Solution of Euler’s Differential Equation and AC-Laplace Transform of Inverse Power Functions and Their Pseudofunctions, in Nonstandard Analysis
Tohru Morita *
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan.
*Author to whom correspondence should be addressed.
Abstract
It is shown that the index law of the Riemann-Liouville fractional derivative is recovered when nonstandard analysis is applied, and then the solutions of Euler’s differential equation are obtained in nonstandard analysis, where infinitesimal number appears. They are given in the form, from which the solutions in distribution theory are obtained. In the derivation, the AC-Laplace transforms of functions tν and tν(loge t) m for complex number ν and positive integer m, are used. By using these formulas, the AC-Laplace transforms of functions t− n + and
t− n +(loge t) m for positive integers n and m, and their pseudofunctions are obtained with the aid of nonstandard analysis.
Keywords: Riemann-Liouvil le fractional derivative, Euler’s differential equation, Laplace transform, AC-Laplace transform, nonstandard analysis, distribution theory, pseudofunction.