Construction of an Explicit Solution of a Time-Fractional Multidimensional Differential Equation

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.

Inverse coefficient problem for the time‐fractional diffusion equation with Hilfer operator

“…The works [10][11][12][13][14] are concerned with inverse memory recovery problems from parabolic integro-differential equations of the second order with integral term of convolution type. In [15,16], there were shown that if the kernel of convolution integral in a classical integro-differential diffusion equation coincides with the two-parametric Mittag-Leffler function of the special argument, then these equations describe the anomalously diffusive transport of solute in heterogeneous porous media [17]. To get acquainted with the methods for solving various initial-boundary value problems for differential equations with fractional time derivatives in the sense of Riemann-Liouville and Caputo using functions of the Mittag-Leffler type, we point out the works [18,19] (see, also, references therein).…”

Inverse coefficient problem for a time‐fractional wave equation with initial‐boundary and integral type overdetermination conditions

“…Boundary value problems and inverse coefficient problems for parabolic equations with periodic boundary conditions are investigated in [25,26]. In the works [27,28], it is noted that if the kernel in the integrals of the integro-differential equation of heat propagation and the wave equation is chosen in the forms…”

“…Boundary value problems and inverse coefficient problems for parabolic equations with periodic boundary conditions are investigated in [25, 26]. In the works [27, 28], it is noted that if the kernel in the integrals of the integro‐differential equation of heat propagation and the wave equation is chosen in the forms $$$ {t}^{k-1-\alpha }{E}_{k-\alpha, k-\alpha}\left({t}^{k-\alpha}\right) $$$, where $$$ {E}_{\alpha, \beta}\left(\cdotp \right) $$$ is the Mittag‐Liffler function, $$$ k=1,2 $$$, then these equations are equivalent to the fractional diffusion equation for $$$ k=1 $$$ and the wave equation for $$$ k=2 $$$ with the Caputo derivative in time. In [29–31], both the existence and uniqueness of a solution to the inverse problem are proved.…”

“…𝜏) 𝛼−1 |q(𝜏) − q(𝜏)|d𝜏, t ∈ [0, T],(28) where 𝜌 0 , 𝜌 1 depends only on 𝜈 0 , 𝜈 1 , T, l, 𝛼, and Γ(𝛼), B(𝛼, 1 − 𝛾). From (28) using Lemma 1, we get the estimate||q − q|| C[0,T] ≤ 𝜌 0 ( ||𝜑 − φ|| C[0,l] + ||𝜓 − ψ|| C[0,l] + ||g − g|| C 𝛼 𝛾 [0,T] + ||𝑓 − f || C 𝛾 ( ΩT )) × × E 𝛼,1 (𝜌 1 Γ(𝛼)T 𝛼 ) .…”

Parallel Algorithm for Solving the Inverse Two-Dimensional Fractional Diffusion Problem of Identifying the Source Term

“…The solution to direct and inverse problems for differential equations with fractional derivatives typically incurs substantial computational costs due to their nonlocal characteristics. Different numerical techniques are available for solving approximate initial-boundary problems for fractional differential equations [11][12][13][14], for example, the finite difference method. An established approach to enhance computational efficiency involves the utilization of parallel computing [15][16][17].…”