Abstract
We consider a diffusion–wave equation with fractional derivative with respect to the time variable, defined on infinite interval, and with the starting point at minus infinity. For this equation, we solve an asympotic boundary value problem without initial conditions, construct a representation of its solution, find out sufficient conditions providing solvability and solution uniqueness, and give some applications in fractional electrodynamics.
Highlights
Consider the equation where ∂α ∂tα− ∆x u( x, t) = f ( x, t), (1)denotes a fractional derivative with respect to t of order α ∈ (0, 2), and ∆x = n ∂2∑ ∂x2, j =1 j is the Laplace operator with respect to x = ( x1, x2, ..., xn ) ∈ S ⊂ Rn .If α = 1, Equation (1) coincides with the diffusion equation, and when α tends to 2, this equation turns to the wave equation
The overwhelming majority of works devoted to fractional differential equations consider fractional derivatives that are defined on finite intervals
We construct a representation of solutions to an asympotic boundary value problem for a diffusion-wave equation with fractional derivative with respect to the time variable
Summary
Interest in the study of this equation is caused by numerous applications fractional calculus in modeling and various fields of natural science In this regard, we recall the works [35,36,37,38,39,40]. The overwhelming majority of works devoted to fractional differential equations consider fractional derivatives that are defined on finite intervals. Starting points of these derivatives, at which initial conditions are specified, are finite. We consider Equation (1) with the Caputo-type fractional derivative with the starting point at minus infinity. We solve an asympotic boundary value problem for this equation, construct a representation of its solution, find out sufficient conditions providing solvability and solution uniqueness, and give some applications in fractional electrodynamics
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