Reconstruction of pointwise sources in a time-fractional diffusion equation

• Published in: Fractional calculus & applied analysis Vol. 26; no. 1; pp. 193 - 219
• Main Authors: Hrizi, Mourad, Hassine, Maatoug, Novotny, Antonio André
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.02.2023
• Subjects: Abstract Harmonic Analysis; Analysis; Functional Analysis; Integral Transforms; Mathematics; Mathematics and Statistics; Operational Calculus; Original Paper; Pointwise source reconstruction; non-iterative reconstruction method; time-fractional diffusion problem; 35R11; 49M15; 49N45; 35Q93
• ISSN: 1311-0454; 1314-2224
• DOI: 10.1007/s13540-022-00127-y

This paper is concerned with an inverse pointwise source problem for the time-fractional diffusion equation in the two-dimensional case. The source term to be identified models the action of a finite number of small particles. Each particle is assumed to be no larger than a single point, characterized by its location and intensity. Both theoretical and numerical aspects are discussed. In the theoretical part, we analyse the well-posedness of the Dirac time-fractional diffusion problem. For the inverse problem, we establish that the unknown point sources can be uniquely identified from local measured data and we derive a local Lipschitz stability result. In the numerical part, we develop a fast and accurate reconstruction approach. The unknown pointwise sources are characterized as solution to an optimization problem minimizing a tracking-type functional. A noniterative reconstruction algorithm is devised, allowing us to determine the number, locations and intensities of the pointwise sources. The efficiency of the proposed approach is confirmed by some numerical examples.