Fractional Differential Equations and Control Problems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 June 2023) | Viewed by 21516

Special Issue Editor


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Guest Editor
Faculty of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Olsztyn, Poland
Interests: fractional calculus; fractional differential equations; Mittag-Leffler function; optimal control; differential game

Special Issue Information

Dear Colleagues,

Differential equations of non-integer order arise in a natural manner as mathematical models of dynamic systems that exhibit such properties as long-term memory and self-similarity. Such systems are then said to have fractional dynamics. Applications and examples of systems with fractional dynamics are ubiquitous in a diverse set of areas. The growing demand for efficient and high-performance systems exhibiting fractional dynamics induces growth in the demand for optimal control theories and methods.

In this Special Issue, we aim to present the recent developments in the theory and applications of any types of fractional differential equations and inclusions, with a special emphasis on control problems for fractional ordinary and partial differential equations.

This Special Issue will accept high-quality papers containing original research results and survey articles of exceptional merit in the following fields:

Fractional differential equations and their applications;

Stability analysis and control design for fractional order systems;

Fractional optimal control;

Fractional calculus of variations;

Differential games with fractional derivatives.

Prof. Dr. Ivan Matychyn
Guest Editor

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Keywords

  • fractional differential equations
  • fractional derivative
  • Mittag-Leffler function
  • fractional optimal control
  • fractional PID controller

Published Papers (13 papers)

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Research

12 pages, 885 KiB  
Article
Exact Solution of Non-Homogeneous Fractional Differential System Containing 2n Periodic Terms under Physical Conditions
by Laila F. Seddek, Abdelhalim Ebaid, Essam R. El-Zahar and Mona D. Aljoufi
Mathematics 2023, 11(15), 3308; https://doi.org/10.3390/math11153308 - 27 Jul 2023
Cited by 2 | Viewed by 899
Abstract
This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve [...] Read more.
This paper solves a generalized class of first-order fractional ordinary differential equations (1st-order FODEs) by means of Riemann–Liouville fractional derivative (RLFD). The principal incentive of this paper is to generalize some existing results in the literature. An effective approach is applied to solve non-homogeneous fractional differential systems containing 2n periodic terms. The exact solutions are determined explicitly in a straightforward manner. The solutions are expressed in terms of entire functions with fractional order arguments. Features of the current solutions are discussed and analyzed. In addition, the existing solutions in the literature are recovered as special cases of our results. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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16 pages, 301 KiB  
Article
A Generalized Lyapunov Inequality for a Pantograph Boundary Value Problem Involving a Variable Order Hadamard Fractional Derivative
by John R. Graef, Kadda Maazouz and Moussa Daif Allah Zaak
Mathematics 2023, 11(13), 2984; https://doi.org/10.3390/math11132984 - 4 Jul 2023
Cited by 1 | Viewed by 649
Abstract
The authors obtain existence and uniqueness results for a nonlinear fractional pantograph boundary value problem containing a variable order Hadamard fractional derivative. This type of model is appropriate for applications involving processes that occur in strongly anomalous media. They also derive a generalized [...] Read more.
The authors obtain existence and uniqueness results for a nonlinear fractional pantograph boundary value problem containing a variable order Hadamard fractional derivative. This type of model is appropriate for applications involving processes that occur in strongly anomalous media. They also derive a generalized Lyapunov-type inequality for the problem considered. Their results are obtained by the fractional calculus and Krasnosel’skii’s fixed point theorem. An example is given to illustrate their approach. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
21 pages, 430 KiB  
Article
Unilateral Laplace Transforms on Time Scales
by Müfit Şan and Manuel D. Ortigueira
Mathematics 2022, 10(23), 4552; https://doi.org/10.3390/math10234552 - 1 Dec 2022
Cited by 1 | Viewed by 1311
Abstract
We review the direct and inverse Laplace transforms on non-uniform time scales. We introduce full backward-compatible unilateral Laplace transforms and studied their properties. We also present the corresponding inverse integrals and some examples. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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24 pages, 508 KiB  
Article
Fractional Systems’ Identification Based on Implicit Modulating Functions
by Oliver Stark, Marius Eckert, Albertus Johannes Malan and Sören Hohmann
Mathematics 2022, 10(21), 4106; https://doi.org/10.3390/math10214106 - 3 Nov 2022
Cited by 3 | Viewed by 929
Abstract
This paper presents a new method for parameter identification based on the modulating function method for commensurable fractional-order models. The novelty of the method lies in the automatic determination of a specific modulating function by controlling a model-based auxiliary system, instead of applying [...] Read more.
This paper presents a new method for parameter identification based on the modulating function method for commensurable fractional-order models. The novelty of the method lies in the automatic determination of a specific modulating function by controlling a model-based auxiliary system, instead of applying and parameterizing a generic modulating function. The input signal of the model-based auxiliary system used to determine the modulating function is designed such that a separate identification of each individual parameter of the fractional-order model is enabled. This eliminates the shortcomings of the common modulating function method in which a modulating function must be adapted to the investigated system heuristically. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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16 pages, 2337 KiB  
Article
Reachability and Observability of Positive Linear Electrical Circuits Systems Described by Generalized Fractional Derivatives
by Tong Yuan, Hongli Yang and Ivan Ganchev Ivanov
Mathematics 2021, 9(22), 2856; https://doi.org/10.3390/math9222856 - 10 Nov 2021
Cited by 3 | Viewed by 1510
Abstract
Positive linear electrical circuits systems described by generalized fractional derivatives are studied in this paper. We mainly focus on the reachability and observability of linear electrical circuits systems. Firstly, generalized fractional derivatives and ρ-Laplace transform of f is presented and some preliminary [...] Read more.
Positive linear electrical circuits systems described by generalized fractional derivatives are studied in this paper. We mainly focus on the reachability and observability of linear electrical circuits systems. Firstly, generalized fractional derivatives and ρ-Laplace transform of f is presented and some preliminary results are provided. Secondly, the positivity of linear electrical circuits systems described by generalized fractional derivatives is investigated and conditions for checking positivity of the systems are derived. Thirdly, reachability and observability of the generalized fractional derivatives systems are studied, in which the ρ-Laplace transform of a Mittag-Leffler function plays an important role. At the end of the paper, illustrative electrical circuits systems are presented, and conclusions of the paper are presented. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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10 pages, 304 KiB  
Article
On the Initial Value Problems for Caputo-Type Generalized Proportional Vector-Order Fractional Differential Equations
by Mohamed I. Abbas and Snezhana Hristova
Mathematics 2021, 9(21), 2720; https://doi.org/10.3390/math9212720 - 26 Oct 2021
Cited by 4 | Viewed by 1460
Abstract
A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of [...] Read more.
A generalized proportional vector-order fractional derivative in the Caputo sense is defined and studied. Two types of existence results for the mild solutions of the initial value problem for nonlinear Caputo-type generalized proportional vector-order fractional differential equations are obtained. With the aid of the Leray–Schauder nonlinear alternative and the Banach contraction principle, the main results are established. In the case of a local Lipschitz right hand side part function, the existence of a bounded mild solution is proved. Some examples illustrating the main results are provided. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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21 pages, 1113 KiB  
Article
Series Solutions of High-Dimensional Fractional Differential Equations
by Jing Chang, Jin Zhang and Ming Cai
Mathematics 2021, 9(17), 2021; https://doi.org/10.3390/math9172021 - 24 Aug 2021
Cited by 2 | Viewed by 1360
Abstract
In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). [...] Read more.
In the present paper, the series solutions and the approximate solutions of the time–space fractional differential equations are obtained using two different analytical methods. One is the homotopy perturbation Sumudu transform method (HPSTM), and another is the variational iteration Laplace transform method (VILTM). It is observed that the approximate solutions are very close to the exact solutions. The solutions obtained are very useful and significant to analyze many phenomena, and the solutions have not been reported in previous literature. The salient feature of this work is the graphical presentations of the third approximate solutions for different values of order α. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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11 pages, 278 KiB  
Article
Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative
by Mohamed Jleli, Bessem Samet and Calogero Vetro
Mathematics 2021, 9(16), 1866; https://doi.org/10.3390/math9161866 - 6 Aug 2021
Cited by 3 | Viewed by 1807
Abstract
Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present [...] Read more.
Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
16 pages, 328 KiB  
Article
Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies
by Mikhail I. Gomoyunov
Mathematics 2021, 9(14), 1667; https://doi.org/10.3390/math9141667 - 15 Jul 2021
Cited by 6 | Viewed by 2015
Abstract
The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α(0,1) and a Bolza-type cost functional. A relationship between the differential game [...] Read more.
The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α(0,1) and a Bolza-type cost functional. A relationship between the differential game and the Cauchy problem for the corresponding Hamilton–Jacobi–Bellman–Isaacs equation with fractional coinvariant derivatives of the order α and the natural boundary condition is established. An emphasis is given to construction of optimal positional (feedback) strategies of the players. First, a smooth case is studied when the considered Cauchy problem is assumed to have a sufficiently smooth solution. After that, to cope with a general non-smooth case, a generalized minimax solution of this problem is involved. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
19 pages, 448 KiB  
Article
Parameter and Order Identification of Fractional Systems with Application to a Lithium-Ion Battery
by Oliver Stark, Martin Pfeifer and Sören Hohmann
Mathematics 2021, 9(14), 1607; https://doi.org/10.3390/math9141607 - 8 Jul 2021
Cited by 3 | Viewed by 1865
Abstract
This paper deals with a method for the parameter and order identification of a fractional model. In contrast to existing approaches that can either handle noisy observations of the output signal or systems that are not at rest, the proposed method does not [...] Read more.
This paper deals with a method for the parameter and order identification of a fractional model. In contrast to existing approaches that can either handle noisy observations of the output signal or systems that are not at rest, the proposed method does not have to compromise between these two characteristics. To handle systems that are not at rest, the parameter, as well as the order identification, are based on the modulating function method. The novelty of the proposed method is that an optimization-based approach is used for the order identification. Thus, even if only noisy observations of the output signal are available, an approximate identification can be performed. The proposed identification method is, then, applied to identify the parameters and orders of a lithium-ion battery model. The experimental results illustrate the practical usefulness and verify the validity of our approach. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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14 pages, 2097 KiB  
Article
Finite-Time Projective Synchronization of Caputo Type Fractional Complex-Valued Delayed Neural Networks
by Shuang Wang, Hai Zhang, Weiwei Zhang and Hongmei Zhang
Mathematics 2021, 9(12), 1406; https://doi.org/10.3390/math9121406 - 17 Jun 2021
Cited by 12 | Viewed by 1725
Abstract
This paper focuses on investigating the finite-time projective synchronization of Caputo type fractional-order complex-valued neural networks with time delay (FOCVNNTD). Based on the properties of fractional calculus and various inequality techniques, by constructing suitable the Lyapunov function and designing two new types controllers, [...] Read more.
This paper focuses on investigating the finite-time projective synchronization of Caputo type fractional-order complex-valued neural networks with time delay (FOCVNNTD). Based on the properties of fractional calculus and various inequality techniques, by constructing suitable the Lyapunov function and designing two new types controllers, i.e., feedback controller and adaptive controller, two sufficient criteria are derived to ensure the projective finite-time synchronization between drive and response systems, and the synchronization time can effectively be estimated. Finally, two numerical examples are presented to verify the effectiveness and feasibility of the proposed results. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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12 pages, 275 KiB  
Article
Multiple Capture in a Group Pursuit Problem with Fractional Derivatives and Phase Restrictions
by Nikolay Nikandrovich Petrov
Mathematics 2021, 9(11), 1171; https://doi.org/10.3390/math9111171 - 22 May 2021
Cited by 5 | Viewed by 1281
Abstract
The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the [...] Read more.
The problem of conflict interaction between a group of pursuers and an evader in a finite-dimensional Euclidean space is considered. All participants have equal opportunities. The dynamics of all players are described by a system of differential equations with fractional derivatives in the form D(α)zi=azi+uiv,ui,vV, where D(α)f is a Caputo derivative of order α of the function f. Additionally, it is assumed that in the process of the game the evader does not move out of a convex polyhedral cone. The set of admissible controls V is a strictly convex compact and a is a real number. The goal of the group of pursuers is to capture of the evader by no less than m different pursuers (the instants of capture may or may not coincide). The target sets are the origin. For such a conflict-controlled process, we derive conditions on its parameters and initial state, which are sufficient for the trajectories of the players to meet at a certain instant of time for any counteractions of the evader. The method of resolving functions is used to solve the problem, which is used in differential games of pursuit by a group of pursuers of one evader. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
14 pages, 3223 KiB  
Article
Analytical Solutions of the Fractional Mathematical Model for the Concentration of Tumor Cells for Constant Killing Rate
by Najma Ahmed, Nehad Ali Shah, Farman Ali, Dumitru Vieru and F.D. Zaman
Mathematics 2021, 9(10), 1156; https://doi.org/10.3390/math9101156 - 20 May 2021
Cited by 4 | Viewed by 2409
Abstract
Two generalized mathematical models with memory for the concentration of tumor cells have been analytically studied using the cylindrical coordinate and the integral transform methods. The generalization consists of the formulating of two mathematical models with Caputo-time fractional derivative, models that are suitable [...] Read more.
Two generalized mathematical models with memory for the concentration of tumor cells have been analytically studied using the cylindrical coordinate and the integral transform methods. The generalization consists of the formulating of two mathematical models with Caputo-time fractional derivative, models that are suitable to highlight the influence of the history of tumor evolution on the present behavior of the concentration of cancer cells. The time-oscillating concentration of cancer cells has been considered on the boundary of the domain. Analytical solutions of the fractional differential equations of the mathematical models have been determined using the Laplace transform with respect to the time variable and the finite Hankel transform with respect to the radial coordinate. The positive roots of the transcendental equation with Bessel function J0(r)=0, which are needed in our study, have been determined with the subroutine rn=root(J0(r),r,(2n1)π/4,(2n+3)π/4),n=1,2, of the Mathcad 15 software. It is found that the memory effects are stronger at small values of the time, t. This aspect is highlighted in the graphical illustrations that analyze the behavior of the concentration of tumor cells. Additionally, the concentration of cancer cells is symmetric with respect to radial angle, and its values tend to be zero for large values of the time, t. Full article
(This article belongs to the Special Issue Fractional Differential Equations and Control Problems)
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