Exact solution method for Fredholm integro-differential equations with multipoint and integral boundary conditions. Part 2. Decomposition-extension method for squared operators

Abstract

Introduction: In Part 1 of this article, a direct method was presented for examining the solvability and uniqueness problem, and for obtaining a closed-form solution of boundary value problems which incorporate an mth order linear ordinary Fredholm integro-differential operator, or a differential operator, along with multipoint and integral boundary conditions. Here, we focus on a special class of boundary value problems including the composite square of an integro-differential operator and the corresponding non-local boundary conditions. Purpose : To investigate the construction of the unique solution of 2mth order boundary value problems in the special case of an operator which can be presented as composite squares of lower m th order ones, and to develop an algorithm for constructing an exact solution for this special case. Results: By decomposition and applying the extension method explicated in Part 1, we provide a formula for obtaining an exact solution of boundary value problems for squared integro-differential operators, or differential operators, with multipoint and integral boundary conditions. This method is simple to use and can be easily incorporated to any Computer Algebra System.