Research Article | Open Access
The Combined RKM and ADM for Solving Nonlinear Weakly Singular Volterra Integrodifferential Equations
The reproducing kernel method (RKM) and the Adomian decomposition method (ADM) are applied to solve th-order nonlinear weakly singular Volterra integrodifferential equations. The numerical solutions of this class of equations have been a difficult topic to analyze. The aim of this paper is to use Taylor’s approximation and then transform the given th-order nonlinear Volterra integrodifferential equation into an ordinary nonlinear differential equation. Using the RKM and ADM to solve ordinary nonlinear differential equation is an accurate and efficient method. Some examples indicate that this method is an efficient method to solve th-order nonlinear Volterra integro-differential equations.
In this paper, we consider the following th-order nonlinear weakly singular Volterra integro-differential equation of the following form [1–4]: where , is a given function, is the unknown function, and is the kernel of the integro equation. We usually assume that the function and are continuous or square integrable on .
Some problems of mathematical physics are described in terms of (1.1) which has been studied by different methods including the spline collocation method , piecewise polynomials , Haar wavelets , the homotopy perturbation method (HPM) [8, 9], the wavelet-Galerkin method , Taylor polynomials , the Tau method , the sinc-collocation method , the combined Laplace transform-Adomian decomposition method , and the Adomian’s asymptotic decomposition method  to determine exact and approximate solutions. But to our knowledge there is still no viable analytic approach for solving weakly singular Volterra integro-differential equations. The present work is motivated by the desire to obtain approximate solution to th-order nonlinear weakly singular Volterra integro-differential equation, where the integrand is weakly singular in the sense that its integral is continuous at the singular point, that is, its kernel is singular as .
Reproducing kernel theory has important application in numerical analysis, differential equation, probability and statistics and so on [16, 17]. And the RKM has been applied successfully to solving linear and nonlinear problems [18–20].
The rest of the paper is organized as follows. In the Section 2, transforming (1.1) into an differential equation by Taylor’s approximation. In Section 3, the RKM is introduced. Applying RKM and ADM to solving (1.1) is discussed in Section 4. The numerical examples are presented in Section 5. Finally, a brief conclusion is stated in last section.
2. Taylor’s Approximation
Consider the following th-order nonlinear weakly singular Volterra integro-different equation: We have by setting Rewriting (2.3) as where the solution under the integral has been replaced by . Thus, So that or equivalently We use the following Taylor’s approximation of degree of about : Thus, Substituting the approximate relation (2.9) into the right hand side of (2.7) yields Therefore, (2.1) can be approximated by the th-order nonlinear differential equation (2.10).
3. Analysis of Reproducing Kernel Hilbert Space
Definition 3.1 (reproducing kernel Space , see ). and endowed it with the inner product and norm, respectively,
Theorem 3.2. The space is a reproducing kernel space. That is, there exists a function , for each fixed , and for any , satisfying the reproducing kernel can be denoted by where are known coefficients.
Theorem 3.3. Let be a reproducing kernel space and . If converges to in the sense of , then converges to uniformly.
Property 1. If is a reproducing kernel space, the reproducing kernel function in is unique.
Definition 3.4 (reproducing kernel Space , see ). and endowed it with the inner product and norm, respectively,
There exists a unique reproducing kernel function , and can be denoted by
4. Combined ADM and RKM
4.1. Representation of the Inverse Operator
Here, we propose a new differential operator, as follows: letting where , then we convert (2.10) as follows: We now introduce how to determine the inverse operator of . Obviously, is a bounded linear operator.
We choose as any dense set in , and let , where is the conjugate operator of and is given by (3.7). Furthermore, for simplicity, let denote , namely,
Now, several Lemmas are given.
Lemma 4.1. is the complete system of .
Proof. For , let , that is,
Note that is the dense set in , therefore . It follows that from the existence of .
Lemma 4.2. The following formula holds where the subscript of operator indicates that the operator applies to functions of .
Proof. Consider This completes the proof.
The orthonormal system of can be derived from Gram-Schmidt orthogonalization process of , where are orthogonal coefficients.
Theorem 4.3. If the inverse operator , exists and is dense on , then the inverse operator can be determined as where .
Proof. From (4.9), it holds that The proof of the theorem is complete.
From Theorem 4.3, obviously, is determined.
4.2. Decomposition Method
By applying to both sides of (4.3), we have The ADM introduces the solution and the nonlinear function by infinite series where are Adomian polynomials for the nonlinear term and can be found from the following formula: Substituting (4.12) and (4.13) into (4.11) yields According to the ADM, the components can be determined as which gives where , .
We obtain approximate solution of the following equation:
The following use some examples to demonstrate the effectiveness of the algorithm.
5. Numerical Examples
To illustrate the applicability and effectiveness of our method, we consider the following examples. Symbolic and numerical computations performed by using Mathematica 5.0.
Example 5.1. Consider the following first-order nonlinear weakly singular Volterra integro-differential equation: with , , and .
Let , , .
On select 100 points and get the approximate solution , the results are shown in Figure 1.
Example 5.2. Consider the following second-order nonlinear weakly singular Volterra integro-differential equation: with , , and .
Let , , , .
On select 100 points and get the approximate solution , the results are shown in Table 1.
Example 5.3. Consider the following third-order nonlinear weakly singular Volterra integro-differential equation: with , , , and .
Let , , , , .
On select 100 points and get the approximate solution , the results are shown in Figure 2.
In this paper, we have reduced the solution of nonlinear weakly singular Volterra integro-differential equations to the solution of ordinary nonlinear differential equations by removing the singularity using an appropriate Taylor’s approximation. Then we have demonstrated the solution of these ordinary nonlinear differential equations by RKM and ADM. The ADM is an accurate and efficient method to solve nonlinear weakly singular Volterra integro-differential equations.
Research supported by the National Natural Science Foundation of China (61071181), the Educational Department Scientific Technology Program of Heilongjiang Province(12512133 and 12521148) and the Scientific Innovation Project for Graduate of Heilongjiang Province (YJSCX2012-184HLJ).
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Copyright © 2012 Xueqin Lv and Sixing Shi. 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.