Numerical Solutions of Caputo-Type Fractional Differential Equations and Derivatives

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: 31 August 2024 | Viewed by 3642

Special Issue Editors


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Guest Editor
Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg 2006, South Africa
Interests: applied mathematics; fluid mechanics; finite element method; fractional differential equations; fractional derivative; nonlinear partial differential equations; numerical analysis; applied and computational mathematics; numerical modeling; numerical simulation

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Guest Editor
Department of Mathematics and Applied Mathematics, University of Johannesburg, Johannesburg 2006, South Africa
Interests: fractional differential equations; computational mathematics; mathematical modeling; Caputo fractional derivative; fractional model; numerical analysis

Special Issue Information

Dear Colleagues,

This Special Issue focuses on numerical solutions to Caputo-type fractional differential equations and derivatives. We will accept papers that provide contributions that delve into developing, analyzing, and applying numerical methods to solve these equations. In the recent past, fractional calculus has witnessed remarkable growth and diversification, yielding an array of definitions and mathematical formulations of the fractional derivative. The practical relevance of fractional calculus has been increasingly evident as it finds applications in a variety of real-life scenarios modeled using fractional differential equations. Crucially, efficient numerical methods have become key to unveiling solutions to these fractional differential equations. As the field of fractional calculus evolves, there is a pressing need for the development of novel numerical methodologies and the refinement of existing techniques.

Dr. Phumlani Dlamini
Dr. Simphiwe Simelane
Guest Editors

Manuscript Submission Information

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Keywords

  • Caputo-type fractional differential equations
  • Caputo-type fractional derivatives
  • generalized fractional derivatives
  • fractional inequalities
  • fractional operator
  • fractional Green’s functions
  • fractional Laplace transform
  • fractional evolution equations
  • finite difference schemes
  • spectral methods
  • existence and uniqueness
  • stability
  • controllability
  • iterative learning controls

Published Papers (4 papers)

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Research

18 pages, 355 KiB  
Article
Constrained State Regulation Problem of Descriptor Fractional-Order Linear Continuous-Time Systems
by Hongli Yang, Xindong Si and Ivan G. Ivanov
Fractal Fract. 2024, 8(5), 255; https://doi.org/10.3390/fractalfract8050255 - 25 Apr 2024
Viewed by 562
Abstract
This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints [...] Read more.
This paper deals with the constrained state regulation problem (CSRP) of descriptor fractional-order linear continuous-time systems (DFOLCS) with order 0<α<1. The objective is to establish the existence of conditions for a linear feedback control law within state constraints and to propose a method for solving the CSRP of DFOLCS. First, based on the decomposition and separation method and coordinate transformation, the DFOLCS can be transformed into an equivalent fractional-order reduced system; hence, the CSRP of the DFOLCS is equivalent to the CSRP of the reduced system. By means of positive invariant sets theory, Lyapunov stability theory, and some mathematical techniques, necessary and sufficient conditions for the polyhedral positive invariant set of the equivalent reduced system are presented. Models and corresponding algorithms for solving the CSRP of a linear feedback controller are also presented by the obtained conditions. Under the condition that the resulting closed system is positive, the given model of the CSRP in this paper for the DFOLCS is formulated as nonlinear programming with a linear objective function and quadratic mixed constraints. Two numerical examples illustrate the proposed method. Full article
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20 pages, 414 KiB  
Article
Contributions to the Numerical Solutions of a Caputo Fractional Differential and Integro-Differential System
by Abdelkader Moumen, Abdelaziz Mennouni and Mohamed Bouye
Fractal Fract. 2024, 8(4), 201; https://doi.org/10.3390/fractalfract8040201 - 29 Mar 2024
Viewed by 689
Abstract
The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, [...] Read more.
The primary goal of this research is to offer an efficient approach to solve a certain type of fractional integro-differential and differential systems. In the Caputo meaning, the fractional derivative is examined. This system is essential for many scientific disciplines, including physics, astrophysics, electrostatics, control theories, and the natural sciences. An effective approach solves the problem by reducing it to a pair of algebraically separated equations via a successful transformation. The proposed strategy uses first-order shifted Chebyshev polynomials and a projection method. Using the provided technique, the primary system is converted into a set of algebraic equations that can be solved effectively. Some theorems are proved and used to obtain the upper error bound for this method. Furthermore, various examples are provided to demonstrate the efficiency of the proposed algorithm when compared to existing approaches in the literature. Finally, the key conclusions are given. Full article
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14 pages, 285 KiB  
Article
Bang-Bang Property and Time-Optimal Control for Caputo Fractional Differential Systems
by Shimaa H. Abel-Gaid, Ahlam Hasan Qamlo and Bahaa Gaber Mohamed
Fractal Fract. 2024, 8(2), 84; https://doi.org/10.3390/fractalfract8020084 - 26 Jan 2024
Viewed by 885
Abstract
In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. [...] Read more.
In this paper, by using the controllability method, a bang-bang property and a time optimal control problem for time fractional differential systems (FDS) are considered. First, we formulate our problem and prove the existence theorem. We then state and prove the bang-bang theorem. Finally, we state the optimality conditions that characterize the optimal control. Some application examples are given to illustrate our results. Full article
15 pages, 371 KiB  
Article
An Efficient Numerical Method Based on Bell Wavelets for Solving the Fractional Integro-Differential Equations with Weakly Singular Kernels
by Yanxin Wang and Xiaofang Zhou
Fractal Fract. 2024, 8(2), 74; https://doi.org/10.3390/fractalfract8020074 - 23 Jan 2024
Viewed by 1150
Abstract
A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The [...] Read more.
A novel numerical scheme based on the Bell wavelets is proposed to obtain numerical solutions of the fractional integro-differential equations with weakly singular kernels. Bell wavelets are first proposed and their properties are studied, and the fractional integration operational matrix is constructed. The convergence analysis of Bell wavelets approximation is discussed. The fractional integro-differential equations can be simplified to a system of algebraic equations by using a truncated Bell wavelets series and the fractional operational matrix. The proposed method’s efficacy is supported via various examples. Full article
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