Abstract and Applied Analysis

Volume 2014 (2014), Article ID 615840, 5 pages

http://dx.doi.org/10.1155/2014/615840

## Solving a Class of Singularly Perturbed Partial Differential Equation by Using the Perturbation Method and Reproducing Kernel Method

^{1}Department of Mathematics, Inner Mongolia University of Technology, Hohhot 010051, China^{2}Jining Teachers College, Wulanchabu, Inner Mongolia 012000, China

Received 28 March 2014; Revised 14 May 2014; Accepted 28 May 2014; Published 17 June 2014

Academic Editor: Dragos-Patru Covei

Copyright © 2014 Yu-Lan Wang 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 give the analytical solution and the series expansion solution of a class of singularly perturbed partial differential equation (SPPDE) by combining traditional perturbation method (PM) and reproducing kernel method (RKM). The numerical example is studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

#### 1. Introduction

Singularly perturbed problems (SPPs) arise very frequently in many branches of mathematics such as fluid mechanics and chemical reactor theory. It is well known that the solutions of SPPs exhibit a multiscale character. So there are some major computation difficulties. In recent years, many special methods have been developed to deal with SPPs. Many papers [1–4] are devoted to SPPs of ordinary differential equation and the authors discussed the situation and width of boundary layer(s) and give some effective numerical algorithms. But few papers [5–7] deal with SPPDE.

The reproducing kernel Hilbert function space has been shown in [8–10] to solve a large class of linear and nonlinear problems effectively. However, in [8–10], it cannot be used directly to SPPs. The aim of this work is to fill this gap. In this paper, we solve a class of SPPs in reproducing kernel space. By using a traditional perturbation method and RKM, the series expansion solution of a class of SPPDE is given. The main contribution of this paper is to use RKM in SPPDE. The reason why we use this method is that we aim to solve some problems in many areas of science and improve high precision.

Let us consider the following SPPDE: where is a positive number, functions and are sufficiently smooth, and . Under suitable continuity and compatibility conditions, the problem (1) has a unique solution . In [5–7], we notice that a small variation in the parameter produces a large variation in the solution. It is quite well known that solution of such problems involves boundary layers.

#### 2. Perturbation Method

Let ; (1) can be equivalently turned into where In view of the traditional perturbation method [11], we use the parameter to expand the solution Substituting (4) into (2), we get and equating coefficients of the identical powers of yields the following equations:

Next, we use the reproducing kernel method to solve each of the equations above, after obtaining all of , from (6), (7), (8),…, because of ; therefore, the analytical solution of (2) is obtained. Now, let us introduce how to use the reproducing kernel method to solve (6), (7), (8),….

#### 3. Reproducing Kernel Method

For getting from (6), (7), (8),…, we let Equation (6) can be converted into the following equivalent form: Equations (7), (8), can be converted into the following equivalent form:

Be aimed at with the purpose of solving (11) and (12), we need to introduce the reproducing kernel space, previously. Like in [12], we give the reproducing kernel spaces : Then, we define the inner product of . Consider the following: From [13], we can prove is a reproducing kernel Hilbert space, and the reproducing kernel of it is After all of these, we introduce the reproducing kernel space [14] and the inner product of it; see [15], and the reproducing kernel of is

Similar to the definition of , we can define and it is the reproducing kernel , where are also a reproducing kernel space with the reproducing kernel (see [16–18]).

It is easy to prove is a linear bounded operator, because the problem (1) has a unique solution ; in other words, is also a invertible operator, so [19] if , where and , is existent and is countable dense points in . Let , where the are the coefficients resulting from Gram-Schmidt orthonormalization and , ; then is an analytical solution of equation .

*(i) Linear Problem.* Suppose equation is a linear problem; that is, ; we define an approximate solution by

Theorem 1 (see [20–22] convergence analysis). *Let ; then the sequence of real numbers is monotonously decreasing and and the sequence is convergent uniformly to .*

*(ii) Nonlinear Problem (see* [23]*).* Suppose equation is a nonlinear problem; that is, , where is a nonlinear operator; we give an iterative sequence : is the solution of the linear equation ; is the solution of the linear equation .

Lemma 2. *If , then is the solution of equation .*

Theorem 3. *Suppose the nonlinear operator satisfies contractive mapping principle; that is,
**
and then is convergent.*

Using reproducing kernel method, we can get

Therefore, the analytical solution of (2) is obtained.

In calculation, we use as the approximation solution of (2).

#### 4. Numerical Experiment

*Example 4. *Considering a nonlinear advection equation with perturbation term
where , is the true solution, and is the approximate solution (Table 2). When we take , , and , the numerical results are given in Table 1.

#### 5. Conclusions and Remarks

In this paper, the combination of traditional perturbation and reproducing kernel space methods was employed successfully for solving nonlinear advection equation with singular term. The numerical results show that the present method is an accurate and reliable. Moreover, the method is also effective solving other nonlinear singular perturbation problems.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This paper is supported by the Natural Science Foundation of China (11361037), the Natural Science Foundation of Inner Mongolia (2013MS0109), and the Project Application Technology Research and Development Foundation of Inner Mongolia (no. 20120312).

#### References

- Y. Han, X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of singular nonlinear Sturm-Liouville problems with Carathéodory perturbed term,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 160891, 23 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Z. Geng and S. P. Qian, “Solving singularly perturbed multipantograph delay equations based on the reproducing kernel method,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 794716, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - M. Stojanović, “Splines difference methods for a singular perturbation problem,”
*Applied Numerical Mathematics*, vol. 21, no. 3, pp. 321–333, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Bushnaq, B. Maayah, S. Momani, and A. Alsaedi, “A Reproducing Kernel Hilbert Space Method for Solving Systems of Fractional Integrodifferential Equations,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 103016, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - F. O. Ilicasu and D. H. Schultz, “High-order finite-difference techniques for linear singular perturbation boundary value problems,”
*Computers & Mathematics with Applications*, vol. 47, no. 2-3, pp. 391–417, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Boglaev, “Domain decomposition in boundary layers for singularly perturbed problems,”
*Applied Numerical Mathematics*, vol. 34, no. 2-3, pp. 145–166, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Phang, Y. Wu, and B. Wiwatanapataphee, “Computation of the domain of attraction for suboptimal immunity epidemic models using the maximal Lyapunov function method,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 508794, 7 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Inc, A. Akgül, and A. Kiliçman, “A novel method for solving KdV equation based on reproducing kernel Hilbert space method,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 578942, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cui and F. Geng, “A computational method for solving one-dimensional variable-coefficient Burgers equation,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1389–1401, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Wang, T. Chaolu, and Z. Chen, “Using reproducing kernel for solving a class of singular weakly nonlinear boundary value problems,”
*International Journal of Computer Mathematics*, vol. 87, no. 1–3, pp. 367–380, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Boglaev, “Monotone iterative algorithms for a nonlinear singularly perturbed parabolic problem,”
*Journal of Computational and Applied Mathematics*, vol. 172, no. 2, pp. 313–335, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Y. Li, B. Y. Wu, and R. T. Wang, “Reproducing kernel method for fractional riccati differential equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 970967, 6 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - H. Du and M. Cui, “Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space,”
*Applied Mathematics Letters*, vol. 21, no. 6, pp. 617–623, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Wang, X. Cao, and X. Li, “A new method for solving singular fourth-order boundary value problems with mixed boundary conditions,”
*Applied Mathematics and Computation*, vol. 217, no. 18, pp. 7385–7390, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Inc, A. Akgül, and A. Kiliçman, “Explicit solution of telegraph equation based on reproducing kernel method,”
*Journal of Function Spaces and Applications*, vol. 2012, Article ID 984682, 23 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Akram and H. U. Rehman, “Numerical solution of eighth order boundary value problems in reproducing kernel space,”
*Numerical Algorithms*, vol. 62, no. 3, pp. 527–540, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Yao and Y. Lin, “Solving singular boundary-value problems of higher even-order,”
*Journal of Computational and Applied Mathematics*, vol. 223, no. 2, pp. 703–713, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Lin and J. Lin, “A numerical algorithm for solving a class of linear nonlocal boundary value problems,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 997–1002, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Y. Wu and X. Y. Li, “A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method,”
*Applied Mathematics Letters*, vol. 24, no. 2, pp. 156–159, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Z. Geng and X. M. Li, “A new method for Riccati differential equations based on reproducing kernel and quasilinearization methods,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 603748, 8 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Mokhtari, F. T. Isfahani, and M. Mohammadi, “Reproducing kernel method for solving nonlinear differential-difference equations,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 514103, 10 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Inc and A. Akgül, “The reproducing kernel Hilbert space method for solving Troeschs problem,”
*Journal of the Association of Arab Universities for Basic and Applied Sciences*, no. 1, pp. 19–27, 2013. View at Publisher · View at Google Scholar - M. Inc and A. Akgül, “Numerical solution of seventh-order boundary value problems by a novel m ethod,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 745287, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet