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Solution of Inverse Problem for Diffusion Equation with Fractional Derivatives Using Metaheuristic Optimization Algorithm
Volume 35, Issue 3 (2024), pp. 453–481
Pub. online: 16 July 2024      Type: Research Article      Open accessOpen Access
Received
1 March 2024
Accepted
1 June 2024
Published
16 July 2024

Abstract

The article focuses on the presentation and comparison of selected heuristic algorithms for solving the inverse problem for the anomalous diffusion model. Considered mathematical model consists of time-space fractional diffusion equation with initial boundary conditions. Those kind of models are used in modelling the phenomena of heat flow in porous materials. In the model, Caputo’s and Riemann-Liouville’s fractional derivatives were used. The inverse problem was based on identifying orders of the derivatives and recreating fractional boundary condition. Taking into consideration the fact that inverse problems of this kind are ill-conditioned, the problem should be considered as hard to solve. Therefore,to solve it, metaheuristic optimization algorithms popular in scientific literature were used and their performance were compared: Group Teaching Optimization Algorithm (GTOA), Equilibrium Optimizer (EO), Grey Wolf Optimizer (GWO), War Strategy Optimizer (WSO), Tuna Swarm Optimization (TSO), Ant Colony Optimization (ACO), Jellyfish Search (JS) and Artificial Bee Colony (ABC). This paper presents computational examples showing effectiveness of considered metaheuristic optimization algorithms in solving inverse problem for anomalous diffusion model.

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Biographies

Brociek Rafał
https://orcid.org/0000-0002-7255-6951
rafal.brociek@polsl.pl

R. Brociek obtained the MSc degree in mathematics from the Silesian University of Technology (in 2013) and the PhD in technical sciences from Czestochowa University (in 2019). He is an adjunct professor at the Department of Mathematics Applications and Methods for Artificial Intelligence, Silesian University of Technology, Gliwice, Poland. His research interests include artificial intelligence, application of computational methods to various problems in engineering and mathematical simulation. He has experience in mathematical modelling, applying of fractional calculus in engineering, as well as the application of artificial intelligence methods in optimization problems.

Goik Mateusz

M. Goik is a student at the Faculty of Applied Mathematics at the Silesian University of Technology. He is currently employed as a software developer at a company that specializes in industrial automation systems. His interests include algorithms, artificial intelligence, and embedded systems. During his free time, he enjoys participating in coding competitions and staying up-to-date with the latest advancements in technology.

Miarka Jakub

J. Miarka is a sophomore majoring in computer science at the Faculty of Applied Mathematics at the Silesian University of Technology. He is a participant of the Silesian University of Technology’s mentoring program. His research interests include: practical application of mathematics, automation and optimization problems and machine learning.

Pleszczyński Mariusz

M. Pleszczyński received the MSc degree in mathematics and the PhD degree in applied sciences, in the area of computer science from the Czestochowa University of Technology, Czestochowa, Poland, in 2001 and 2009, respectively. He is an adjunct professor with the Faculty of Applied Mathematics, Silesian University of Technology. He has authored/coauthored more than 30 research papers in international conferences and journals in the area of applied computing. He is currently working on numerical methods, particularly, by applying mathematics, computer tomography.

Napoli Christian

C. Napoli is an associate professor with the Department of Computer, Control, and Management Engineering “Antonio Ruberti”, Sapienza University of Rome, since 2019, where he also collaborates with the department of Physics and the Faculty of Medicine and Psychology, as well as holding the office of scientific director of the International School of Advanced and Applied Computing (ISAAC). He received the BSc degree in physics from the Department of Physics and Astronomy, University of Catania, in 2010, where he also got the MSc degree in astrophysics in 2012 and the PhD in computer science in 2016 from the Department of Mathematics and Computer Science. Christian Napoli has been a research associate with the Department of Mathematics and Computer Science, University of Catania, from 2018 to 2019, and, previously, a research fellow and an adjunct professor with the same department from 2015 to 2018. He has been a student research fellow with the Department of Electrical, Electronics, and Informatics Engineering, University of Catania, from 2009 to 2016, a collaborator of the Astrophysical Observatory of Catania and the National Institute for Nuclear Physics, since 2010. He has been invited as a professor to the Silesian University of Technology several times, a visiting academic at the New York University, and responsible of many different institutional topics from 2011 until now for undegraduate, graduate and PhD students in computer science, computer engineering and electronics engineering. His teaching activity focused on artificial intelligence, neural networks, machine learning, computing systems, computer architectures, distributed systems, and high performance computing. He is involved in several international research projects, serves as a reviewer and member of the board program committee for major international journals and international conferences. His current research interests include neural networks, artificial intelligence, human-computer interaction and computational neuropsychology.


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Keywords
metaheuristic algorithms inverse problem fractional derivative time-space fractional diffusion equation fractional boundary condition identifying parameters numerical computation

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