Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation $\partial_t^\beta u= -\big(-\Delta\big)^{\alpha/2}u-\big(-\Delta\big)^{\gamma/2}u$

Abstract

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x) \ \ t> 0, \ -1<x<1, $$ where $\partial_t^\beta$ is the Caputo fractional derivative of order $\beta \in (0,1)$ and $0<\alpha\leq \gamma<2.$ We consider a nonlocal inverse problem and show that the fractional exponents $\beta$, $\alpha$ and $\gamma$ are determined uniquely by the data $u(t, 0) = g(t), \ 0 < t \leq T.$ The existence of the solution for the inverse problem is proved using the quasi-solution method which is based on minimizing an error functional between the output data and the additional data. In this context, an input-output mapping is defined and its continuity is established. The uniqueness of the solution for the inverse problem is proved by means of eigenfunction expansion of the solution of the forward problem and some basic properties of fractional Laplacian. A numerical method based on discretization of the minimization problem, namely the steepest descent method and a least squares approach, is proposed for the solution of the inverse problem. The numerical method determines the fractional exponents simultaneously. Finally, numerical examples with noise-free and noisy data illustrate applicability and high accuracy of the proposed method.