A New Numerical Method to Solve Nonlinear Volterra-Fredholm Integro-Diﬀerential Equations

. In this paper, a new method combining the simpliﬁed reproducing kernel method (SRKM) and the homotopy perturbation method (HPM) to solve the nonlinear Volterra-Fredholm integro-diﬀerential equations (V-FIDE) is proposed. Firstly the HPM can convert nonlinear problems into linear problems. After that we use the SRKM to solve the linear problems. Secondly, we prove the uniform convergence of the approximate solution. Finally, some numerical calculations are proposed to verify the eﬀectiveness of the approach.

The parameters λ 1 , λ 2 are constants. F (u(x)) and G(u(x)) are constant coefficient polynomials of u(x). The V-FIDE has been widely used in physics, biological and engineering [1,2,3,10,16]. In order to obtain accurate numerical solutions more quickly, many methods for solving such problems have been proposed in recent years. Maleknejad [10] introduced the hybrid functions method. Babolian [2,3] proposed the triangular functions method and the operational matrix with block-pulse functions. Hybrid Legendre polynomials and block-pulse functions approach were used by Maleknejad [9]. Bakodah [4] discussed the Laplace discrete Adomian decomposition method over the integro-differential equation. Biazar and Ghanbari [5] presented He's homotopy perturbation method. Bildik [6] used the modified decomposition method to obtain the approximate solution of nonlinear V-FIDE. Ghasemi [8] formulated homotopy perturbation method for solving nonlinear equations. In recent years, with the development of reproducing kernel space theory, many scholars have successfully applied reproducing kernel method to solve problems [12,13,14,15,17,18]. But the traditional reproducing kernel method [11] is difficult to deal with the integral term, while the HPM can be effectively dealt with the integral term. Because the traditional reproducing kernel method needs orthogonalization, the calculation method is complex and time-consuming. Our method avoids the Smith orthogonalization process in order to save the calculation time and running memory. This article discusses the nonlinear V-FIDE by using SRKM and HPM in the reproducing kernel space, so that the equation can achieve higher accuracy. In this paper, we describe the homotopy perturbation theory in Section 2. The reproducing kernel theory will be shown in Sections 3 and 4. The last part presents some numerical examples. In the end, we have the conclusions.

Homotopy perturbation method
For Equation (1.1), we first have to solve the nonlinear part. The homotopy perturbation method provides a good theoretical basis, we embed a small parameter p (p ∈ [0, 1]) by constructing a homotopy map when p = 0, the Equation (2.1) is an initial value problem: In this way, when p → 1, the approximate solution of the nonlinear operator equation is obtained Take the k derivatives of F, G and set p = 0, then substitute the type into Equation (2.1): Comparing the coefficients of p i on both sides of equation and setting them equal, we can get for k = 0, Adding the solution u k of Equations (2.3)-(2.4), we obtain the true solution to nonlinear equations

Reproducing kernel Hilbert space
We will discuss the Equation (1.1) with support of the reproducing kernel space theory.
Definition 1. ( [7]) Let H be the Hilbert space, and the elements in H are complex-valued functions on X. If there is a unique function K s (t) for ∀s ∈ X that satisfies Then H is defined as reproducing kernel space, K(s, t) = K s (t) is defined as reproducing kernel function.
is an absolutely continuous real value function, is an absolutely continuous real value function, The combination of HPM and SRKM Now we introduce reproducing kernel method. Firstly, we define a linear operator L : Therefore, Equation (4.1) can be expressed as It's easy to prove that L is a bounded linear operator. L * is the adjoint operator of L.
] is a set of mutually distinct dense points, then Let ψ 1 (x) = R(x, a) and S n+1 = span{φ 1 (x), φ 2 (x), . . . , φ n (x), ψ 1 (x)}. We can obtain the following conclusions: , then u n = P n u satisfies: Proof. Proof. u n − u → 0 holds as n → ∞ in W 2 2 [a, b]. According to the reproducibility of the reproducing kernel function, we have because of the boundedness of the reproducing kernel function, R x (y) ≤ M 0 , then Consequently, the exact solution u n ∈ S n+1 of Equation (4.2) can be developed as follows Then applying the form Equation (4.4) to Equation (4.3), we can obtain Note that . . .

Numerical examples
The main methods used in this work have been described in the previous sections, some numerical examples are given to illustrate its effectiveness. Meanwhile, the red lines in the figure represent the approximate solutions and the blue dots represent the exact solutions. The absolute errors of the exact, the approximate solutions and CPU time (seconds) are listed in the tables. We also used the following formula to calculate the convergence rate r: Example 1. For the following nonlinear V-FIDE: where y(x) = 2x + x 2 + 1 10 x 6 − 1 32 , the exact solution is u(x) = x 2 (see Figure 1(a)). The comparison of the numerical results and the absolute error are listed in Table 1, we get an exact solution with higher precision than the method of hybrid Legendre polynomials and Block-Pulse functions [9] for n = 12.
The exact solution is u(x) = sin(x) (see Fig. 1 Table 2 illustrates the numerical results and the absolute error. From the Table 2 results, we can see that our method approximates the exact solution more closely than the hybrid Legendre polynomials and Block-Pulse functions [9] for n = 12.
Example 3. Consider the nonlinear V-FIDE: with the exact solution given by u(x) = x 2 . The comparison of the numerical results and the absolute error are listed in Table 3, our method is more accurate than the method of Laplace discrete adomian decomposition in [4] for n = 4.

Conclusions
In this article, the SRKM and the HPM were successfully applied to figure out the nonlinear V-FIDE by getting the uniform approximate solution. Besides, compared with the method of Hybrid Legendre polynomials [9], Laplace discrete adomian decomposition method [4], the convergence speed and accuracy of solution were better.