### Date:

**Speaker : **K. Van Bockstal, Ghent Analysis & PDE Center, Department of Mathematics: Analysis, Logic and Discrete, Mathematics, Ghent University, Belgium

**Time : **15:00 - 16:00 CEST (Rome/Paris)

**Hosted at: **SISSA, International School of Advanced Studies, Trieste, Italy

**YouTube** : https://youtu.be/QEzLVtZxnhI

**Organizers :** Pavan Pranjivan Mehta* (pavan.mehta@sissa.it) and Arran Fernandez** (arran.fernandez@emu.edu.tr)

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

*Keywords: inverse source problems, Rothe’s method, energy estimates*

**Abstract: **This talk is based on the papers [3, 4, 2, 1] focused on inverse source problems (ISPs) for time-fractional diffusion equations with time-dependent coefficients in the governing second-order linear elliptic operator. Specifically, I will examine the uniqueness of a solution in determining the space-dependent part of the source from the final-in-time measurement and time-averaged measurement, as presented in [3, 4]. I will also present counterexamples that demonstrate the failure of the uniqueness of a solution when the conditions for uniqueness are not met. Next, I will discuss the reconstruction of the time-dependent part of the source from the knowledge of the space-averaged measurement, as considered in [2]. The main research questions addressed include: (i) the existence and uniqueness of a (weak) solution to the ISP for exact data, and (ii) the numerical reconstruction of the unknown source. Finally, I will discuss the extension of the results to multiterm time-fractional diffusion equation, as presented in [1], and conclude with some potential directions for future work

**Biography: **Dr K. Van Bockstal obtained his PhD (in mathematical engineering) in 2015 at Ghent University, Belgium, and is currently a postdoctoral researcher (Ghent Analysis & PDE Center) at the Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University. His research interests are related to the mathematical analysis of evolutionary partial differential equations as well as to the development of numerical algorithms and their numerical implementation. This research focus concerns direct and inverse problems in heat transfer, elasticity, electromagnetism and thermo-elasticity. He was awarded the EAIP Young Scientist Award of the 8th International Conference “Inverse Problems: Modelling and Simulation”, May 2016

**Bibliography**

[1] A. S. Hendy and K. Van Bockstal. “A solely time-dependent source reconstruction in a multiterm time-fractional order diffusion equation with non-smooth solutions”. In:

Numer. Algorithms 90.2 (2022), pp. 809–832. issn: 1572-9265. doi: 10.1007/s11075-021-01210-w. url: https://doi.org/10.1007/s11075-021-01210-w.

[2] A. S. Hendy and K. Van Bockstal. “On a reconstruction of a solely time-dependent source in a time-fractional diffusion equation with non-smooth solutions”. English. In:

J. Sci. Comput. 90.1 (2022). Id/No 41, p. 33. issn: 0885-7474. doi: 10.1007/s10915-021-01704-8.

[3] M. Slodiˇcka, K. ˇSiˇskov´a, and K. Van Bockstal. “Uniqueness for an inverse source problem of determining a space dependent source in a time-fractional diffusion equation”. In: Applied Mathematics Letters 91 (2019), pp. 15–21. issn: 0893-9659.

[4] K. Van Bockstal. “Uniqueness for Inverse Source Problems of Determining a Space-Dependent Source in Time-Fractional Equations with Non-Smooth Solutions”. In: Fractal and Fractional 5.4 (2021). issn: 2504-3110. doi: 10.3390/fractalfract5040169.

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