Existence of a solution for a general order boundary value problem using the Leray-Schauder fixed point theorem

Abstract

Introduction/purpose: This paper illustrates the existence of a generic Green’s function for a boundary value problem of arbitrary order that appears in many phenomena of heat convection, e.g. in the atmosphere, in the oceans, and on the Sun’s surface. Methods: A fixed point theorem in the Leray–Schauder form has been used to establish the existence of a fixed point in the problem. Results: The existence of a solution has been shown for an arbitrary order of the problem. Some practical examples are proposed. Conclusions: The boundary problem has a solution for an arbitrary order n.