A high order approach for nonlinear Volterra-Hammerstein integral equations

Abstract: Here a scheme for solving the nonlinear integral equation of Volterra-Hammerstein type is given. We combine the related theories of homotopy perturbation method (HPM) with the simplified reproducing kernel method (SRKM). The nonlinear system can be transformed into linear equations by utilizing HPM. Based on the SRKM, we can solve these linear equations. Furthermore, we discuss convergence and error analysis of the HPM-SRKM. Finally, the feasibility of this method is verified by numerical examples.


Introduction
As a classical model of nonlinear integral equation, the nonlinear Volterra-Hammerstein type equations [1][2][3] can be used in biological models, fluid mechanics, communication theory, etc. In this article, we primarily concentrate on numerical solutions for Volterra-Hammerstein. Generally, these systems can be characterized by: where nonlinear function F is known. λ is a constant, G : C n [a, b] → C[a, b] is a linear operator with boundedness. The choice of constraints H are satisfied that G(u(x)) = f have a unique solution. Several numerical methods of the nonlinear Volterra-Hammerstein type equations have been proffered, for instance, continuous interpolation method [1], iteration method [2], Galerkin method [4], and other methods [5][6][7]. Mirzaee [8] approximated the nonlinear Hammerstein integral equation by utilizing the least square method based on the Legendre-Bernstein. In [9], Ordokhani proposed a configuration method based on a Walsh function that converts the Hammerstein equations into algebraic equations. Normally, in applying these methods, we have to calculate substantially integrals or employ the iterative method, which is computationally complicated. To address these problems, in [4], Mandal proposed Galerkin methods and acquired the superconvergence results in the uniform norm. Recently, reproducing kernel space theory has commonly applied to solve the nonlinear boundary value problems [10,11], heat conduction equation [12], interfacial issues [13,14], the Allen-Cahn equation [15], fractional-order Boussinesq equation [16], and other functional equation models [17][18][19][20][21]. Several methods [22][23][24] are also proposed for different kinds of integral equations. It is worth mentioning that many scholars improved the RKMs to study various kind of equations which is the case of [25][26][27][28][29][30]. However, the traditional reproducing kernel method [31] requires orthogonalization in the solution process, and the calculation process is complex and time-consuming. In this work, we apply HPM to do away with the integral term conveniently. We utilize SRKM to effectively avoid the Smith orthogonalization process and economizes the calculation time.
The outline of the work is as follows: We introduce the reproducing kernel theory and the homotopy perturbation theory in section 2. In section 3, we display the HPM-SRKM. Then in section 4, some numerical experiments are presented. Finaly, a conclusion is generalized in the final section.

Reproducing kernel Hilbert space
The inner product and the norm of two reproducing kernel spaces related to this model are introduced.
Definition 1. ( [32]) Let H be the Hilbert space, and the elements in H be complex-valued functions on X. If there is a unique function K s (t) for ∀s ∈ X that satisfies Then H is defined as a reproducing kernel space, K(s, t) = K s (t) is defined as a reproducing kernel function.

Introduction to Homotopy perturbation method (HPM)
The HPM ( [33]) is implemented by embedding a small perturbation operator p(p ∈ [0, 1]) and a homotopy path is constructed: (2.1) When the operator p = 0, the Eq (2.1) is equivalent to the subsequent initial value problem: The Eq (2.1) is the original problem when p = 1. The Eqs (2.1) and (2.2) are also subject to condition H. When the operator p changes from 0 to 1, the solution u(x) of Eq (1.1) follows the homotopy path from the initial value Eq (2.2) to the original problem. From the perturbation parameter theory [34], the solution satisfying the homotopy path can be extended into the form of Maclaurin series of p: Therefore, when p → 1, the approximate solution of the homotopy equation is Bring Eq (2.3) back Eq (2.1), and taking the k derivatives of function F, the Eq (2.1) is equaled to and B k is depended on u 0 (t), u 1 (t), · · · , u k (t). By comparing the coefficients of p k , the solution of Eq (2.1) is equivalent to the following system: Through the above calculations, we can get the approximate solution of the Eq (1.1) by adding up the solutions of the Eq (2.6).

HPM-SRKM for solving equation
3.1. Presenting the HPM-SRKM to solve the Eq (2.6) Since the system Eq (2.6) are two linear equations, we can equate them to the following problem . ., then v(x) ≡ 0. This proof is completed.

Convergence and error estimation
In this subsection, we will argue the convergence of the HPM-SRKM.
From the continuity of the reproducing kernel function R x , we obtain where M is a nonnegative constant. This proof is completed.

Numerical examples
The theoretical part of the solution is proved in the previous sections. Some numerical examples are given to illustrate its effectiveness. We operate our programs in MATHMATICA 7.0. Meanwhile, the red lines represent the approximate solutions and the blue dots represent the exact solutions in the figure. The absolute errors e i , the exact and the approximate solutions are listed in the tables. We also use the following formulas to calculate the convergence rate r. r = log 2 e n e 2n .
Example 1. For the following nonlinear HIE: where f (x) = 1 6 (−3 + 8cosx + cos2x), the exact solution is u(x) = cosx. The numerical results are illustrated in Figure 1. The comparison of the numerical results and the absolute error e i are listed in Table 1. We get an exact solution with higher precision than the method of traditional reproducing kernel method [20] for n = 25.
Example 2. Consider the nonlinear HIE: where f (x) = − 15 56 x 8 + 13 14 x 7 − 11 10 x 6 + 9 20 x 5 + x 2 − x. The exact solution is u(x) = x 2 − x. The numerical results are illustrated in Figure 2. Table 2 is illustrated the numerical results and the absolute error e i . From the results of Table 2, we can see that our method approximates the exact solution more closely than the Legendre spectral Galerkin and multi-Galerkin methods [4] for n = 10.

Conclusions
In this article, the SRKM-HPM was smoothly applied to figure out the nonlinear HIE by getting the approximate uniform solution. Besides, compared with the method of traditional reproducing kernel method [31], Legendre spectral Galerkin, and multi-Galerkin methods [4], the convergence speed and accuracy of solution were better.