Abstract
In this work, an explicit solution of the initial-boundary value problem for a multidimensional time-fractional differential equation is constructed. The possibility of obtaining this equation from an integro-differential wave equation with a Mittag–Leffler–type memory kernel is shown. An explicit solution to the problem under consideration is obtained using the Laplace and Fourier transforms, the properties of the Fox H-functions and the convolution theorem.
Highlights
The presence of memory in such models indicates the dependence of its current state on a finite number of its previous states
This means the non-local properties of non-classical mathematical models, for example, in the mechanics of viscoelastic media when describing the action of aftereffect [1,2]
Fractional derivative operators have many definitions and have unique properties, but they all describe to one degree or another memory effect characterizing information about the previous states of the system
Summary
The investigation of many mathematical models that have a fractal structure have numerous applied applications. The presence of memory in such models indicates the dependence of its current state on a finite number of its previous states This means the non-local properties of non-classical mathematical models, for example, in the mechanics of viscoelastic media when describing the action of aftereffect [1,2]. If the memory functions are given and are power-law, we can go to other types of equations that are based on derivatives of fractional orders, properties of which are considered in table books on fractional calculus [23]. Equation (1) for ρ = 1, m = 1 describes the anomalously diffusive transport of solute in heterogeneous porous media [27] This equation containing both the classical and fractional derivatives is more general and is of interest in the theory of the differential equations with fractional derivatives.
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