Solution of Inhomogeneous Fractional Differential Equations with Polynomial Coefficients in Terms of the Green’s Function, in Nonstandard Analysis

Abstract

Discussions are presented by Morita and Sato in Mathematics 2017; 5, 62: 1–24, on the problem of obtaining the particular solution of an inhomogeneous ordinary differential equation with polynomial coefficients in terms of the Green’s function, in the framework of distribution theory. In the present paper, a compact recipe in nonstandard analysis is presented, which is applicable to an inhomogeneous ordinary and also fractional differential equation with polynomial coefficients. The recipe consists of three theorems, each of which provides the particular solution of a differential equation for an inhomogeneous term, satisfying one of three conditions. The detailed derivation of the applications of these theorems is given for a simple fractional differential equation and an ordinary differential equation.

Keywords

• Green’s function
• fractional differential equations with polynomial coefficients
• Kim and O’s differential equation
• nonstandard analysis
• distribution theory
• operational calculus