Abstract
We propose operational method with recourse to generalized forms of orthogonal polynomials for solution of a variety of differential equations of mathematical physics. Operational definitions of generalized families of orthogonal polynomials are used in this context. Integral transforms and the operational exponent together with some special functions are also employed in the solutions. The examples of solution of physical problems, related to such problems as the heat propagation in various models, evolutional processes, Black–Scholes-like equations etc. are demonstrated by the operational technique.
Highlights
Differential equations, besides playing important role in pure mathematics, constitute fundamental part of mathematical description of physical processes
Some fractional type ordinary and partial differential equations involving non-integer derivatives were explored in Demiray et al (2015)
We have demonstrated on simple examples how the usage of inverse derivative together with operational formalism and, in particular, with exponential operator technique, provide elegant and easy way to find solutions in some classes of differential equations
Summary
Differential equations, besides playing important role in pure mathematics, constitute fundamental part of mathematical description of physical processes. With the help of this general method we will obtain exact analytical solutions for a broad class of differential equations, including those with non-integer derivatives, evolution type equations, generalized forms of heat, mass transfer and Black–Scholes type equations, involving the Laguerre derivative operator. This method was applied for solution of some differential equations in Zhukovsky (2014, 2015a) and Dattoli et al (2007).
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