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Mathematics 2016, 4(1), 19; doi:10.3390/math4010019

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

1
Graduate School of Information Sciences, Tohoku University, Sendai 980-8577, Japan
2
College of Engineering, Nihon University, Koriyama 963-8642, Japan
*
Author to whom correspondence should be addressed.
Academic Editor: Hari M. Srivastava
Received: 22 December 2015 / Revised: 6 March 2016 / Accepted: 11 March 2016 / Published: 17 March 2016
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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. View Full-Text
Keywords: Kummer’s differential equation; hypergeometric differential equation; distribution theory; operational calculus; fractional calculus; Laplace transform Kummer’s differential equation; hypergeometric differential equation; distribution theory; operational calculus; fractional calculus; Laplace transform
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Morita, T.; Sato, K.-I. Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform. Mathematics 2016, 4, 19.

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