A new reproducing kernel method for linear nonlocal boundary value problems

Applied Mathematics and Computation 248 (2014) 421–425

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

A new reproducing kernel method for linear nonlocal boundary value problems F.Z. Geng ⇑, S.P. Qian Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, China

a r t i c l e

i n f o

Keywords: Reproducing kernel method Nonlocal boundary value problems Approximation

a b s t r a c t In the previous works, the authors developed the reproducing kernel method (RKM) for nonlocal boundary value problems. A key of the method is the construction of the reproducing kernel (RK) satisfying the homogenous boundary conditions (BCs) of the considered problems. However, it is very difficult to obtain the RK of a reproducing kernel space satisfying nonlocal BCs or nonlinear BCs. Even if the RK is found, its representation is also very complicated compared with the RK without any BCs. In this paper, we will present a new RKM for linear nonlocal boundary value problems. The method can avoid reducing the inhomogeneous BCs to homogeneous BCs and constructing RK satisfying homogeneous nonlocal linear BCs. Numerical examples are provided to show the effectiveness of the new method. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction Nonlocal boundary value problems have applications in a variety of different areas of applied mathematics and physics. The existence, uniqueness and multiplicity of solutions of nonlocal boundary value problems have been discussed by some authors, see [1–6] and the references therein. However, discussion about numerical methods for solving nonlocal boundary value problems is rare. Geng [7] introduced a method for singular second order three-point boundary value problems by eliminating a three-point boundary condition. Wu and Li [8] and Li and Wu [9] gave a method of constructing RK satisfying linear multi-point boundary conditions and applied it to multi-point boundary value problems. Lin and Lin [10] proposed a method for multi-point BVPs based on constructing reproducing kernel satisfying multi-point boundary conditions. However, the form of the obtained reproducing kernel is very complicated and the computational cost is high. Tatari and Dehghan [11] and Saadatmandi and Dehghan [12] introduced the Adomian decomposition method and Sinc-collocation method for solving multi-point boundary value problems. Li and Wu [13] developed a method for solving linear fourth-order multipoint boundary value problems by constructing auxiliary two-point boundary value problems. Geng [14] presented an effective numerical algorithm for nonlinear multi-point boundary value problems. Tirmizi et al. [15] developed a second order method for solving third order three-point boundary value problems based on Pade approximation. In this paper, we consider the error estimation for the reproducing kernel method applied to the following second order two-point boundary value problems:



u00 ðxÞ þ pðxÞu0 ðxÞ þ qðxÞuðxÞ ¼ f ðxÞ; B1 ðuÞ ¼ 0; B2 ðuÞ ¼ 0;

⇑ Corresponding author. E-mail address: [email protected] (F.Z. Geng). http://dx.doi.org/10.1016/j.amc.2014.10.002 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

0 < x < 1;

ð1:1Þ

422

F.Z. Geng, S.P. Qian / Applied Mathematics and Computation 248 (2014) 421–425

where pðxÞ; qðxÞ are continuous and f ðxÞ is given such that (1.1) satisfies the existence and uniqueness of the solutions. Notice that B1 ðuÞ ¼ 0 and B2 ðuÞ ¼ 0 includes several types of boundary conditions: two point conditions, multi-point conditions, or integral conditions. The rest of the paper is organized as follows. In the next section, the new reproducing kernel method for solving (1.1) is introduced. Numerical examples are provided in Section 3. Section 4 ends this paper with a brief conclusion. 2. New method for (1.1) To solve (1.1) based on reproducing kernel theory, we will construct reproducing kernel spaces W m ½0; 1; ðm P 3Þ in which every function is not required to satisfy the BCs of (1.1). Definition 2.1. W m ½0; 1 ¼ fuðxÞj uðm1Þ ðxÞ is an absolutely continuous real value function, uðmÞ ðxÞ 2 L2 ½0; 1g. The inner product and norm in W m ½0; 1 are given respectively by

ðu; v Þm ¼

m1 X

uðiÞ ð0Þv ðiÞ ð0Þ þ

Z

1

uðmÞ ðxÞv ðmÞ ðxÞdx

0

i¼0

and

kukm ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðu; uÞm ;

u; v 2 W m ½0; 1:

By [16–19], it is easy to verify that W m ½0; 1 is a reproducing kernel space and its reproducing kernels Kðx; yÞ can be obtained. Put wi ðxÞ ¼ Ls Kðx; sÞjs¼xi ; ði ¼ 1; 2; . . . ; nÞ; wnþ1 ðxÞ ¼ 1; wnþ2 ðxÞ ¼ x. Let U n be a subspace generated by fwi ðxÞgni¼1 ; V n be a nþ2 nþ2 subspace generated by fwi ðxÞgnþ2 i¼1 . Applying Gram–Schmidt orthogonalization process to fwi ðxÞgi¼1 , one obtains fwi ðxÞgi¼1 which is an orthonormal basis for V n , and

wi ðxÞ ¼

i X bik wk ðxÞ;

ðbii > 0; i ¼ 1; 2; . . . ; n þ 2Þ:

ð2:1Þ

k¼1

We find an approximation to the solution uðxÞ of (1.1) in the space V n in the form

un ðxÞ ¼

n X ðu; wi Þwi þ c1 wnþ1 þ c2 wnþ2 ;

ð2:2Þ

i¼1

where c1 ; c2 are constants to be determined. Theorem 2.1. ðu; wi Þ ¼

un ðxÞ ¼

Pi

k¼1 bik f ðxk Þ

and

n X i X

bik f ðxk Þwi þ c1 wnþ1 þ c2 wnþ2 :

ð2:3Þ

i¼1 k¼1

Proof. Note that

ðu; wi Þ ¼

i i X X bik ðuðxÞ; Ls Kðx; sÞjs¼xk Þ ¼ bik Ls ðuðxÞ; Kðx; sÞÞjs¼xk : k¼1

k¼1

By the reproducing property of Kðx; sÞ and LuðsÞ ¼ f ðsÞ, it follows that

ðu; wi Þ ¼

i i X X bik Ls uðsÞjs¼xk ¼ bik f ðxk Þ: k¼1

k¼1

Therefore,

un ðxÞ ¼

n X i X bik f ðxk Þwi þ c1 wnþ1 þ c2 wnþ2 :



i¼1 k¼1

Imposing the nonlocal BCs of (1.1) on un ðxÞ, we have

B1 un ¼ a1 ;

B2 un ¼ a2 :

ð2:4Þ

These two linear equations serve to determine c1 ; c2 . Theorem 2.2. un satisfies Lun ðxi Þ ¼ f ðxi Þ; i ¼ 1; 2; . . . ; n and the BCs of (1.1), that is, un ðxÞ is an approximate solution of (1.1).

F.Z. Geng, S.P. Qian / Applied Mathematics and Computation 248 (2014) 421–425

423

Proof. Letting

Ai ¼

i X bik f ðxk Þ;

i ¼ 1; 2; . . . ; n;

Anþ1 ¼ c1 ;

Anþ2 ¼ c2 ;

k¼1

un ðxÞ becomes

un ðxÞ ¼

nþ2 X Ai wi : i¼1

By the reproducing and symmetric properties of reproducing kernel function Kðx; yÞ, we can derive that

ðLun Þðxk Þ ¼

nþ2 X

Ai Lx wi ðxÞjx¼xk

i¼1

¼

nþ2 X

  Ai Lx wi ðsÞ; Kðx; sÞ jx¼xk

i¼1 nþ2 X   ¼ Ai wi ðsÞ; Lx Kðx; sÞ jx¼xk i¼1 nþ2   X ¼ Ai wi ðsÞ; Lx Kðx; sÞjx¼xk i¼1 nþ2 X   ¼ Ai wi ðsÞ; wk ðsÞ : i¼1

It is easy to verify that j j nþ2 X X X bjk ðLun Þðxk Þ ¼ Ai wi ; bjk wk k¼1

i¼1

k¼1

! ¼

nþ2 X Ai ðwi ; wj Þ ¼ Aj :

ð2:5Þ

i¼1

Taking j ¼ 1, one gets ðLun Þðx1 Þ ¼ f ðx1 Þ: Taking j ¼ 2, we have ðLun Þðx2 Þ ¼ f ðx2 Þ: Furthermore, it is easy to see by induction that

ðLun Þðxj Þ ¼ f ðxj Þ;

j ¼ 3; 4; . . . ; n:

ð2:6Þ

From (2.4), clearly, un satisfies the BCs of (1.1). Therefore, un ðxÞ is an approximate solution of (1.1). h In fact, un ðxÞ is a collocation solution of (1.1). Compared with collocation methods, the main advantage of the present method is that it can avoid the solution of systems of linear equations. By Theorem 2.2 and [20], we have the following theorem: Theorem 2.3. If un ðxÞ is the approximate solution of (1.1) in space W m ½0; 1ð3 < m < 7Þ; 0 ¼ x1 < x2 <    < xn ¼ 1, and if pðxÞ; qðxÞ; f ðxÞ 2 C 2ðm3Þ ½0; 1, then 2ðm3Þ1

kuðxÞ  un ðxÞk1 6 d h

;

where kuðxÞk1 ¼ maxx2½0;1 juðxÞj; d is a constant, h ¼ max jxiþ1  xi j. 16i6n1

3. Numerical examples

Example 3.1. 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  xÞ x sinh x. It is easy to see that the exact solution is uðxÞ ¼ sinh x. i1 Using the present method, taking n ¼ 11; xi ¼ n1 ; i ¼ 1; 2; . . . ; n, the numerical results are compared with [7] in Table 1. Example 3.2. Consider the following nonlocal boundary value problem

(

u00 ðxÞ þ sin xu0 ðxÞ þ sinh xuðxÞ ¼ f ðxÞ; 0 6 x 6 1 pffiffi R 12 1 e 65 uð0Þ þ uð13Þ þ uð1Þ ¼ 19 þ e þ e3 ; 0 xuðxÞdx ¼ 64  2 9

where f ðxÞ is given such that its solution is uðxÞ ¼ x2 þ ex .

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Table 1 Comparison of relative errors for Example 3.1. x

Exact solution u(x)

Relative error [7]

Present method (W 4 )

Present method (W 5 )

Present method (W 6 )

0.08 0.16 0.24 0.40 0.48 0.56 0.64 0.72 0.80 0.88 0.96

0.080085 0.160684 0.242311 0.410752 0.498646 0.589732 0.684594 0.783840 0.888106 0.998058 0.114400

3.6E04 2.0E-04 1.4E04 9.5E05 7.5E05 5.5E05 3.6E05 1.9E05 1.8E15 4.2E04 5.2E04

8.9E05 5.2E05 3.1E05 1.7E05 2.4E05 1.6E05 1.1E05 6.2E06 5.7E13 2.6E07 2.0E06

9.6E08 4.1E07 1.7E07 1.7E07 2.1E07 2.0E07 1.1E07 5.6E08 4.0E11 1.2E07 7.9E08

9.7E08 8.7E08 6.1E08 3.7E08 3.4E08 2.9E08 1.3E08 1.2E08 2.6E09 3.5E08 6.3E08

1.5 10

6

1. 10

6

5. 10

7. 6. 5. 4. 3. 2. 1.

7

0.2

0.4

0.6

0.8

1.0

10 10 10 10 10 10 10

8 8 8 8 8 8 8

0.2

0.4

0.6

0.8 4

5. 10

9

4. 10

9

3. 10

9

2. 10

9

1. 10

9

1.0 5

0.2

0.4

0.6

0.8

1.0

6

Fig. 1. Absolute errors of u11 ðxÞ in W ; W and W .

Table 2 Comparison of absolute errors for Example 3.3. x

Exact solution

Absolute error [15]

Present method (W 6 )

0.10 0.20 0.30 0.40 0.60 0.70 0.80 0.90 1.00

0.112685 0.009222 0.006466 0.003320 0.0033201 0.0064668 0.0092222 0.0112685 0.1210710

6.50E05 5.25E05 3.63E05 1.87E05 1.73E05 3.40E05 4.98E05 6.20E05 6.34E05

2.90E07 1.90E07 8.79E08 4.14E08 4.08E08 8.43E08 1.77E07 2.68E07 5.47E07

i1 Using the present method, taking n ¼ 11; xi ¼ n1 ; i ¼ 1; 2; . . . ; n, the numerical results are shown in Fig. 1.

Example 3.3. The present method can also be extended to higher order linear nonlocal problems. Consider the following sandwich problem governed by a linear third-order multi-point boundary value problem

(

2

u000 ðxÞ  k u0 ðxÞ þ r ¼ 0; 0 6 x 6 1;  u0 ð0Þ ¼ 0; u 12 ¼ 0; u0 ð1Þ ¼ 0;

r ðkð2x1Þ2 sinhðkxÞþ2 coshðkxÞ tanh ð2kÞÞ The exact solution is given by uðxÞ ¼ . The method was tested for r ¼ 1 and k ¼ 5 and 10. Tak2k3 i1 ing n ¼ 11; xi ¼ n1 ; i ¼ 1; 2; . . . ; n, the numerical results are shown in Table 2.

4. Conclusion In this paper, an improved RKM is proposed for linear nonlocal boundary value problems. The improvement can avoid the construction of RK satisfying homogenous nonlocal BCs and then reduce the computational work of classical RKM. Also, the present method can be extended to boundary value problems with nonlinear BCs. Acknowledgments The author would like to express thanks to the unknown referees for their careful reading and helpful comments. The work was supported by the NSFC (Grant Nos. 11201041, 11026200), the Special Funds of the National Natural Science Foundation of China (Grant No. 11141003) and Qing Lan Project of Jiangsu Province.

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References [1] J. Henderson, C.J. Kunkel, Uniqueness of solution of linear nonlocal boundary value problems, Appl. Math. Lett. 21 (2008) 1053–1056. [2] J. Henderson, Existence and uniqueness of solutions of (k + 2)-point nonlocal boundary value problems for ordinary differential equation, Nonlinear Anal. 74 (2011) 2576–2584. [3] J. Henderson, R. Luca, Existence and multiplicity for positive solutions of a multi-point boundary value problem, Appl. Math. Comput. 218 (2012) 10572–10585. [4] Z.J. Du, L.J. Kong, Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems, Appl. Math. Lett. 23 (2010) 980–983. [5] F.C. Meng, Z.J. Du, Solvability of a second-order multi-point boundary value problem at resonance, Appl. Math. Comput. 208 (2009) 23–30. [6] Z.B. Bai, Positive solutions of some nonlocal fourth-order boundary value problem, Appl. Math. Comput. 215 (2010) 4191–4197. [7] F.Z. Geng, Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. Math. Comput. 215 (2009) 2095–2102. [8] B.Y. Wu, X.Y. Li, Application of reproducing kernel method to third order three-point boundary value problems, Appl. Math. Comput. 217 (2010) 3425– 3428. [9] X.Y. Li, B.Y. Wu, Reproducing kernel method for singular fourth order four-point boundary value problems, Bull. Malays. Math. Sci. Soc. 2 ((1) (2011) 147–151. [10] Y.Z. Lin, J.N. Lin, Numerical algorithm about a class of linear nonlocal boundary value problems, Appl. Math. Lett. 23 (2010) 997–1002. [11] M. Tatari, M. Dehghan, The use of the Adomian decomposition method for solving multipoint boundary value problems, Phys. Scr. 73 (2006) 672–676. [12] A. Saadatmandi, M. Dehghan, The use of Sinc-collocation method for solving multi-point boundary value problems, Commun. Nonlinear Sci. Numer. Simulat. 17 (2012) 593–601. [13] B.Y. Wu, X.Y. Li, A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method, Appl. Math. Lett. 24 (2010) 156–159. [14] F.Z. Geng, A numerical algorithm for nonlinear multi-point boundary value problems, J. Comput. Appl. Math. 236 (2012) 1789–1794. [15] I.A. Tirmizi, E.H. Twizell, S.U. Islam, A numerical method for third-order non-linear boundary-value problems in engineering, Int. J. Comput. Math. 82 (2005) 103–109. [16] M.G. Cui, F.Z. Geng, Solving singular two-point boundary value problem in reproducing kernel space, J. Comput. Appl. Math. 205 (2007) 6–15. [17] F.Z. Geng, M.G. Cui, Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl. 327 (2007) 1167–1181. [18] F.Z. Geng, M.G. Cui, Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl. Math. Comput. 192 (2007) 389–398. [19] M.G. Cui, Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Space, Nova Science Pub Inc, Hauppauge, 2009. [20] X.Y. Li, B.Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math. 243 (2013) 10–15.