Research

12 pages, 885 KiB

Open AccessArticle

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 754

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

2 n

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

Open AccessArticle

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 - 04 Jul 2023

Viewed by 544

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

Open AccessFeature PaperArticle

Unilateral Laplace Transforms on Time Scales

by Müfit Şan and Manuel D. Ortigueira

Mathematics 2022, 10(23), 4552; https://doi.org/10.3390/math10234552 - 01 Dec 2022

Cited by 1 | Viewed by 1119

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

Open AccessArticle

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 - 03 Nov 2022

Cited by 2 | Viewed by 783

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

Open AccessArticle

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 1346

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

Open AccessArticle

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 1263

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

Open AccessArticle

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 1209

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

Open AccessArticle

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 - 06 Aug 2021

Cited by 3 | Viewed by 1700

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

Open AccessArticle

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 1828

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 \in (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 \in (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

Open AccessArticle

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 - 08 Jul 2021

Cited by 3 | Viewed by 1731

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

Open AccessArticle

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 11 | Viewed by 1550

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

Open AccessArticle

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 1167

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)} z_{i} = a z_{i} + u_{i} - v, u_{i}, v \in V,

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

Open AccessArticle

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 2276

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

J_{0} (r) = 0

, which are needed in our study, have been determined with the subroutine

r_{n} = r o o t (J_{0} (r), r, (2 n - 1) π / 4, (2 n + 3) π / 4), n = 1, 2, \dots

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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