Inverse problem for a three-parameter space-time fractional diffusion equation

By Ngartelbaye Guerngar, Erkan Nane, Suleyman Ulusoy. 2018

3 citations recorded by the source corpus.

In this article, we consider the space-time Fractional (nonlocal) diffusion equation $$\partial_t^\beta u(t,x)=\mathcal{L} u(t,x), \ \ t\geq 0, \ -1<x<1, $$ where $\partial_t^\beta$ is the caputo fractional derivative of order $\beta \in (0,1)$ and the differential operator $\mathcal{L}$ is the generator of a Levy process, sum of two symmetric independent $\alpha_1-$stable and $\alpha_2-$stable processes. We consider a nonlocal inverse problem and show that the fractional exponents $\beta$ and $\alpha_i, \ i=1,2$ are determined uniquely by the data $u(t; 0) = g(t), 0 < t < T.$ The uniqueness result is a theoretical background for determining experimentally the order of many anomalous diffusion phenomena, which are important in physics and in environmental engineering.

Subfields: Applied Mathematics, Modeling and Simulation

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Citations in the corpus

fPINNs: Fractional Physics-Informed Neural Networks (2018, background): “For inverse problems, little is known about 3D space-time-fractional ADEs with a fractional Laplacian, despite some existing works on 1D and 2D problems [27, 38, 13].”