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Open AccessArticle
An Inverse Problem for a Fractional Space–Time Diffusion Equation with Fractional Boundary Condition
1
Department of Artificial Intelligence Modelling, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
2
Department of Electrical, Electronics and Informatics Engineering, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy
3
Institute of Energy and Fuel Processing Technology, Zamkowa 1, 41-800 Zabrze, Poland
4
Department of Computer, Control, and Management Engineering, Sapienza University of Rome, Via Ariosto 25, 00185 Roma, Italy
5
Department of Artificial Intelligence, Czestochowa University of Technology, Dabrowskiego 69, 42-201 Czestochowa, Poland
6
Department of Mathematical Methods in Technology and Computer Science, Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
*
Author to whom correspondence should be addressed.
Entropy 2026, 28(1), 81; https://doi.org/10.3390/e28010081 (registering DOI)
Submission received: 30 November 2025
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Revised: 5 January 2026
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Accepted: 7 January 2026
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Published: 10 January 2026
Abstract
This article presents an algorithm for solving the direct and inverse problem for a model consisting of a fractional differential equation with non-integer order derivatives with respect to time and space. The Caputo derivative was taken as the fractional derivative with respect to time, and the Riemann–Liouville derivative in the case of space. On one of the boundaries of the considered domain, a fractional boundary condition of the third kind was adopted. In the case of the direct problem, a differential scheme was presented, and a metaheuristic optimization algorithm, namely the Group Teaching Optimization Algorithm (GTOA), was used to solve the inverse problem. The article presents numerical examples illustrating the operation of the proposed methods. In the case of inverse problem, a function occurring in the fractional boundary condition was identified. The presented approach can be an effective tool for modeling the anomalous diffusion phenomenon.
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MDPI and ACS Style
Brociek, R.; Wajda, A.; Napoli, C.; Capizzi, G.; Słota, D.
An Inverse Problem for a Fractional Space–Time Diffusion Equation with Fractional Boundary Condition. Entropy 2026, 28, 81.
https://doi.org/10.3390/e28010081
AMA Style
Brociek R, Wajda A, Napoli C, Capizzi G, Słota D.
An Inverse Problem for a Fractional Space–Time Diffusion Equation with Fractional Boundary Condition. Entropy. 2026; 28(1):81.
https://doi.org/10.3390/e28010081
Chicago/Turabian Style
Brociek, Rafał, Agata Wajda, Christian Napoli, Giacomo Capizzi, and Damian Słota.
2026. "An Inverse Problem for a Fractional Space–Time Diffusion Equation with Fractional Boundary Condition" Entropy 28, no. 1: 81.
https://doi.org/10.3390/e28010081
APA Style
Brociek, R., Wajda, A., Napoli, C., Capizzi, G., & Słota, D.
(2026). An Inverse Problem for a Fractional Space–Time Diffusion Equation with Fractional Boundary Condition. Entropy, 28(1), 81.
https://doi.org/10.3390/e28010081
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