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Axioms 2016, 5(4), 29; doi:10.3390/axioms5040029

Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations

1
Faculty of Physics, Moscow State University, Leninskie Gory, Moscow 119991, Russia
2
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
*
Author to whom correspondence should be addressed.
Academic Editor: Hans J. Haubold
Received: 27 October 2016 / Revised: 28 November 2016 / Accepted: 6 December 2016 / Published: 13 December 2016
View Full-Text   |   Download PDF [302 KB, uploaded 14 December 2016]

Abstract

A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations. View Full-Text
Keywords: inverse operator; derivative; differential equation; special functions; Hermite and Laguerre polynomials inverse operator; derivative; differential equation; special functions; Hermite and Laguerre polynomials
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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Zhukovsky, K.V.; Srivastava, H.M. Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations. Axioms 2016, 5, 29.

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