Some uniqueness and existence results for the initial-boundary-value problems for the generalized time-fractional diffusion equation

Abstract

a b s t r a c tIn this paper, some uniqueness and existence results for the solutions of the initialboundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain G × (0, T ), G ⊂ R n are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the CaputoDzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.

Existence and uniqueness of the solution for a time-fractional diffusion equation

“…The same author proved in [13] the existence of the solution using the notion of a generalized solution and the Fourier method of the variable separation. It should be noted that the generalized solution defined in [13] is a continuous function up to the boundary, not a generalized one.…”

On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

“…For α = 2, the fractional Laplacian (−∆) α 2 is simply −∆, and thus Equation (1) is a particular case of the time-fractional diffusion-wave equation that was considered in many publications, including, for example, [1, 2,5,[14][15][16][17]. For α = 2 and β = 1, Equation (1) is reduced to the diffusion equation, and for α = 2 and β = 2, it is the wave equation that justifies its denotation as a fractional diffusion-wave equation.…”

“…Now we proceed with the case x = 0 and first discuss the convergence of the integral in the integral representation given by Equation (16). It follows from the asymptotic formulas for the Mittag-Leffler function and the known asymptotic behavior of the Bessel function (see, e.g., [20]) that the integral in Equation (16) converges conditionally in the case n < 2α + 1 and absolutely in the case n < 2α − 1.…”

On the solutions of partial integrodifferential equations of fractional order

“…Some partial dierential equations of fractional order type like one-dimensional time-fractional diusion-wave equation were used for modeling relevant physical processes (see [26]). Regarding fractional partial dierential equations, Luchko [18] used the Fourier transform method of the variable separation to construct a formal solution and under certain condition he showed that the formal solution is the generalized solution of the initial-boundary value problem. To prove the uniqueness he used the maximum principle for generalized time fractional diusion equation [17].…”