An exact solution method for Fredholm integro-differential equations
Resumen
Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom considered with integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocal integral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocal integral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existence criterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditions and obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for the special case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for the existence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct method analytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations. If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existence and uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by two examples. The article uses computer algebra system Mathematica to demonstrate the results. © 2019 Saint Petersburg State University of Aerospace Instrumentation. All rights reserved.