Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform

Tohru Morita and Ken-ichi Sato
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Tohru Morita: Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
Ken-ichi Sato: College of Engineering, Nihon University, Koriyama 963-8642, Japan

Mathematics, 2016, vol. 4, issue 1, 1-18

Abstract: In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivative (fD) and the distribution theory in the space D? R or the AC of Laplace transform. We now obtain the solutions of the hypergeometric DE, which are expressed by the hypergeometric functions, with the aid of the AC of Riemann–Liouville fD, and the distribution theory in the space D? r,R , which is introduced in this paper, or by the term-by-term inverse Laplace transform of AC of Laplace transform of the solution expressed by a series.

Keywords: Kummer’s differential equation; hypergeometric differential equation; distribution theory; operational calculus; fractional calculus; Laplace transform (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2016
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