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.
Highlights
Differential equations (DE) play an important role in pure mathematics and physics
In the present work we advocate the operational approach for solution of linear differential equations (DE) and the use of inverse differential operators, which allow direct and straightforward finding of solutions
The latter include the action of the operator of heat conduction and the operator of shift and dilatation
Summary
Differential equations (DE) play an important role in pure mathematics and physics. They describe a broad range of physical processes and finding their solutions is of great importance. Advanced numerical methods for the solution of fractional differential equations, formulated, for example, in [38,39,40,41], can be effectively executed with modern computers. In this context we note semi-analytical models and numerical simulations of relaxation of hot electrons and holes [42], the diffusion of charge carriers, and the energy relaxation and transfer with respect to the electron excited states in crystals [43,44].
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