Solutions of Some Types of Differential Equations and of Their Associated Physical Problems by Means of Inverse Differential Operators

Abstract

We present an operational method, involving an inverse derivative operator, in order to obtain solutions for differential equations, which describe a broad range of physical problems. 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 the mass, the heat and other processes of propagation are demonstrated by the developed operational technique. In particular, the evolutional type problems, the generalizations of the Black–Scholes, of the heat, of the Fokker–Plank and of the telegraph equations are considered as well as equations, involving the Laguerre derivative operator.