Computational Methods for Parameter Identification in 2D Fractional System with Riemann-Liouville Derivative
Rafał Brociek1, Agata Wajda2, Grazia Lo Sciuto3,4
1Department of Mathematics Applications and Methods for Artificial Intelligence, Faculty of Applied Mathematics, Silesian University of Technology, 44-100 Gliwice, Poland.
Sensors (Basel, Switzerland)
|May 20, 2022
Summary
This study introduces a numerical algorithm for solving two-dimensional fractional differential equations using the alternating direction implicit method (ADIM). It also details methods for solving inverse problems, crucial for engineering applications like anomalous diffusion and control theory.
Area of Science:
- Mathematics
- Numerical Analysis
- Engineering
Background:
- Fractional derivatives offer enhanced modeling capabilities for complex systems.
- Traditional methods may not adequately capture phenomena described by fractional differential equations.
Purpose of the Study:
- To present a numerical algorithm for solving two-dimensional fractional differential equations using the Riemann-Liouville derivative.
- To describe an algorithm for solving the inverse problem to determine unknown model parameters.
- To illustrate the effectiveness and accuracy of these methods with a numerical example.
Main Methods:
- The alternating direction implicit method (ADIM) is employed for solving the forward fractional differential equation.
- Objective function minimization using the ant algorithm and Hooke-Jeeves method is used for inverse problem solutions.
Main Results:
- A numerical example demonstrates the effectiveness and accuracy of the proposed ADIM and inverse problem-solving algorithms.
- Comparison of objective function minimization techniques (ant algorithm vs. Hooke-Jeeves) is presented.
Conclusions:
- The developed numerical algorithms provide effective tools for solving fractional differential equations and their inverse problems.
- These methods are applicable to various engineering fields and can be utilized for parameter training in artificial neural networks.
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