Abstract and Applied AnalysisVolume 2012 (2012), Article ID 414612, 9 pagesdoi:10.1155/2012/414612

Research Article

Approximate Solutions to Three-Point Boundary Value Problems with Two-Space Integral Condition for Parabolic Equations

Jing Niu,1,2 Yingzhen Lin,¹ and Minggen Cui¹

¹Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China²School of Mathematics and Sciences, Harbin Normal University, Harbin 150025, China

Received 24 January 2012; Accepted 16 February 2012

Academic Editor: Chaitan Gupta

Copyright © 2012 Jing Niu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We construct a novel reproducing kernel space and give the expression of reproducing kernel skillfully. Based on the orthogonal basis of the reproducing kernel space, an efficient algorithm is provided firstly to solve a three-point boundary value problem of parabolic equations with two-space integral condition. The exact solution of this problem can be expressed by the series form. The numerical method is supported by strong theories. The numerical experiment shows that the algorithm is simple and easy to implement by the common computer and software.

1. Introduction

Nonclassical boundary value problems with nonlocal boundary conditions arise naturally in various engineering models and physical phenomena, for example, chemical engineering, thermoelasticity, underground water flow, and population dynamics [1–4]. The importance of boundary value problems with integral boundary conditions has been pointed out by Samarskiĭ [5].

Boundary value problems for parabolic equations with an integral boundary condition are investigated in the literature for the development, analysis, and implementation of accurate methods [6–11]. Integral boundary conditions of models emerged in previous literatures can be summed up as However, Marhoune [12] studied the parabolic equation with a generalized integral boundary condition (1.2)–(1.4). This model is more universal, and it extended usual integral boundary conditions. The form is as follows: subject to the initial-boundary value conditions and the integral condition with the function ,  and its derivatives are bounded on the interval : , . In the following, we may assume   because it can be got from homogeneous boundary conditions. The existence and uniqueness of the solution for (1.2)–(1.4) have been proved in [12].

A practical model is typed by various definite conditions under different environment. Investigation about the definite conditions is the key problem to the model. Due to condition (1.4), it is difficult to construct reproducing kernel space, so nobody gives the algorithm for the above problem by applying reproducing kernel theory. In this paper, the author successfully constructs a novel reproducing kernel space which includes boundary conditions (1.3)-(1.4) and gains the expression of the reproducing kernel skillfully. Meanwhile, we provide a simple algorithm for solving (1.2)–(1.4). Based on the orthogonal basis in the reproducing kernel space, the exact solution is given by the form of series. Meanwhile, using a similar process, it is possible to solve other linear ordinary differential equations, partial differential equations with the same boundary value conditions.

2. Constructive Method for the Reproducing Kernel Space

  and   are inner product spaces and , and  they are defined in the following.

Definition 2.1. The inner product space is defined byThe inner product of is defined by And it possesses associated norm .

Lemma 2.2. Inner space is a Hilbert reproducing kernel space. Its reproducing kernel function and for any , where where

This proof can be found in [13–17].

Clearly, one has

Lemma 2.3. Fix an , .

Definition 2.4. The subspace is defined by Obviously, is a closed subspace of the reproducing kernel .  It is very important to obtain the representation of reproducing kernel in ,  which is the base of our algorithm. Therefore, our work begins with some lemmas to provide constructive method for reproducing kernel in .

Lemma 2.5. Fix an , .

Proof. Otherwise, for all , by (2.4), then , which is contradictory.

Lemma 2.6. Fix an , , namely, .

Proof. Otherwise, ,  then which implies that , and this contradicts with Lemma 2.5.

Consider a function and one can check carefully that   is the reproducing kernel of .

3. An Orthogonal Basis of

Let an operator ,  putting rewriting (1.2)–(1.4) as

Lemma 3.1. is a bounded linear operator.

Proof. Noting that where are positive real numbers, Combining with the bounded function , it holds that is a bounded linear operator.

We will choose and fix a countable dense subset    and define by

Lemma 3.2. Consider the following:  ,  (see [11]).

Lemma 3.3. The function system is a complete system of the space .

Proof. For every , we have which shows that   due to the denseness of . It follows that   from the existence of .

Applying Gram-Schmidt process, we obtain an orthogonal basis   of , such that where are orthogonal coefficients.

4. Numerical Algorithm

In this section, it is explained how to deduce the exact solution from the orthogonal basis   of .

Theorem 4.1. The exact solution of (3.2) can be expressed by

Proof. The exact solution   can be expanded to a Fourier series in terms of normal orthogonal basis in ,

We obtain the -truncation approximate solution of (3.2), which is -truncation Fourier series of the exact solution in (3.2), so ,  as .

Theorem 4.2. Consider the following:  ,  as  .

Proof. Since   is continuous on , it follows that , for all ,  where is a constant. When ,

5. Numerical Example

In this section, a numerical example is studied to demonstrate the accuracy of the present algorithm. The example is computed by Mathematica 5.0. Results obtained by the algorithm are compared with the analytical solution and are found to be in good agreement.

Example 5.1. Consider a three-point boundary value problem of parabolic equations with two-space integral condition The exact solution is . The numerical results are collected in Table 1.

Table 1: Numerical results.

(0.00,0.0001)
0.000700035
0.000000000

(0.95,0.05)
0.187934
0.000230442
0.0000112112

(0.85,0.15)
−0.241376
0.0000224685

(0.75,0.25)
−1.36687
0.00086302
0.0000715374

(0.65,0.35)
−2.7296
0.00322583
0.000250607

(0.55,0.45)
−3.87049
0.00287117
0.000571766

(0.45,0.55)
−4.34049
0.00517334
0.000894053

(0.35,0.65)
−3.71389
0.00142632
0.000913571

(0.25,0.75)
−1.60569
0.00213525
0.000480841

(0.15,0.85)
2.3062
0.00450865
0.000593452

(0.05,0.95)
8.25283
0.0020922
0.00048678

6. Conclusion

In this paper, we construct a reproducing kernel space by new method, in which each function satisfies boundary value conditions of considered problems. In this space, a numerical algorithm is presented for solving a class of parabolic equations with two-space integral boundary condition. Exact solution with series form is given. Approximate solution obtained by present algorithm converges to exact solution uniformly.

Acknowledgments

The authors appreciate the constructive comments and suggestions provided from the kind referees and editor. This work was supported by Youth Foundation of Heilongjiang Province under Grant no. QC2010036, Fundamental Research Funds for the Central Universities under Grant no. HIT.NSRIF.2009050, and Academic Foundation for Youth of Harbin Normal University (11KXQ-04) and also by 10KXQ-05.

References

Y. S. Choi and K.-Y. Chan, “A parabolic equation with nonlocal boundary conditions arising from electrochemistry,” Nonlinear Analysis: Theory, Methods & Applications, vol. 18, no. 4, pp. 317–331, 1992.

R. E. Ewing and T. Lin, “A class of parameter estimation techniques for fluid flow in porous media,” Advances in Water Resources, vol. 14, no. 2, pp. 89–97, 1991.

P. Shi, “Weak solution to an evolution problem with a nonlocal constraint,” SIAM Journal on Mathematical Analysis, vol. 24, no. 1, pp. 46–58, 1993.

A. V. Kartynnik, “A three-point mixed problem with an integral condition with respect to the space variable for second-order parabolic equations,” Differentsial'nye Uravneniya, vol. 26, no. 9, pp. 1568–1575, 1990.

A. A. Samarskiĭ, “Some problems of the theory of differential equations,” Differentsial'nye Uravneniya, vol. 16, no. 11, pp. 1925–1935, 1980.

M. Dehghan, “Numerical solution of a parabolic equation with non-local boundary specifications,” Applied Mathematics and Computation, vol. 145, no. 1, pp. 185–194, 2003.

M. Dehghan, “Efficient techniques for the second-order parabolic equation subject to nonlocal specifications,” Applied Numerical Mathematics, vol. 52, no. 1, pp. 39–62, 2005.

M. Dehghan, “Numerical techniques for a parabolic equation subject to an overspecified boundary condition,” Applied Mathematics and Computation, vol. 132, no. 2-3, pp. 299–313, 2002.

M. Denche and A. L. Marhoune, “A three-point boundary value problem with an integral condition for parabolic equations with the Bessel operator,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 13, no. 6, pp. 85–89, 2000.

M. Tatari and M. Dehghan, “On the solution of the non-local parabolic partial differential equations via radial basis functions,” Applied Mathematical Modelling, vol. 33, no. 3, pp. 1729–1738, 2009.

Y. Zhou, M. Cui, and Y. Lin, “Numerical algorithm for parabolic problems with non-classical conditions,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 770–780, 2009.

A. L. Marhoune, “A three-point boundary value problem with an integral two-space-variables condition for parabolic equations,” Computers & Mathematics with Applications, vol. 53, no. 6, pp. 940–947, 2007.

F. Geng and M. Cui, “Solving singular nonlinear two-point boundary value problems in the reproducing kernel space,” Journal of the Korean Mathematical Society, vol. 45, no. 3, pp. 631–644, 2008.

Y. Lin, J. Niu, and M. Cui, “A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7362–7368, 2012.

W. Jiang, M. Cui, and Y. Lin, “Anti-periodic solutions for Rayleigh-type equations via the reproducing kernel Hilbert space method,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 7, pp. 1754–1758, 2010.

Y. Lin and J. Lin, “Numerical method for solving the nonlinear four-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3855–3864, 2010.

M. G. Cui and Y. Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Hilbert Space, Nova, New York, NY, USA, 2009.