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The Scientific World Journal
Volume 2014 (2014), Article ID 454865, 8 pages
http://dx.doi.org/10.1155/2014/454865
Research Article

Solution of Some Types of Differential Equations: Operational Calculus and Inverse Differential Operators

Faculty of Physics, M. V. Lomonosov Moscow State University, Leninskie Gory, Moscow 119899, Russia

Received 23 January 2014; Accepted 23 February 2014; Published 30 April 2014

Academic Editors: D. Baleanu and H. Jafari

Copyright © 2014 K. Zhukovsky. 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.

Abstract

We present a general method of operational nature to analyze and obtain solutions for a variety of equations of mathematical physics and related mathematical problems. We construct inverse differential operators and produce operational identities, involving inverse derivatives and families of generalised orthogonal polynomials, such as Hermite and Laguerre polynomial families. We develop the methodology of inverse and exponential operators, employing them for the study of partial differential equations. Advantages of the operational technique, combined with the use of integral transforms, generating functions with exponentials and their integrals, for solving a wide class of partial derivative equations, related to heat, wave, and transport problems, are demonstrated.