Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂tβ u = −(− Δ)α/2 u − (− Δ)γ/2 u

Ngartelbaye Guerngar; Erkan Nane; Ramazan Tinaztepe; Suleyman Ulusoy; Hans Werner Van Wyk

doi:10.1515/fca-2021-0035

Published by De Gruyter June 23, 2021

Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂_t^β u = −(− Δ)^α/2 u − (− Δ)^γ/2 u

Ngartelbaye Guerngar , Erkan Nane , Ramazan Tinaztepe , Suleyman Ulusoy and Hans Werner Van Wyk

From the journal Fractional Calculus and Applied Analysis

https://doi.org/10.1515/fca-2021-0035

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Abstract

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called “double-scale” anomalous diffusion

∂tβu(t,x)=−(−Δ)α/2u(t,x)−(−Δ)γ/2u(t,x),t>0,−1<x<1,

where ∂tβ is the Caputo fractional derivative of order β ∈ (0, 1) and 0 < α ≤ γ < 2. We consider a nonlocal inverse problem and show that the fractional exponents β, α and γ are determined uniquely by the data u(t, 0) = g(t), 0 < t ≤ 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.

Key Words and Phrases: fractional derivative; two-term fractional Laplacian; weak solution; inverse problem; Mittag-Leffler function; Cauchy problem

MSC 2010: Primary 45K05; 35R30; 65M32; Secondary 35B

Acknowledgements

The authors thank the referee and the editor for the valuable suggestions that improved the presentation of the original manuscript.

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Received: 2020-05-19

Revised: 2021-05-09

Published Online: 2021-06-23

Published in Print: 2021-06-25

Simultaneous inversion for the fractional exponents in the space-time fractional diffusion equation ∂_t^β u = −(− Δ)^α/2 u − (− Δ)^γ/2 u

Abstract

Acknowledgements

References

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