Operational solution for some types of second order differential equations and for relevant physical problems

Abstract

We present an operational method to obtain solutions for differential equations, describing a broad range of physical problems, including ordinary non-integer order and high order partial differential equations. Inverse differential operators are proposed to solve a variety of differential equations. Integral transforms and the operational exponent are used to obtain the solutions. Generalized families of orthogonal polynomials and special functions are also employed with recourse to their operational definitions. Examples of solutions of physical problems, related to propagation of the heat and other quantities are demonstrated by the developed operational technique. In particular, the evolution type problems, the generalizations of the Black–Scholes, of the heat conduction, of the Fokker–Planck equations are considered as well as equations, involving the Laguerre derivative operator.