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

Discussions are presented by Morita and Sato on the problem of obtaining the particular solution of an inhomogeneous differential equation with polynomial coefficients in terms of the Green's function. In a paper, the problem is treated in distribution theory, and in another paper, the formulation is given on the basis of nonstandard analysis, where fractional derivative of degree, which is a complex number added by an infinitesimal number, is used. In the present paper, a simple recipe based on nonstandard analysis, which is closely related with distribution theory, is presented, where in place of Heaviside's step function H(t) and Dirac's delta function delta(t) in distribution theory, functions H & varepsilon;(t):=1 Gamma(1+& varepsilon;)t & varepsilon;H(t) and delta & varepsilon;(t):=ddtH & varepsilon;(t)=1 Gamma(& varepsilon;)t & varepsilon;-1H(t) for a positive infinitesimal number & varepsilon;, are used. As an example, it is applied to Kummer's differential equation.