Nonlenear Integro-differential Equations and Splines of the Fifth Order of Approximation

Abstract

In this paper, we consider the solution of nonlinear Volterra–Fredholm integro-differential equation, which contains the first derivative of the function. Our method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations. The method based on the application of the local polynomial splines of the fifth order of approximation is proposed. Theorems about the errors of the approximation of a function and its first derivative by these splines are given. With the help of the proposed splines, the function and the derivative are replaced by the corresponding approximation. Note that at the beginning, in the middle and at the end of the interval of the definition of the integro-differential equation, the corresponding types of splines are used: the left, the right or the middle splines of the fifth order of approximation. When using the spline approximations, we also obtain the corresponding formulas for numerical differentiation. which we also apply for the solution of integro-differential equations. The formulas for approximation of the function and its derivative are presented. The results of the numerical solution of several integro-differential equations are presented. The proposed method is shown that it can be applied to solve integro-differential equations containing the second derivative of the solution.