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).