Generalized Differential and Integral EquationsView this Special Issue
Solving a System of Linear Volterra Integral Equations Using the Modified Reproducing Kernel Method
A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation , is bounded and its image space is the reproducing kernel space . It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. The n-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient.
The purpose of this paper is to solve a system of linear Volterra integral equations where
In (1), the functions and are given, and is the solution to be determined. We assume that (1) has a unique solution. Volterra integral equation arises in many physical applications, for example, potential theory and Dirichlet problems, electrostatics, mathematical problems of radiative equilibrium, the particle transport problems of astrophysics and reactor theory, and radiative heat transfer problems [1–5]. Several valid methods for solving Volterra integral equation have been developed in recent years, including power series method , Adomain’s decomposition method , homotopy perturbation method [8, 9], block by block method , and expansion method .
Since the reproducing kernel space , which is a special Hilbert space, is constructed in 1986 , the reproducing kernel theory has been applied successfully to many linear and nonlinear problems, such as differential equation, population model, and many other equations appearing in physics and engineering [12–21]. The traditional reproducing kernel method is limited, because it requires that the image space of operator in linear operator equation is and operator must be bounded. In order to enlarge its application range, the MRKM removes the boundedness of and weakens its image space to . Subsequently, we apply the MRKM to obtain the series expression of the exact solution for (1). The n-term approximation solution is provided by truncating the series. The final numerical comparisons between our method and other methods show the efficiency of the proposed method. It is worth to mention that the MRKM can be generalized to solve other system of linear equations.
2.1. The Reproducing Kernel Space
The reproducing kernel space consists of all absolute continuous real-valued functions, which defined on the closed interval , and the first derivative functions belong to .
The inner product and the norm are equipped with
Theorem 1. is a reproducing kernel space with reproducing kernel  that is, for every and , it follows that
2.2. The Reproducing Kernel Space
The reproducing kernel space consists of all real-valued functions in which the first derivative functions are absolute continuous on the closed interval and the second derivative functions belong to .
The inner product and the norm are equipped with
Theorem 2. is a reproducing kernel space with reproducing kernel  that is, for every and , it follows that
2.3. Hilbert Space
Hilbert space is defined by The inner product and the norm are given by
It is easy to prove that is a Hilbert space.
3. The Exact Solution of (1)
3.1. Identical Transformation of (1)
Consider the th equation of (1):
Define operator , where . Then, (1) can be turned into where .
3.2. The Exact solution of (1)
Let be a dense subset of interval , and define for every ; the subscript of means that the operator acts on the function of . It is easy to prove that .
Theorem 3. is complete in .
Proof. Take such that for every .
From this fact, it holds that
for every . The dense assumes that
Since (16) has a unique solution, it follows that . This completes the proof.
We arrange , ,, , , , , , , , ,, ,, denoted by ; that is, . In a general way, . The orthogonal basis in from Gram-Schmidt orthogonalization of is as follows:
Theorem 4. The exact solution of (1) can be expressed by where ; if , then .
Proof. Assume that is the exact solution of (1). can be expanded to Fourier series in terms of normal orthogonal basis in : if , then When , it holds that
Corollary 5. The approximate solution of (1) is and converges uniformly to on as for every .
Proof. Obviously, holds as ; that is, is the approximate solution of (1).
Note that . Combining with the expression of , we have It shows that converges uniformly to on as for every . So the proof is complete.
Remark 6. If and in (1), then it is reasonable to regard the unknown functions as the elements of .
4. Numerical Examples
Example 7. Consider the following system of Volterra integral equations of the second kind : where and are chosen such that the exact solution is . The numerical results obtained by using the present method are compared with  in Table 1.
Example 8. Consider the following system of linear Volterra integral equations of the second kind : where and are chosen such that the exact solution is . The numerical results obtained by using the present method are compared with  in Table 2.
In this paper, we modify the traditional reproducing kernel method to enlarge its application range. The new method named MRKM is applied successfully to solve a system of linear Volterra integral equations. The numerical results show that our method is effective. It is worth to be pointed out that the MRKM is still suitable for solving other systems of linear equations.
The research was supported by the Fundamental Research Funds for the Central Universities.
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