A method of solving a nonlinear boundary value problem for the Fredholm integro-differential equation

Abstract

We propose a method to solve a nonlinear boundary value problem for a Fredholm integro-differential equation on a finite interval. By introducing additional parameters chosen as the values of the solution at the left-end points of the partition subintervals, the problem under consideration is transformed into an equivalent boundary value problem for a system of nonlinear integro-differential equations with parameters on the subintervals. For fixed parameters, we obtain a special Cauchy problem for this system, which is represented as a nonlinear operator equation and solved by an iterative method. By substitution of the solution to the special Cauchy problem with parameters into the boundary condition and the continuity conditions of the solution to the original problem at the interior partition points, we construct a system of nonlinear algebraic equations in parameters. It is proved that the solvability of this system provides the existence of a solution to the original boundary value problem.

The algorithm for solving the special Cauchy problem includes two auxiliary problems: the Cauchy problems for ordinary differential equations and the evaluation of definite integrals. The accuracy of the method that we propose to solve the boundary value problem depends on the accuracy of methods applied to the auxiliary problems and does not depend on the number of the partition subintervals. Since iterative methods are used to solve both the constructed system of algebraic equations and the special Cauchy problem, we offer an approach to find an initial guess for the solutions to these problems.