knowledge, there have been no methods for solving singular multi-point boundary value problems. In this paper, we will give a method for solving a class of singular second order three-point boundary value problems. The rest of the paper is organized as follows. In the next section, (1.1) is converted into an equivalent integro-differential equation. The equivalent integro-differential equation is solved using reproducing kernel Hilbert space method (RKHSM) in Section 3. The numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion. q
This work was supported by the Scientific Research Project of Heilongjiang Education Office (Grant No. 2009-11541098). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.002
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2. The equivalent equation of (1.1) (1.1) cannot be solved directly using RKHSM, since it is impossible to obtain the reproducing kernel satisfying the threepoint boundary conditions of (1.1). So, we will make great efforts to convert (1.1) into an equivalent equation, which is easily solved using RKHSM. Integration of both sides of (1.1) from 1 to x yields
8 0 0 0 0 > < aðxÞuR ðxÞ að1Þu ð1Þ þ ½bðxÞ a ðxÞuðxÞ ½bð1Þ a ð1Þuð1Þ x 00 0 þ 1 ½a ðsÞ b ðsÞ þ cðsÞuðsÞds ¼ f0 ðx; uÞ; 0 6 x 6 1; > : uð0Þ ¼ 0;
ð2:1Þ
Rx where f0 ðx; uÞ ¼ 1 hðs; uÞds. Substituting uð1Þ ¼ auðgÞ þ c into (2.1) leads to
8 0 0 0 0 > < aðxÞuR ðxÞ að1Þu ð1Þ þ ½bðxÞ a ðxÞuðxÞ a½bð1Þ a ð1ÞuðgÞ x 00 0 þ 1 ½a ðsÞ b ðsÞ þ cðsÞuðsÞds ¼ f ðx; uÞ; 0 6 x 6 1; > : uð0Þ ¼ 0;
ð2:2Þ
where f ðx; uÞ ¼ g 0 ðxÞ þ c½bð1Þ a0 ð1Þ. Letting x ¼ 1 in (2.2), from bð1Þ a0 ð1Þ–0, one obtains
uð1Þ ¼ auðgÞ þ c: Obviously, (1.1) and (2.2) are equivalent. Therefore, it suffices for us to solve (2.2). 3. Solving (2.2) using RKHSM Reproducing kernel theory has important application in numerical analysis, differential equations, probability and statistics and so on [25,26]. Recently, using the RKHSM, the authors discussed singular linear two-point boundary value problems, singular nonlinear two-point periodic boundary value problems, nonlinear systems of boundary value problems and nonlinear partial differential equations and so on [10–12,27–31]. In order to solve (2.2) using RKHSM, we first construct a reproducing kernel Hilbert space W 22 ½0; 1 in which every function satisfies the boundary condition of (2.2). Definition 3.1 (Reproducing kernel). Let E be a nonempty abstract set. A function K : E E ! C is a reproducing kernel of the Hilbert space H if and only if
ðaÞ 8t 2 E;
Kð; tÞ 2 H
ðbÞ 8t 2 E;
8u 2 H;
ðu; Kð; tÞÞ ¼ uðtÞ:
The last condition is called ‘‘the reproducing property”: the value of the function u at the point t is reproduced by the inner product of u with Kð; tÞ. A Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS). 3.1. The reproducing kernel Hilbert space W 22 ½0; 1 The inner product space W 22 ½0; 1 is defined as W 22 ½0; 1 ¼ fuðxÞ; u0 ðxÞ are absolutely continuous real value functions, u ðxÞ 2 L2 ½0; 1; uð0Þ ¼ 0g. The inner product and norm in W 22 ½0; 1 are given, respectively, by 00
ðuðyÞ; v ðyÞÞW 2 ¼ uð0Þv ð0Þ þ uð1Þv ð1Þ þ 2
Z
1
u00 v 00 dy
0
and
kukW 2 ¼ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu; uÞW 2 ; 2
u; v 2 W 22 ½0; 1:
Theorem 3.1. The space W 22 ½0; 1 is a reproducing kernel Hilbert space. That is, there exists Rx ðyÞ 2 W 22 ½0; 1, for any uðyÞ 2 W 22 ½0; 1 and each fixed x 2 ½0; 1; y 2 ½0; 1, such that ðuðyÞ; Rx ðyÞÞW 2 ¼ uðxÞ. The reproducing kernel Rx ðyÞ can be denoted by
Rx ðyÞ ¼
8 2 2 > < yðxð8 3x þ x Þ þ ð1 þ xÞy Þ ;
y 6 x; 6 2 2 > : xðx ð1 þ yÞ þ yð8 3y þ y ÞÞ ; y > x: 6
For the proof of Theorem 3.1, see [10].
2
ð3:1Þ
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3.2. The reproducing kernel Hilbert space W 12 ½0; 1 The inner product space W 12 ½0; 1 is defined by W 12 ½0; 1 ¼ fuðxÞ j u is an absolutely continuous real value function, u0 2 L2 ½0; 1g. The inner product and norm in W 12 ½0; 1 are given, respectively, by
ðuðxÞ; v ðxÞÞW 1 ¼ 2
where uðxÞ; v ðxÞ 2 kernel is
Rx ðyÞ ¼
Z
1
ðuv þ u0 v 0 Þdx;
0
W 12 ½0; 1.
kukW 1 ¼ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu; uÞW 1 ; 2
In [31], Li and Cui proved that W 12 ½0; 1 is a reproducing kernel Hilbert space and its reproducing
1 ½coshðx þ y 1Þ þ coshðjx yj 1Þ: 2 sinhð1Þ
3.3. The solution of (2.2) Rx 0 In (1.1), let LuðxÞ aðxÞu0 ðxÞ að1Þu0 ð1Þ þ ½bðxÞ a0 ðxÞuðxÞ a½bð1Þ a0 ð1ÞuðgÞ þ 1 ½a00 ðsÞ b ðsÞ þ cðsÞuðsÞds. It is clear 2 1 that L : W 2 ½0; 1 ! W 2 ½0; 1 is a bounded linear operator. Put ui ðxÞ ¼ Rxi ðxÞ and wi ðxÞ ¼ L ui ðxÞ where L is the adjoint operator i ðxÞg1 of W 2 ½0; 1 can be derived from Gram–Schmidt orthogonalization process of of L. The orthonormal system fw 2 i¼1 , fwi ðxÞg1 i¼1
i ðxÞ ¼ w
i X
bik wk ðxÞ; ðbii > 0; i ¼ 1; 2; . . .Þ:
ð3:2Þ
k¼1
1 2 Theorem 3.2. For (2.2), if fxi g1 i¼1 is dense on [0,1], then fwi ðxÞgi¼1 is the complete system of W 2 ½0; 1 and wi ðxÞ ¼ Ly Rx ðyÞjy¼xi . The subscript y by the operator L indicates that the operator L applies to the function of y.
Proof. Note that
wi ðxÞ ¼ ðL ui ÞðxÞ ¼ ððL ui ÞðyÞ; Rx ðyÞÞ ¼ ðui ðyÞ; Ly Rx ðyÞÞ ¼ Ly Rx ðyÞjy¼xi : Clearly, wi ðxÞ 2 W 22 ½0; 1. For each fixed uðxÞ 2 W 22 ½0; 1, let ðuðxÞ; wi ðxÞÞ ¼ 0; ði ¼ 1; 2; . . .Þ, which means that,
ðuðxÞ; ðL ui ÞðxÞÞ ¼ ðLuðÞ; ui ðÞÞ ¼ ðLuÞðxi Þ ¼ 0: fxi g1 i¼1
Note that complete. h
ð3:3Þ 1
is dense on [0, 1], hence, ðLuÞðxÞ ¼ 0. It follows that u 0 from the existence of L . So the proof is
Theorem 3.3. If fxi g1 i¼1 is dense on [0, 1] and the solution of (2.2) is unique, then the solution of (2.2) satisfies the form
uðxÞ ¼ L1 f ðx; uðxÞÞ ¼
1 X i X
i ðxÞ: bik f ðxk ; uðxk ÞÞw
ð3:4Þ
i¼1 k¼1 2 Proof. Applying Theorem 3.2, it is easy to see that fwi ðxÞg1 i¼1 is the complete orthonormal basis of W 2 ½0; 1.
Note that ðv ðxÞ; ui ðxÞÞ ¼ v ðxi Þ for each v ðxÞ 2 W 12 ½0; 1. Hence we have
uðxÞ ¼
1 1 X i 1 X i X X X i ðxÞÞw i ðxÞ ¼ i ðxÞ ¼ i ðxÞ ðuðxÞ; w bik ðuðxÞ; L uk ðxÞÞw bik ðLuðxÞ; uk ðxÞÞw i¼1
¼
1 X i X
i¼1 k¼1
i ðxÞ ¼ bik ðf ðx; uðxÞÞ; uk ðxÞÞw
i¼1 k¼1
i¼1 k¼1 1 X i X
i ðxÞ bik f ðxk ; uðxk ÞÞw
ð3:5Þ
i¼1 k¼1
and the proof of the theorem is complete. h Remark. Case(i): (2.2) is linear, that is, f ðx; uðxÞÞ ¼ f ðxÞ. Then the analytical solution to (1.1) can be obtained directly from (3.4). Case(ii): (2.2) is nonlinear. In this case, the approximate solution to (2.2) can be obtained using the following method. According to (3.4), we construct the following iteration formula:
8 > < u0 ðxÞ ¼ 0; 1 > : unþ1 ðxÞ ¼ L f ðx; un ðxÞÞ ¼
1 P i P i¼1 k¼1
i ðxÞ; bik f ðxk ; un ðxk ÞÞw
n ¼ 0; 1; . . . :
ð3:6Þ
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Remark. In the iteration process of (3.6), we can guarantee that the approximation un ðtÞ satisfies the boundary conditions of (1.1). 3.3.1. The convergence and (3.6) In fact, the solution of problem (1.1) is considered as the fixed point of the following functional under the suitable choice of the initial term u0 ðxÞ:
unþ1 ðxÞ ¼ L1 f ðx; un ðxÞÞ ¼
1 X i X
i ðxÞ: bik f ðxk ; un ðxk ÞÞw
ð3:7Þ
i¼1 k¼1
As a well known powerful tool, we have Theorem 3.4 (Banach’s fixed point theorem). Assume that X is a Banach space and
A:X!X is a nonlinear mapping, and suppose that
kA½u A½v k 6 aku v k;
u; v 2 X
for some constants a < 1. Then A has a unique fixed point. Furthermore, the sequence
unþ1 ¼ A½un ; with an arbitrary choice of u0 2 X, converges to the fixed point of A. According to Theorem 3.4, for the nonlinear mapping
A½uðxÞ ¼ L1 f ðx; uðxÞÞ ¼
1 X i X
i ðxÞ; bik f ðxk ; uðxk ÞÞw
i¼1 k¼1
a sufficient condition for convergence of the present iteration method is strictly contraction of A. Furthermore, the sequence (3.6) converges to the fixed point of A which is also the solution of problem (1.1). Now, the approximate solution uNn ðxÞ can be obtained by taking finitely many terms in the series representation of un ðxÞ and
uNn ðxÞ ¼
N X i X
i ðxÞ: bik f ðxk ; un1 ðxk ÞÞw
i¼1 k¼1
4. Numerical examples In this section, four numerical examples are studied to demonstrate the accuracy of the present method. The examples are computed using Mathematica 5.0. Results obtained by the method are compared with the analytical solution of each example and are found to be in good agreement with each other. Example 4.1. Consider the following singular three-point boundary value problem:
(
xu00 ðxÞ þ 2u0 ðxÞ ¼ f ðxÞ; uð0Þ ¼ 0;
uð1Þ ¼
1 uð12Þ 2
0 6 x 6 1; 12 ;
Table 1 Numerical results for Example 4.1 (n = 1, N = 11). x
True solution u(x)
Approximate solution u11 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.248690 0.481754 0.684547 0.844328 0.951057 0.998027 0.982287 0.904827 0.770513 0.587785 0.368125 0.125333
0.247643 0.480385 0.682673 0.841789 0.947664 0.994087 0.978457 0.901389 0.767705 0.586084 0.367952 0.126626
4.2E03 2.8E03 2.7E03 3.0E03 3.5E03 3.9E03 3.8E03 3.7E03 3.6E03 2.8E03 4.6E04 1.0E02
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F. Geng / Applied Mathematics and Computation 215 (2009) 2095–2102 Table 2 Numerical results for Example 4.1 (n = 1, N = 51). x
True solution u(x)
Approximate solution u51 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.248690 0.481754 0.684547 0.844328 0.951057 0.998027 0.982287 0.904827 0.770513 0.587785 0.368125 0.125333
0.248680 0.481718 0.684479 0.844227 0.950927 0.997878 0.982136 0.904690 0.770404 0.587718 0.368110 0.125378
4.1E05 7.3E05 9.8E05 1.1E04 1.3E04 1.4E04 1.5E04 1.5E04 1.4E04 1.4E04 4.0E05 3.5E04
Table 3 Numerical results for Example 4.2 (n = 1, N = 11). x
True solution u(x)
Approximate solution u11 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 0.114400
0.0800565 0.160650 0.242274 0.325451 0.410713 0.498608 0.589699 0.684569 0.783825 0.888106 0.998484 0.114990
3.6E04 2.0E04 1.4E04 1.1E04 9.5E05 7.5E05 5.5E05 3.6E05 1.9E05 1.8E15 4.2E04 5.2E04
Table 4 Numerical results for Example 4.2 (n = 1, N = 51). x
True solution u(x)
Approximate solution u51 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 0.114400
0.0800849 0.160683 0.24231 0.325488 0.410751 0.498644 0.589731 0.684593 0.783840 0.888106 0.998074 0.11443
6.0E06 4.6E06 4.0E06 3.4E06 3.0E06 2.4E06 2.0E06 1.3E06 6.8E07 1.7E13 1.5E05 2.7E05
Table 5 Numerical results for Example 4.3 (n = 5, N = 11). x
True solution u(x)
Approximate solution u11 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 1.114400
0.0800853 0.160683 0.242310 0.325489 0.410752 0.498645 0.589731 0.684593 0.783843 0.888119 0.997949 1.11486
3.4E07 1.0E06 1.1E06 9.5E07 1.3E06 1.8E06 1.7E06 2.0E06 3.0E06 1.5E05 1.5E05 4.1E04
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Table 6 Numerical results for Example 4.3 (n = 5, N = 21). x
True solution u(x)
Approximate solution u21 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 1.114400
0.0800854 0.160684 0.242311 0.325489 0.410752 0.498645 0.589732 0.684594 0.783840 0.888108 0.997061 1.11443
3.0E08 6.7E08 2.6E07 2.8E07 3.1E07 4.0E07 3.0E07 3.0E07 5.2E09 1.8E06 2.7E06 2.2E05
Table 7 Numerical results for Example 4.4 (n = 5, N = 11). x
True solution u(x)
Approximate solution u11 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770739 0.819192
0.079914 0.159318 0.237702 0.314567 0.389418 0.461779 0.531186 0.597195 0.659384 0.717355 0.770787 0.818854
1.3E07 5.6E07 5.6E07 3.5E07 4.6E07 8.0E07 9.2E07 2.2E07 7.5E07 1.9E06 6.2E05 4.1E04
where f ðxÞ ¼ pð2 cosðpxÞ px sinðpxÞÞ. The exact solution is given by uðxÞ ¼ sinðpxÞ. Using our method, taking n ¼ 1; xi ¼ i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 51, the numerical results are given in Tables 1 and 2. n1 Example 4.2. Consider the following singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ ð1 xÞu0 ðxÞ þ uðxÞ ¼ f ðxÞ; sinh4 uð0Þ ¼ 0; uð1Þ þ 12 u 45 ¼ 2 5 þ sinh 1
06x61
where f ðxÞ ¼ ð1 xÞ cosh x þ sinh x þ ð1 xÞx sinh x. The exact solution is given by uðxÞ ¼ sinh x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 51, the numerical results are given in Tables 3 and 4. n ¼ 1; xi ¼ n1 Example 4.3. Consider the following singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ 6u0 ðxÞ þ 2uðxÞ þ u2 ðxÞ ¼ f ðxÞ; sinh4 uð0Þ ¼ 0; uð1Þ þ 12 u 45 ¼ 2 5 þ sinh 1
0 6 x 6 1;
where f ðxÞ ¼ 6 cosh x þ sinh xð2 þ x x2 þ sinh xÞ. The exact solution is given by uðxÞ ¼ sinh x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 21, the numerical results are given in Tables 5 and 6. n ¼ 5 xi ¼ n1 Example 4.4. Consider the following nonlinear singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ 10u0 ðxÞ þ 2uðxÞ þ u5 ðxÞ ¼ f ðxÞ; sin5 uð0Þ ¼ 0; uð1Þ þ 12 u 56 ¼ 2 6 þ sin 1 5
0 6 x 6 1;
where f ðxÞ ¼ sin x ð1 xÞx sin x þ 2 sin x þ 10 cos x. The exact solution is given by uðxÞ ¼ sin x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 21, the numerical results are given in Tables 7 and 8. n ¼ 5; xi ¼ n1
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F. Geng / Applied Mathematics and Computation 215 (2009) 2095–2102 Table 8 Numerical results for Example 4.4 (n = 5, N = 21). x
True solution u(x)
Approximate solution u21 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770739 0.819192
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770735 0.819168
4.0E08 3.4E08 1.0E07 1.0E07 1.2E07 1.6E07 1.1E07 1.3E07 8.4E08 1.7E08 5.2E06 2.8E05
5. Conclusion In this paper, we introduce a method for solving a class of singular second order three-point boundary value problems. The numerical results show that only a few number of iteration steps can be used for numerical purpose with a high degree of accuracy. Therefore, the present method is an accurate and reliable analytical technique for singular second order threepoint boundary value problems. Acknowledgments The authors would like to express their thanks to unknown referees for their careful reading and helpful comments. References [1] R.P. Agarwal, D. O’Regan, I. Rachunkova, S. Stanek, Two-point higher-order BVPs with singularities in phase variables, Computer and Mathematics with Applications 46 (2003) 1799–1826. [2] R.P. Agarwal, S. Stanek, Nonnegative solutions of singular boundary value problems with sigh changing nonlinearities, Computer and Mathematics with Applications 46 (2003) 1827–1837. [3] M.K. Kadalbajoo, V.K. Aggarwal, Numerical solution of singular boundary value problems via Chebyshev polynomial and B-spline, Applied Mathematics and Computation 160 (2005) 851–863. [4] A.S.V. Ravi Kanth, K. Aruna, Solution of singular two-point boundary value problems using differential transformation method, Physics Letters A 372 (2008) 4671–4673. [5] A.S.V. Ravi Kanth, Y.N. Reddy, Higher order finite difference method for a class of singular boundary value problems, Applied Mathematics and Computation 155 (2004) 249–258. [6] A.S.V. Ravi Kanth, Y.N. Reddy, Cubic spline for a class of singular boundary value problems, Applied Mathematics and Computation 170 (2005) 733– 740. [7] A.S.V. Ravi Kanth, Vishnu Bhattacharya, Cubic spline for a class of non-linear singular boundary value problems arising in physiology, Applied Mathematics and Computation 189 (2) (2007) 2017–2022. [8] A.S.V. Ravi Kanth, Cubic spline polynomial for non-linear singular two-point boundary value problems, Applied Mathematics and Computation 174 (1) (2006) 768–774. 4 [9] R.K. Mohanty, P.L. Sachder, N. Jha, An Oðh Þ accurate cubic spline TAGE method for nonlinear singular two point boundary value problem, Applied Mathematics and Computation 158 (2004) 853–868. [10] M.G. Cui, F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, Journal of Computational and Applied Mathematics 205 (2007) 6–15. [11] F.Z. Geng, M.G. Cui, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Applied Mathematics and Computation 192 (2007) 389–398. [12] F.Z. Geng, M.G. Cui, Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, Journal of the Korean Mathematical Society 45 (3) (2008) 77–87. [13] R.P. Agarwal, H.B. Thompson, C.C. Tisdell, Three-point boundary value problems for second-order discrete equations, Computer and Mathematics with Applications 45 (2003) 1429–1435. [14] H.B. Thompson, C. Tisdell, Three-point boundary value problems for second-order, ordinary, differential equation, Mathematical and Computer Modelling 34 (2001) 311–318. [15] A.Ya. Lepin, V.D. Ponomarev, On a positive solution of a three-point boundary value problem, Differential Equations 42 (2) (2006) 291–293. [16] Z.G. Zhang, L.S. Liu, C.X. Wu, Nontrival solution of third-order nonlinear eigenvalue problems, Applied Mathematics and Computation 176 (2006) 714– 721. [17] M.H. Pei, S.K. Chang, A quasilinearization method for second-order four-point boundary value problems, Applied Mathematics and Computation 202 (2008) 54–66. [18] X.F. Li, Multiple positive solutions for some four-point boundary value problems with p-Laplacian, Applied Mathematics and Computation 202 (2008) 413–426. [19] R.P. Agarwal, I. Kiguradze, On multi-point boundary value problems for linear ordinary differential equations with singularities, Journal of Mathematical Analysis and Applications 297 (2004) 131–151. [20] Z.X. Zhang, J.Y. Wang, The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, Journal of Computational and Applied Mathematics 147 (2002) 41–52. [21] R.Y. Ma, D. O’Regan, Solvability of singular second order m-point boundary value problems, Journal of Mathematical Analysis and Applications 301 (2005) 124–134. [22] Q.M. Zhang, D.Q. Jiang, Upper and lower solutions method and a second order three-point singular boundary value problems, Computer and Mathematics with Applications 56 (2008) 1059–1070.
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[23] X.S. Du, Z.Q. Zhao, Existence and uniqueness of positive solutions to a class of singular m-point boundary value problems, Applied Mathematics and Computation 198 (2008) 487–493. [24] B.M. Liu, L.S. Liu, Y.H. Wu, Positive solutions for a singular second-order three-point boundary value problem, Applied Mathematics and Computation 196 (2008) 532–541. [25] A. Daniel, Reproducing Kernel Spaces and Applications, Springer, 2003. [26] A. Berlinet, Christine Thomas-Agnan, Reproducing Kernel Hilbert Space in Probability and Statistics, Kluwer Academic Publishers, 2004. [27] F.Z. Geng, M.G. Cui, Solving a nonlinear system of second order boundary value problems, Journal of Mathematical Analysis and Applications 327 (2007) 1167–1181. [28] M.G. Cui, F.Z. Geng, A computational method for solving one-dimensional variable-coefficient Burgers equation, Applied Mathematics and Computation 188 (2007) 1389–1401. [29] M.G. Cui, Y.Z. Lin, A new method of solving the coefficient inverse problem of differential equation, Science in China Series A 50 (4) (2007) 561–572. [30] M.G. Cui, Z. Chen, The exact solution of nonlinear age-structured population model, Nonlinear Analysis: Real World Applications 8 (2007) 1096–1112. [31] C.L. Li, M.G. Cui, The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Applied Mathematics and Computation 143 (2-3) (2003) 393–399.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method q Fazhan Geng Department of Mathematics, Changshu Institute of Technology, Changshu, Jiangsu 215500, PR China
a r t i c l e
i n f o
Keywords: Nonlinear Singular three-point boundary value problem Reproducing kernel Hilbert space method
a b s t r a c t This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction In this paper, we consider the following singular second order three-point boundary value problems:
aðxÞu00 ðxÞ þ bðxÞu0 ðxÞ þ cðxÞuðxÞ ¼ hðx; uÞ; uð0Þ ¼ 0;
0 6 x 6 1;
uð1Þ ¼ auðgÞ þ c;
ð1:1Þ
where g 2 ð0; 1Þ; aðxÞ 2 C 2 ½0; 1; bðxÞ 2 C 1 ½0; 1; cðxÞ 2 C½0; 1; að0Þ ¼ 0 or að1Þ ¼ 0; aðxÞ–0; x 2 ð0; 1Þ; bð0Þ–0; cð0Þ–0; bð1Þ–0; cð1Þ–0 and bð1Þ a0 ð1Þ–0. That is, the equation may be singular at x ¼ 0; 1. We consider uð0Þ ¼ 0 since the boundary conditions uð0Þ ¼ a0 can be reduced to uð0Þ ¼ 0. Singular multi-point boundary value problems arise in a variety of applied mathematics and physics. Singular two-point boundary value problems have been extensively studied in the literature, see [1–12]. Also, the existence and multiplicity of solutions of nonsingular multi-point boundary value problems have been studied by many authors, see [13–18] and the references therein. However, research for singular multi-point boundary value problems has proceeded very slowly. Recently, the existence and multiplicity of solutions of singular multi-point boundary value problems have been studied by some authors, see [19–24]. However, to the best of our knowledge, there have been no methods for solving singular multi-point boundary value problems. In this paper, we will give a method for solving a class of singular second order three-point boundary value problems. The rest of the paper is organized as follows. In the next section, (1.1) is converted into an equivalent integro-differential equation. The equivalent integro-differential equation is solved using reproducing kernel Hilbert space method (RKHSM) in Section 3. The numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion. q
This work was supported by the Scientific Research Project of Heilongjiang Education Office (Grant No. 2009-11541098). E-mail address: [email protected]
0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.08.002
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2. The equivalent equation of (1.1) (1.1) cannot be solved directly using RKHSM, since it is impossible to obtain the reproducing kernel satisfying the threepoint boundary conditions of (1.1). So, we will make great efforts to convert (1.1) into an equivalent equation, which is easily solved using RKHSM. Integration of both sides of (1.1) from 1 to x yields
8 0 0 0 0 > < aðxÞuR ðxÞ að1Þu ð1Þ þ ½bðxÞ a ðxÞuðxÞ ½bð1Þ a ð1Þuð1Þ x 00 0 þ 1 ½a ðsÞ b ðsÞ þ cðsÞuðsÞds ¼ f0 ðx; uÞ; 0 6 x 6 1; > : uð0Þ ¼ 0;
ð2:1Þ
Rx where f0 ðx; uÞ ¼ 1 hðs; uÞds. Substituting uð1Þ ¼ auðgÞ þ c into (2.1) leads to
8 0 0 0 0 > < aðxÞuR ðxÞ að1Þu ð1Þ þ ½bðxÞ a ðxÞuðxÞ a½bð1Þ a ð1ÞuðgÞ x 00 0 þ 1 ½a ðsÞ b ðsÞ þ cðsÞuðsÞds ¼ f ðx; uÞ; 0 6 x 6 1; > : uð0Þ ¼ 0;
ð2:2Þ
where f ðx; uÞ ¼ g 0 ðxÞ þ c½bð1Þ a0 ð1Þ. Letting x ¼ 1 in (2.2), from bð1Þ a0 ð1Þ–0, one obtains
uð1Þ ¼ auðgÞ þ c: Obviously, (1.1) and (2.2) are equivalent. Therefore, it suffices for us to solve (2.2). 3. Solving (2.2) using RKHSM Reproducing kernel theory has important application in numerical analysis, differential equations, probability and statistics and so on [25,26]. Recently, using the RKHSM, the authors discussed singular linear two-point boundary value problems, singular nonlinear two-point periodic boundary value problems, nonlinear systems of boundary value problems and nonlinear partial differential equations and so on [10–12,27–31]. In order to solve (2.2) using RKHSM, we first construct a reproducing kernel Hilbert space W 22 ½0; 1 in which every function satisfies the boundary condition of (2.2). Definition 3.1 (Reproducing kernel). Let E be a nonempty abstract set. A function K : E E ! C is a reproducing kernel of the Hilbert space H if and only if
ðaÞ 8t 2 E;
Kð; tÞ 2 H
ðbÞ 8t 2 E;
8u 2 H;
ðu; Kð; tÞÞ ¼ uðtÞ:
The last condition is called ‘‘the reproducing property”: the value of the function u at the point t is reproduced by the inner product of u with Kð; tÞ. A Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS). 3.1. The reproducing kernel Hilbert space W 22 ½0; 1 The inner product space W 22 ½0; 1 is defined as W 22 ½0; 1 ¼ fuðxÞ; u0 ðxÞ are absolutely continuous real value functions, u ðxÞ 2 L2 ½0; 1; uð0Þ ¼ 0g. The inner product and norm in W 22 ½0; 1 are given, respectively, by 00
ðuðyÞ; v ðyÞÞW 2 ¼ uð0Þv ð0Þ þ uð1Þv ð1Þ þ 2
Z
1
u00 v 00 dy
0
and
kukW 2 ¼ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu; uÞW 2 ; 2
u; v 2 W 22 ½0; 1:
Theorem 3.1. The space W 22 ½0; 1 is a reproducing kernel Hilbert space. That is, there exists Rx ðyÞ 2 W 22 ½0; 1, for any uðyÞ 2 W 22 ½0; 1 and each fixed x 2 ½0; 1; y 2 ½0; 1, such that ðuðyÞ; Rx ðyÞÞW 2 ¼ uðxÞ. The reproducing kernel Rx ðyÞ can be denoted by
Rx ðyÞ ¼
8 2 2 > < yðxð8 3x þ x Þ þ ð1 þ xÞy Þ ;
y 6 x; 6 2 2 > : xðx ð1 þ yÞ þ yð8 3y þ y ÞÞ ; y > x: 6
For the proof of Theorem 3.1, see [10].
2
ð3:1Þ
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3.2. The reproducing kernel Hilbert space W 12 ½0; 1 The inner product space W 12 ½0; 1 is defined by W 12 ½0; 1 ¼ fuðxÞ j u is an absolutely continuous real value function, u0 2 L2 ½0; 1g. The inner product and norm in W 12 ½0; 1 are given, respectively, by
ðuðxÞ; v ðxÞÞW 1 ¼ 2
where uðxÞ; v ðxÞ 2 kernel is
Rx ðyÞ ¼
Z
1
ðuv þ u0 v 0 Þdx;
0
W 12 ½0; 1.
kukW 1 ¼ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu; uÞW 1 ; 2
In [31], Li and Cui proved that W 12 ½0; 1 is a reproducing kernel Hilbert space and its reproducing
1 ½coshðx þ y 1Þ þ coshðjx yj 1Þ: 2 sinhð1Þ
3.3. The solution of (2.2) Rx 0 In (1.1), let LuðxÞ aðxÞu0 ðxÞ að1Þu0 ð1Þ þ ½bðxÞ a0 ðxÞuðxÞ a½bð1Þ a0 ð1ÞuðgÞ þ 1 ½a00 ðsÞ b ðsÞ þ cðsÞuðsÞds. It is clear 2 1 that L : W 2 ½0; 1 ! W 2 ½0; 1 is a bounded linear operator. Put ui ðxÞ ¼ Rxi ðxÞ and wi ðxÞ ¼ L ui ðxÞ where L is the adjoint operator i ðxÞg1 of W 2 ½0; 1 can be derived from Gram–Schmidt orthogonalization process of of L. The orthonormal system fw 2 i¼1 , fwi ðxÞg1 i¼1
i ðxÞ ¼ w
i X
bik wk ðxÞ; ðbii > 0; i ¼ 1; 2; . . .Þ:
ð3:2Þ
k¼1
1 2 Theorem 3.2. For (2.2), if fxi g1 i¼1 is dense on [0,1], then fwi ðxÞgi¼1 is the complete system of W 2 ½0; 1 and wi ðxÞ ¼ Ly Rx ðyÞjy¼xi . The subscript y by the operator L indicates that the operator L applies to the function of y.
Proof. Note that
wi ðxÞ ¼ ðL ui ÞðxÞ ¼ ððL ui ÞðyÞ; Rx ðyÞÞ ¼ ðui ðyÞ; Ly Rx ðyÞÞ ¼ Ly Rx ðyÞjy¼xi : Clearly, wi ðxÞ 2 W 22 ½0; 1. For each fixed uðxÞ 2 W 22 ½0; 1, let ðuðxÞ; wi ðxÞÞ ¼ 0; ði ¼ 1; 2; . . .Þ, which means that,
ðuðxÞ; ðL ui ÞðxÞÞ ¼ ðLuðÞ; ui ðÞÞ ¼ ðLuÞðxi Þ ¼ 0: fxi g1 i¼1
Note that complete. h
ð3:3Þ 1
is dense on [0, 1], hence, ðLuÞðxÞ ¼ 0. It follows that u 0 from the existence of L . So the proof is
Theorem 3.3. If fxi g1 i¼1 is dense on [0, 1] and the solution of (2.2) is unique, then the solution of (2.2) satisfies the form
uðxÞ ¼ L1 f ðx; uðxÞÞ ¼
1 X i X
i ðxÞ: bik f ðxk ; uðxk ÞÞw
ð3:4Þ
i¼1 k¼1 2 Proof. Applying Theorem 3.2, it is easy to see that fwi ðxÞg1 i¼1 is the complete orthonormal basis of W 2 ½0; 1.
Note that ðv ðxÞ; ui ðxÞÞ ¼ v ðxi Þ for each v ðxÞ 2 W 12 ½0; 1. Hence we have
uðxÞ ¼
1 1 X i 1 X i X X X i ðxÞÞw i ðxÞ ¼ i ðxÞ ¼ i ðxÞ ðuðxÞ; w bik ðuðxÞ; L uk ðxÞÞw bik ðLuðxÞ; uk ðxÞÞw i¼1
¼
1 X i X
i¼1 k¼1
i ðxÞ ¼ bik ðf ðx; uðxÞÞ; uk ðxÞÞw
i¼1 k¼1
i¼1 k¼1 1 X i X
i ðxÞ bik f ðxk ; uðxk ÞÞw
ð3:5Þ
i¼1 k¼1
and the proof of the theorem is complete. h Remark. Case(i): (2.2) is linear, that is, f ðx; uðxÞÞ ¼ f ðxÞ. Then the analytical solution to (1.1) can be obtained directly from (3.4). Case(ii): (2.2) is nonlinear. In this case, the approximate solution to (2.2) can be obtained using the following method. According to (3.4), we construct the following iteration formula:
8 > < u0 ðxÞ ¼ 0; 1 > : unþ1 ðxÞ ¼ L f ðx; un ðxÞÞ ¼
1 P i P i¼1 k¼1
i ðxÞ; bik f ðxk ; un ðxk ÞÞw
n ¼ 0; 1; . . . :
ð3:6Þ
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Remark. In the iteration process of (3.6), we can guarantee that the approximation un ðtÞ satisfies the boundary conditions of (1.1). 3.3.1. The convergence and (3.6) In fact, the solution of problem (1.1) is considered as the fixed point of the following functional under the suitable choice of the initial term u0 ðxÞ:
unþ1 ðxÞ ¼ L1 f ðx; un ðxÞÞ ¼
1 X i X
i ðxÞ: bik f ðxk ; un ðxk ÞÞw
ð3:7Þ
i¼1 k¼1
As a well known powerful tool, we have Theorem 3.4 (Banach’s fixed point theorem). Assume that X is a Banach space and
A:X!X is a nonlinear mapping, and suppose that
kA½u A½v k 6 aku v k;
u; v 2 X
for some constants a < 1. Then A has a unique fixed point. Furthermore, the sequence
unþ1 ¼ A½un ; with an arbitrary choice of u0 2 X, converges to the fixed point of A. According to Theorem 3.4, for the nonlinear mapping
A½uðxÞ ¼ L1 f ðx; uðxÞÞ ¼
1 X i X
i ðxÞ; bik f ðxk ; uðxk ÞÞw
i¼1 k¼1
a sufficient condition for convergence of the present iteration method is strictly contraction of A. Furthermore, the sequence (3.6) converges to the fixed point of A which is also the solution of problem (1.1). Now, the approximate solution uNn ðxÞ can be obtained by taking finitely many terms in the series representation of un ðxÞ and
uNn ðxÞ ¼
N X i X
i ðxÞ: bik f ðxk ; un1 ðxk ÞÞw
i¼1 k¼1
4. Numerical examples In this section, four numerical examples are studied to demonstrate the accuracy of the present method. The examples are computed using Mathematica 5.0. Results obtained by the method are compared with the analytical solution of each example and are found to be in good agreement with each other. Example 4.1. Consider the following singular three-point boundary value problem:
(
xu00 ðxÞ þ 2u0 ðxÞ ¼ f ðxÞ; uð0Þ ¼ 0;
uð1Þ ¼
1 uð12Þ 2
0 6 x 6 1; 12 ;
Table 1 Numerical results for Example 4.1 (n = 1, N = 11). x
True solution u(x)
Approximate solution u11 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.248690 0.481754 0.684547 0.844328 0.951057 0.998027 0.982287 0.904827 0.770513 0.587785 0.368125 0.125333
0.247643 0.480385 0.682673 0.841789 0.947664 0.994087 0.978457 0.901389 0.767705 0.586084 0.367952 0.126626
4.2E03 2.8E03 2.7E03 3.0E03 3.5E03 3.9E03 3.8E03 3.7E03 3.6E03 2.8E03 4.6E04 1.0E02
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F. Geng / Applied Mathematics and Computation 215 (2009) 2095–2102 Table 2 Numerical results for Example 4.1 (n = 1, N = 51). x
True solution u(x)
Approximate solution u51 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.248690 0.481754 0.684547 0.844328 0.951057 0.998027 0.982287 0.904827 0.770513 0.587785 0.368125 0.125333
0.248680 0.481718 0.684479 0.844227 0.950927 0.997878 0.982136 0.904690 0.770404 0.587718 0.368110 0.125378
4.1E05 7.3E05 9.8E05 1.1E04 1.3E04 1.4E04 1.5E04 1.5E04 1.4E04 1.4E04 4.0E05 3.5E04
Table 3 Numerical results for Example 4.2 (n = 1, N = 11). x
True solution u(x)
Approximate solution u11 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 0.114400
0.0800565 0.160650 0.242274 0.325451 0.410713 0.498608 0.589699 0.684569 0.783825 0.888106 0.998484 0.114990
3.6E04 2.0E04 1.4E04 1.1E04 9.5E05 7.5E05 5.5E05 3.6E05 1.9E05 1.8E15 4.2E04 5.2E04
Table 4 Numerical results for Example 4.2 (n = 1, N = 51). x
True solution u(x)
Approximate solution u51 1
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 0.114400
0.0800849 0.160683 0.24231 0.325488 0.410751 0.498644 0.589731 0.684593 0.783840 0.888106 0.998074 0.11443
6.0E06 4.6E06 4.0E06 3.4E06 3.0E06 2.4E06 2.0E06 1.3E06 6.8E07 1.7E13 1.5E05 2.7E05
Table 5 Numerical results for Example 4.3 (n = 5, N = 11). x
True solution u(x)
Approximate solution u11 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 1.114400
0.0800853 0.160683 0.242310 0.325489 0.410752 0.498645 0.589731 0.684593 0.783843 0.888119 0.997949 1.11486
3.4E07 1.0E06 1.1E06 9.5E07 1.3E06 1.8E06 1.7E06 2.0E06 3.0E06 1.5E05 1.5E05 4.1E04
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Table 6 Numerical results for Example 4.3 (n = 5, N = 21). x
True solution u(x)
Approximate solution u21 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.080085 0.160684 0.242311 0.325489 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 1.114400
0.0800854 0.160684 0.242311 0.325489 0.410752 0.498645 0.589732 0.684594 0.783840 0.888108 0.997061 1.11443
3.0E08 6.7E08 2.6E07 2.8E07 3.1E07 4.0E07 3.0E07 3.0E07 5.2E09 1.8E06 2.7E06 2.2E05
Table 7 Numerical results for Example 4.4 (n = 5, N = 11). x
True solution u(x)
Approximate solution u11 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770739 0.819192
0.079914 0.159318 0.237702 0.314567 0.389418 0.461779 0.531186 0.597195 0.659384 0.717355 0.770787 0.818854
1.3E07 5.6E07 5.6E07 3.5E07 4.6E07 8.0E07 9.2E07 2.2E07 7.5E07 1.9E06 6.2E05 4.1E04
where f ðxÞ ¼ pð2 cosðpxÞ px sinðpxÞÞ. The exact solution is given by uðxÞ ¼ sinðpxÞ. Using our method, taking n ¼ 1; xi ¼ i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 51, the numerical results are given in Tables 1 and 2. n1 Example 4.2. Consider the following singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ ð1 xÞu0 ðxÞ þ uðxÞ ¼ f ðxÞ; sinh4 uð0Þ ¼ 0; uð1Þ þ 12 u 45 ¼ 2 5 þ sinh 1
06x61
where f ðxÞ ¼ ð1 xÞ cosh x þ sinh x þ ð1 xÞx sinh x. The exact solution is given by uðxÞ ¼ sinh x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 51, the numerical results are given in Tables 3 and 4. n ¼ 1; xi ¼ n1 Example 4.3. Consider the following singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ 6u0 ðxÞ þ 2uðxÞ þ u2 ðxÞ ¼ f ðxÞ; sinh4 uð0Þ ¼ 0; uð1Þ þ 12 u 45 ¼ 2 5 þ sinh 1
0 6 x 6 1;
where f ðxÞ ¼ 6 cosh x þ sinh xð2 þ x x2 þ sinh xÞ. The exact solution is given by uðxÞ ¼ sinh x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 21, the numerical results are given in Tables 5 and 6. n ¼ 5 xi ¼ n1 Example 4.4. Consider the following nonlinear singular three-point boundary value problem:
(
xð1 xÞu00 ðxÞ þ 10u0 ðxÞ þ 2uðxÞ þ u5 ðxÞ ¼ f ðxÞ; sin5 uð0Þ ¼ 0; uð1Þ þ 12 u 56 ¼ 2 6 þ sin 1 5
0 6 x 6 1;
where f ðxÞ ¼ sin x ð1 xÞx sin x þ 2 sin x þ 10 cos x. The exact solution is given by uðxÞ ¼ sin x. Using our method, taking i1 ; i ¼ 1; 2; . . . ; N; N ¼ 11; 21, the numerical results are given in Tables 7 and 8. n ¼ 5; xi ¼ n1
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F. Geng / Applied Mathematics and Computation 215 (2009) 2095–2102 Table 8 Numerical results for Example 4.4 (n = 5, N = 21). x
True solution u(x)
Approximate solution u21 5
Relative error
0.08 0.16 0.24 0.32 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770739 0.819192
0.079914 0.159318 0.237703 0.314567 0.389418 0.461779 0.531186 0.597195 0.659385 0.717356 0.770735 0.819168
4.0E08 3.4E08 1.0E07 1.0E07 1.2E07 1.6E07 1.1E07 1.3E07 8.4E08 1.7E08 5.2E06 2.8E05
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