Inverse Problems for Fractional Differential Equations

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (31 May 2024) | Viewed by 2421

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Special Issue Information

Dear Colleagues,

Real practical scenarios are ubiquitously complex in inverse problems, and the goodness of their modelling depends on the tools used, as well as the quality of the observation dataset. Notwithstanding, clean and perfect data collections are seldom available. In that sense, fractional calculus is a powerful tool to describe the real behavior of practical and complex systems, unless one considers nested or merged simple models. Walking in this avenue, recent reports in the literature have demonstrated the potential of the fractional differential equations for describing the dynamic behavior of geological, mechanical, thermal, electronic, and chemical systems from observations with a certain degree of external and undesired disturbances. Therefore, we happily welcome high-quality manuscripts that address inverse problems using or related to non-integer operators, such as fractional differential equations, fractional derivatives, fractional transforms, and fractional models, to mention a few. We hope that this initiative is of interest to you, and we encourage you to submit your current research to be included in the Special Issue.

Dr. Jorge Mario Cruz-Duarte
Guest Editor

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Keywords

  • inverse problems
  • fractional differential equations
  • fractional derivatives
  • non-integer derivatives
  • fractional models
  • chaotic systems
  • time-fractional diffusion
  • spatial-fractional diffusion
  • non-linear fractional differential equations
  • pseudo-differential equations
  • fractional reactions

Published Papers (2 papers)

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Research

21 pages, 4157 KiB  
Article
Designing Heuristic-Based Tuners for Fractional-Order PID Controllers in Automatic Voltage Regulator Systems Using a Hyper-Heuristic Approach
by Daniel Fernando Zambrano-Gutierrez, Gerardo Humberto Valencia-Rivera, Juan Gabriel Avina-Cervantes, Ivan Amaya and Jorge Mario Cruz-Duarte
Fractal Fract. 2024, 8(4), 223; https://doi.org/10.3390/fractalfract8040223 - 12 Apr 2024
Viewed by 975
Abstract
This work introduces an alternative approach for developing a customized Metaheuristic (MH) tailored for tuning a Fractional-Order Proportional-Integral-Derivative (FOPID) controller within an Automatic Voltage Regulator (AVR) system. Leveraging an Automated Algorithm Design (AAD) methodology, our strategy generates MHs by utilizing a population-based Search [...] Read more.
This work introduces an alternative approach for developing a customized Metaheuristic (MH) tailored for tuning a Fractional-Order Proportional-Integral-Derivative (FOPID) controller within an Automatic Voltage Regulator (AVR) system. Leveraging an Automated Algorithm Design (AAD) methodology, our strategy generates MHs by utilizing a population-based Search Operator (SO) domain, thus minimizing human-induced bias. This approach eliminates the need for manual coding or the daunting task of selecting an optimal algorithm from a vast collection of the current literature. The devised MH consists of two distinct SOs: a dynamic swarm perturbator succeeded by a Metropolis-type selector and a genetic crossover perturbator, followed by another Metropolis-type selector. This MH fine-tunes the FOPID controller’s parameters, aiming to enhance control performance by reducing overshoot, rise time, and settling time. Our research includes a comparative analysis with similar studies, revealing that our tailored MH significantly improves the FOPID controller’s speed by 1.69 times while virtually eliminating overshoot. Plus, we assess the tuned FOPID controller’s resilience against internal disturbances within AVR subsystems. The study also explores two facets of control performance: the impact of fractional orders on conventional PID controller efficiency and the delineating of a confidence region for stable and satisfactory AVR operation. This work’s main contributions are introducing an innovative method for deriving efficient MHs in electrical engineering and control systems and demonstrating the substantial benefits of precise controller tuning, as evidenced by the superior performance of our customized MH compared to existing solutions. Full article
(This article belongs to the Special Issue Inverse Problems for Fractional Differential Equations)
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39 pages, 2298 KiB  
Article
Efficient Inverse Fractional Neural Network-Based Simultaneous Schemes for Nonlinear Engineering Applications
by Mudassir Shams and Bruno Carpentieri
Fractal Fract. 2023, 7(12), 849; https://doi.org/10.3390/fractalfract7120849 - 29 Nov 2023
Cited by 3 | Viewed by 919
Abstract
Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors [...] Read more.
Finding all the roots of a nonlinear equation is an important and difficult task that arises naturally in numerous scientific and engineering applications. Sequential iterative algorithms frequently use a deflating strategy to compute all the roots of the nonlinear equation, as rounding errors have the potential to produce inaccurate results. On the other hand, simultaneous iterative parallel techniques require an accurate initial estimation of the roots to converge effectively. In this paper, we propose a new class of global neural network-based root-finding algorithms for locating real and complex polynomial roots, which exploits the ability of machine learning techniques to learn from data and make accurate predictions. The approximations computed by the neural network are used to initialize two efficient fractional Caputo-inverse simultaneous algorithms of convergence orders ς+2 and 2ς+4, respectively. The results of our numerical experiments on selected engineering applications show that the new inverse parallel fractional schemes have the potential to outperform other state-of-the-art nonlinear root-finding methods in terms of both accuracy and elapsed solution time. Full article
(This article belongs to the Special Issue Inverse Problems for Fractional Differential Equations)
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