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Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 20123

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Special Issue Editors

Prof. Dr. Cemil Tunç

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Guest Editor

Department of Mathematics, Faculty of Science, Van Yuzuncu Yil University, 65080 Van, Turkey
Interests: mathematics; differential equations; integral equations; partial differential equations

Prof. Dr. Jen-Chih Yao

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Guest Editor

Center for General Education, China Medical University, Taichung 40402, Taiwan
Interests: vector optimization; fixed point theory; variational inequalities; complementarity problems; variational analysis; equilibrium problems; optimal control; generalized convexity and generalized monotonicity
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Prof. Dr. Mouffak Benchohra

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Laboratory of Mathematics, Djillali Liabes University of Sidi Bel Abbes, P.O. Box 89, Sidi Bel Abbes 22000, Algeria
Interests: fractional differential equations; abstract differential equations; functional differential equations with delay
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Prof. Dr. Ahmed M. A. El-Sayed

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Faculty of Science, Alexandria University, 21500 Alexandria, Egypt
Interests: fractional calculus; applied functional analysis; functional differential and functional integral equations; dynamical system

Special Issue Information

Dear Colleagues,

In the last three decades, fractional calculus, fractional differential and fractional integro-differential equations, and qualitative theory of these equations have been broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the theory of fractional calculus, the qualitative theory of fractional differential and fractional integro-differential equations, their numerical simulations, and symmetry analysis are mathematical analysis tools applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. Fractional order operators are nonlinear operators that are more useful than classical formulations. Qualitative theory of fractional differential equations, fractional integro-differential equations, and fractional order operators can occur in numerous scientific fields, such as fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing, entropy theory, and so on. This is why the applications of theory of fractional calculus and qualitative theory of the mentioned equations have become a focus of international academic research, and a lot of researchers have adopted them in their new studies. One of the most recently developed studies is the use of different types of kernels. Singular and non-singular kernels have been used in recent studies for the analysis of dynamical models, and their results are comparable to those of classical work.

Prof. Dr. Cemil Tunç
Prof. Dr. Jen-Chih Yao
Prof. Dr. Mouffak Benchohra
Prof. Dr. Ahmed M. A. El-Sayed
Guest Editors

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Keywords

fractional calculus
dynamical models of fractional orders
fractional order modeling with Mittag–Leffler kernel of two parameters
fixed point theorems with applications to the fractional differential equations
discrete fractional differential equations
numerical methods and their applications
existence of solutions for fractional differential equations
hybrid fractional differential equations Qualitative properties fractional integral equations
qualitative properties of fractional integro-differential equations
stability of symmetry solutions of fractional ordinary differential equation
symmetry analysis of fractional ordinary differential equations
symmetry analysis of conformable differential equation

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Published Papers (12 papers)

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Editorial

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4 pages, 168 KiB

Open AccessEditorial

Editorial for the Special Issue of “Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis”

by Cemil Tunç, Jen-Chih Yao, Mouffak Benchohra and Ahmed M. A. El-Sayed

Symmetry 2024, 16(9), 1193; https://doi.org/10.3390/sym16091193 - 11 Sep 2024

Viewed by 596

Abstract

The fractional calculus is a specific case of classical calculus, as is well known [...] Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

Research

Jump to: Editorial

17 pages, 414 KiB

Open AccessArticle

A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

by Zahra Sahraee and Maryam Arabameri

Symmetry 2023, 15(9), 1759; https://doi.org/10.3390/sym15091759 - 13 Sep 2023

Cited by 1 | Viewed by 1358

Abstract

The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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15 pages, 315 KiB

Open AccessArticle

Systems of Sequential ψ₁-Hilfer and ψ₂-Caputo Fractional Differential Equations with Fractional Integro-Differential Nonlocal Boundary Conditions

by Surang Sitho, Sotiris K. Ntouyas, Chayapat Sudprasert and Jessada Tariboon

Symmetry 2023, 15(3), 680; https://doi.org/10.3390/sym15030680 - 8 Mar 2023

Viewed by 1169

Abstract

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type

ψ_{1}

-Hilfer and

ψ_{2}

-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As [...] Read more.

In this paper, we introduce and study a new class of coupled and uncoupled systems, consisting of mixed-type

ψ_{1}

-Hilfer and

ψ_{2}

-Caputo fractional differential equations supplemented with asymmetric and symmetric integro-differential nonlocal boundary conditions (systems (2) and (13), respectively). As far as we know, this combination of

ψ_{1}

-Hilfer and

ψ_{2}

-Caputo fractional derivatives in coupled systems is new in the literature. The uniqueness result is achieved via the Banach contraction mapping principle, while the existence result is established by applying the Leray–Schauder alternative. Numerical examples illustrating the obtained results are also presented. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

15 pages, 543 KiB

Open AccessArticle

Numerical Analysis of the Time-Fractional Boussinesq Equation in Gradient Unconfined Aquifers with the Mittag-Leffler Derivative

by Safyan Mukhtar

Symmetry 2023, 15(3), 608; https://doi.org/10.3390/sym15030608 - 27 Feb 2023

Cited by 1 | Viewed by 1466

Abstract

In this study, two numerical methods—the variational iteration transform method (VITM) and the Adomian decomposition (ADM) method—were used to solve the second- and fourth-order fractional Boussinesq equations. Both methods are helpful in approximating non-linear problems effectively, easily, and accurately. The fractional Atangana–Baleanu operator and ZZ transform were utilized to derive solutions for the equation. Two examples are discussed to validate the methods and solutions. The results demonstrate that both the VITM and ADM methods are effective in obtaining accurate and reliable solutions for the time-fractional Boussinesq equation. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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13 pages, 5849 KiB

Open AccessArticle

Fractional Order Operator for Symmetric Analysis of Cancer Model on Stem Cells with Chemotherapy

by Muhammad Azeem, Muhammad Farman, Ali Akgül and Manuel De la Sen

Symmetry 2023, 15(2), 533; https://doi.org/10.3390/sym15020533 - 16 Feb 2023

Cited by 6 | Viewed by 1640

Abstract

Cancer is dangerous and one of the major diseases affecting normal human life. In this paper, a fractional-order cancer model with stem cells and chemotherapy is analyzed to check the effects of infection in individuals. The model is investigated by the Sumudu transform and a very effective numerical method. The positivity of solutions with the ABC operator of the proposed technique is verified. Fixed point theory is used to derive the existence and uniqueness of the solutions for the fractional order cancer system. Our derived solutions analyze the actual behavior and effect of cancer disease in the human body using different fractional values. Modern mathematical control with the fractional operator has many applications including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions of a given fractional differential equation in a fairly systematic way. Such an analysis will provide a better understanding to control the of cancer disease in the human body. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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15 pages, 294 KiB

Open AccessArticle

On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations

by Cemil Tunç, Osman Tunç and Jen-Chih Yao

Symmetry 2023, 15(1), 109; https://doi.org/10.3390/sym15010109 - 30 Dec 2022

Cited by 7 | Viewed by 1482

Abstract

In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the kernel of the equations. Using the Lyapunov–Krasovski functional method, we give various sufficient conditions of stability, asymptotic stability, uniform stability of zero solution, convergence and boundedness, and square integrability of nonzero solutions in relation to the considered scalar nonlinear integro-differential equations for various cases. As the novel contributions of this article, the new scalar nonlinear integro-differential equation with the fading memory is firstly investigated in the literature, and seven theorems, which have novel sufficient qualitative conditions, are provided on the qualitative behaviors of solutions called boundedness, convergence, stability, integrability, asymptotic stability and uniform stability of solutions. The novel outcomes and originality of this article are that the considered integro-differential equations are new mathematical models, they include former mathematical models in relation to the mathematical models of this paper as well as the given main seven qualitative results are also new. The outcomes of this paper enhance some present results and provide new contributions to the relevant literature. The results of the article have complementary properties for the symmetry of integro-differential equations. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

13 pages, 2120 KiB

Open AccessArticle

A Comparative Study of the Genetic Deep Learning Image Segmentation Algorithms

by Wenbo Wang, Muhammad Yousaf, Ding Liu and Ayesha Sohail

Symmetry 2022, 14(10), 1977; https://doi.org/10.3390/sym14101977 - 21 Sep 2022

Cited by 12 | Viewed by 2151

Abstract

Medical optical imaging, with the aid of the “terahertz tomography”, is a novel medical imaging technique based on the electromagnetic waves. Such advanced imaging techniques strive for the detailed theoretical and computational analysis for better verification and validation. Two important aspects, the analytic approach for the understanding of the Schrodinger transforms and machine learning approaches for the understanding of the medical images segmentation, are presented in this manuscript. While developing an AI algorithm for complex datasets, the computational speed and accuracy cannot be overlooked. With the passage of time, machine learning approaches have been further modified using the Bayesian, genetic and quantum approaches. These strategies have boosted the efficiency of the machine learning, and specifically the deep learning tools, by taking into account the probabilistic, evolutionary and quantum qubits hypothesis and operations, respectively. The current research encompasses the detailed analysis of image segmentation algorithms based on the evolutionary approach. The image segmentation algorithm that converts the color model from RGB to HSI and the image segmentation algorithm that uses the clustering technique are discussed in detail, and further extensions of these genetic algorithms to quantum algorithms are proposed. Based on the genetic algorithm, the optimal selection of parameters is realized so as to achieve a better segmentation effect. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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15 pages, 2212 KiB

Open AccessArticle

Dynamic Analysis and Control for a Bioreactor in Fractional Order

by Angelo M. Tusset, Danilo Inacio, Maria E. K. Fuziki, Priscilla M. L. Z. Costa and Giane G. Lenzi

Symmetry 2022, 14(8), 1609; https://doi.org/10.3390/sym14081609 - 4 Aug 2022

Cited by 7 | Viewed by 2648

Abstract

In this paper, a mathematical model was developed to describe the dynamic behavior of a bioreactor in which a fermentation process takes place. The analysis took into account the bioreactor temperature controlled by the refrigerant fluid flow through the reactor jacket. An optimal LQR control acting in the water flow through a jacket was used in order to maintain the reactor temperature during the process. For the control design, a reduced-order model of the system was considered. Given the heat transfer asymmetry observed in reactors, a model considering the fractional order heat exchange between the reactor and the jacket using the Riemann–Liouville differential operators was proposed. The numerical simulation demonstrated that the proposed control was efficient in maintaining the temperature at the desired levels and was robust for disturbances in the inlet temperature reactor. Additionally, the proposed control proved to be easy to apply in real life, bypassing the singularity problem and the difficulty of initial conditions for real applications that can be observed when considering Riemann–Liouville differential operators. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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25 pages, 394 KiB

Open AccessArticle

Analysis of Tempered Fractional Calculus in Hölder and Orlicz Spaces

by Hussein A. H. Salem and Mieczysław Cichoń

Symmetry 2022, 14(8), 1581; https://doi.org/10.3390/sym14081581 - 1 Aug 2022

Cited by 12 | Viewed by 1584

Abstract

Here, we propose a general framework covering a wide variety of fractional operators. We consider integral and differential operators and their role in tempered fractional calculus and study their analytic properties. We investigate tempered fractional integral operators acting on subspaces of [...] Read more.

L_{1} [a, b]

, such as Orlicz or Hölder spaces. We prove that in this case, they map Orlicz spaces into (generalized) Hölder spaces. In particular, they map Hölder spaces into the same class of spaces. The obtained results are a generalization of classical results for the Riemann–Liouville fractional operator and constitute the basis for the use of generalized operators in the study of differential and integral equations. However, we will show the non-equivalence differential and integral problems in the spaces under consideration. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

22 pages, 355 KiB

Open AccessArticle

Formulation, Solution’s Existence, and Stability Analysis for Multi-Term System of Fractional-Order Differential Equations

by Dildar Ahmad, Ravi P. Agarwal and Ghaus ur Rahman

Symmetry 2022, 14(7), 1342; https://doi.org/10.3390/sym14071342 - 29 Jun 2022

Cited by 10 | Viewed by 1525

Abstract

In the recent past, multi-term fractional equations have been studied using symmetry methods. In some cases, many practical test problems with some symmetries are provided to demonstrate the authenticity and utility of the used techniques. Fractional-order differential equations can be formulated by using two types of differential operators: single-term and multi-term differential operators. Boundary value problems with single- as well as multi-term differential operators have been extensively studied, but several multi-term fractional differential equations still need to be formulated, and examination should be done with symmetry or any other feasible techniques. Therefore, the purpose of the present research work is the formulation and study of a new couple system of multi-term fractional differential equations with delay, as well as supplementation with nonlocal boundary conditions. After model formulation, the existence of a solution and the uniqueness conditions will be developed, utilizing fixed point theory and functional analysis. Moreover, results related to Ulam’s and other types of functional stability will be explored, and an example is carried out to illustrate the findings of the work. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

13 pages, 294 KiB

Open AccessArticle

On Fractional Hybrid Non-Linear Differential Equations Involving Three Mixed Fractional Orders with Boundary Conditions

by Fang Li, Liping Zhang and Huiwen Wang

Symmetry 2022, 14(6), 1189; https://doi.org/10.3390/sym14061189 - 9 Jun 2022

Cited by 1 | Viewed by 1534

Abstract

In this paper, we study a class of non-linear fractional hybrid differential equations involving three mixed fractional orders with boundary conditions. Under weak assumptions, a formula of solutions is constructed and the existence results of the solutions for the problem are established. The results can be used to solve more general fractional hybrid equations, such as the general variable coefficient fractional hybrid Langevin equations. Moreover, the form of the solution for this kind of equation can provide a theoretical basis for the further study of the positive solution and its symmetry. We provide an example to support our main result. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

16 pages, 302 KiB

Open AccessArticle

Qualitative Study for a Delay Quadratic Functional Integro-Differential Equation of Arbitrary (Fractional) Orders

by Ahmed M. A. El-Sayed, Eman M. A. Hamdallah and Malak M. S. Ba-Ali

Symmetry 2022, 14(4), 784; https://doi.org/10.3390/sym14040784 - 9 Apr 2022

Cited by 6 | Viewed by 1339

Abstract

Symmetry analysis has been applied to solve many differential equations, although determining the symmetries can be computationally intensive compared to other solution methods. In this work, we study some operators which keep the set of solutions invariant. We discuss the existence of solutions for two initial value problems of a delay quadratic functional integro-differential equation of arbitrary (fractional) orders and its corresponding integer orders equation. The existence of the maximal and the minimal solutions is proved. The sufficient condition for the uniqueness of the solutions is given. The continuous dependence of the unique solution on some data is studied. The continuation of the arbitrary (fractional) orders problem to the integer order problem is investigated. Full article

(This article belongs to the Special Issue Fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis)

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