Solution of Euler’s Differential Equation in Terms of Distribution Theory and Fractional Calculus

Abstract

For Euler’s differential equation of order n, a theorem is presented to give n solutions, by modifying a theorem given in a recent paper of the present authors in J. Adv. Math. Comput. Sci. 2018; 28(3): 1–15, and then the corresponding theorem in distribution theory is given. The latter theorem is compared with recent studies on Euler’s differential equation in distribution theory. A supplementary argument is provided on the solutions expressed by nonregular distributions, on the basis of nonstandard analysis and Laplace transform.

Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis

“…Remark 1.1. It was mentioned in [5], that this observation justifies the proposal of Ghil and Kim given in [6], in which the inverse Laplace transform of C is given by t −1 , where C is a constant; although their choice C = −1 cannot be justified.…”

“…The following two theorems taking account of nonstandard analysis corresponds to Theorems 7 and 8 in [5].…”

“…In Section 5, we recall the therems on the solution of Euler's differential equation, which are given in [5]. We then show that the solutions of Euler's differential equation in nonstandard analysis can be given in the form, from which the solutions in distribution theory are obtained, with the aid of Remark 1.2.…”