# Numerical Solution of Functional Volterra–Hammerstein Integro-differential Equations of Order Two

Research Paper
Part of the following topical collections:
1. Mathematics

## Abstract

In this paper, the functional Volterra–Hammerstein integro-differential equations (FVHIDEs) of order two with variable coefficients are studied. First, the existence and uniqueness of the solution to the mentioned problem are proved. Then a numerical method is proposed to solve the FVHIDEs. The proposed method is a combination of the well-known finite difference method with successive approximations method, trapezoidal quadrature rule and natural cubic spline interpolation method. An error bound is obtained, and the convergence of the method is proved. The numerical stability of the method is also proved by a suitable way. Finally, some examples are given to show the accuracy and efficiency of the proposed method.

## Keywords

Volterra–Hammerstein integro-differential equations Successive approximations Finite difference Spline interpolation Trapezoidal quadrature rule

65R20 97N40

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