Fractional Differential Equations: Computation and Modelling with Applications

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

Deadline for manuscript submissions: 15 July 2024 | Viewed by 7924

Special Issue Editors


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Guest Editor
Department of Mathematics, National Institute of Technology Rourkela, Odisha 769008, India
Interests: differential equations (fractional, partial and ordinary); numerical analysis; applied mathematics; computational methods; mathematical modeling; soft computing

E-Mail Website
Guest Editor
Department of Mathematics, National Institute of Technology Rourkela, Odisha 769008, India
Interests: differential equations; numerical analysis; computational methods; structural dynamics (FGM, Nano); mathematical modeling; neural networks, robotics, and uncertainty modeling

Special Issue Information

Dear Colleagues,

Nowadays, many researchers from various fields have become interested in the topic of fractional calculus based on integrals and derivatives of fractional order. It has numerous applications in the widespread field of science and engineering, including wave and fluid dynamics, mathematical biology, financial systems, structural dynamics, robotics, artificial intelligence, etc. Therefore, fractional models have become relevant when dealing with phenomena with memory effects instead of relying on ordinary or partial differential equations. Fractional calculus offers superior tools to cope with the time-dependent effects noticed compared to integer-order calculus, which forms the mathematical foundation of most mathematical systems. As a result, fractional calculus is crucial to model real-life problems and finding mathematical solutions is a great challenge. Since fractional differential equations are used to model real-life problems, many mathematical methods (numerical/analytical/exact) are being developed to obtain the solutions to fractional differential equations/models/systems.

In this Special Issue, we invite review and original research papers dealing with recent developments in fractional calculus along with all theoretical/analytical/numerical, as well as practical developments in various science and engineering, including mathematics and physics.

This Special Issue will be focused upon, but not limited to, the following:

  • Fractional-order differential/partial/integral equations;
  • New fractional-order operators and their properties;
  • Existence and uniqueness of solutions;
  • Computational efficient methods (analytical/numerical) for fractional order systems;
  • Special functions in fractional calculus;
  • Fractional models in physics, biology, medicine, engineering, etc.;
  • Neural computations with fractional calculus;
  • Bifurcation and chaos;
  • Artificial Intelligence;
  • Fuzzy fractional calculus;
  • Mathematical modelling of fractional complex systems;
  • Deterministic and stochastic fractional differential equations;
  • Fractional calculus with uncertainties and modelling;
  • Fractional delay differential equations;
  • Fractal-fractional models.

Dr. Rajarama Mohan Jena
Prof. Dr. Snehashish Chakraverty
Guest Editors

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • fractional derivatives and integrals
  • fractional operators
  • fractal-fractional derivatives and integrals
  • mathematical modelling
  • computational efficient methods
  • uncertainty and artificial intelligence
  • nonsingular kernels

Published Papers (8 papers)

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Research

41 pages, 3964 KiB  
Article
Radial Basis Functions Approximation Method for Time-Fractional FitzHugh–Nagumo Equation
by Mehboob Alam, Sirajul Haq, Ihteram Ali, M. J. Ebadi and Soheil Salahshour
Fractal Fract. 2023, 7(12), 882; https://doi.org/10.3390/fractalfract7120882 - 13 Dec 2023
Cited by 1 | Viewed by 1013
Abstract
In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate [...] Read more.
In this paper, a numerical approach employing radial basis functions has been applied to solve time-fractional FitzHugh–Nagumo equation. Spatial approximation is achieved by combining radial basis functions with the collocation method, while temporal discretization is accomplished using a finite difference scheme. To evaluate the effectiveness of this method, we first conduct an eigenvalue stability analysis and then validate the results with numerical examples, varying the shape parameter c of the radial basis functions. Notably, this method offers the advantage of being mesh-free, which reduces computational overhead and eliminates the need for complex mesh generation processes. To assess the method’s performance, we subject it to examples. The simulated results demonstrate a high level of agreement with exact solutions and previous research. The accuracy and efficiency of this method are evaluated using discrete error norms, including L2L, and Lrms. Full article
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16 pages, 320 KiB  
Article
Qualitative Aspects of a Fractional-Order Integro-Differential Equation with a Quadratic Functional Integro-Differential Constraint
by Ahmed M. A. El-Sayed, Antisar A. A. Alhamali, Eman M. A. Hamdallah and Hanaa R. Ebead
Fractal Fract. 2023, 7(12), 835; https://doi.org/10.3390/fractalfract7120835 - 24 Nov 2023
Viewed by 873
Abstract
This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution xC[0,T] to the problem. Moreover, we [...] Read more.
This manuscript investigates a constrained problem of an arbitrary (fractional) order quadratic functional integro-differential equation with a quadratic functional integro-differential constraint. We demonstrate that there is at least one solution xC[0,T] to the problem. Moreover, we outline the necessary demands for the solution’s uniqueness. In addition, the continuous dependence of the solution and the Hyers–Ulam stability of the problem are analyzed. In order to illustrate our results, we provide some particular cases and instances. Full article
20 pages, 1176 KiB  
Article
Computational Analysis of Fractional-Order KdV Systems in the Sense of the Caputo Operator via a Novel Transform
by Mashael M. AlBaidani, Abdul Hamid Ganie and Adnan Khan
Fractal Fract. 2023, 7(11), 812; https://doi.org/10.3390/fractalfract7110812 - 09 Nov 2023
Cited by 2 | Viewed by 931
Abstract
The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy [...] Read more.
The main features of scientific efforts in physics and engineering are the development of models for various physical issues and the development of solutions. In order to solve the time-fractional coupled Korteweg–De Vries (KdV) equation, we combine the novel Yang transform, the homotopy perturbation approach, and the Adomian decomposition method in the present investigation. KdV models are crucial because they can accurately represent a variety of physical problems, including thin-film flows and waves on shallow water surfaces. The fractional derivative is regarded in the Caputo meaning. These approaches apply straightforward steps through symbolic computation to provide a convergent series solution. Different nonlinear time-fractional KdV systems are used to test the effectiveness of the suggested techniques. The symmetry pattern is a fundamental feature of the KdV equations and the symmetrical aspect of the solution can be seen from the graphical representations. The numerical outcomes demonstrate that only a small number of terms are required to arrive at an approximation that is exact, efficient, and trustworthy. Additionally, the system’s approximative solution is illustrated graphically. The results show that these techniques are extremely effective, practically applicable for usage in such issues, and adaptable to other nonlinear issues. Full article
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18 pages, 578 KiB  
Article
Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method
by Naveen S., Parthiban V. and Mohamed I. Abbas
Fractal Fract. 2023, 7(11), 804; https://doi.org/10.3390/fractalfract7110804 - 04 Nov 2023
Viewed by 910
Abstract
This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and [...] Read more.
This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω(1,2) for the RLC model. Based on Schaefer’s fixed point theorem and Banach’s contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples. Full article
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16 pages, 442 KiB  
Article
The Müntz–Legendre Wavelet Collocation Method for Solving Weakly Singular Integro-Differential Equations with Fractional Derivatives
by Haifa Bin Jebreen
Fractal Fract. 2023, 7(10), 763; https://doi.org/10.3390/fractalfract7100763 - 17 Oct 2023
Cited by 1 | Viewed by 1053
Abstract
We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional [...] Read more.
We offer a wavelet collocation method for solving the weakly singular integro-differential equations with fractional derivatives (WSIDE). Our approach is based on the reduction of the desired equation to the corresponding Volterra integral equation. The Müntz–Legendre (ML) wavelet is introduced, and a fractional integration operational matrix is constructed for it. The obtained integral equation is reduced to a system of nonlinear algebraic equations using the collocation method and the operational matrix of fractional integration. The presented method’s error bound is investigated, and some numerical simulations demonstrate the efficiency and accuracy of the method. According to the obtained results, the presented method solves this type of equation well and gives significant results. Full article
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33 pages, 7785 KiB  
Article
Modeling and Dynamical Analysis of a Fractional-Order Predator–Prey System with Anti-Predator Behavior and a Holling Type IV Functional Response
by Baiming Wang and Xianyi Li
Fractal Fract. 2023, 7(10), 722; https://doi.org/10.3390/fractalfract7100722 - 30 Sep 2023
Viewed by 686
Abstract
We here investigate the dynamic behavior of continuous and discrete versions of a fractional-order predator–prey system with anti-predator behavior and a Holling type IV functional response. First, we establish the non-negativity, existence, uniqueness and boundedness of solutions to the system from a mathematical [...] Read more.
We here investigate the dynamic behavior of continuous and discrete versions of a fractional-order predator–prey system with anti-predator behavior and a Holling type IV functional response. First, we establish the non-negativity, existence, uniqueness and boundedness of solutions to the system from a mathematical analysis perspective. Then, we analyze the stability of its equilibrium points and the possibility of bifurcations using stability analysis methods and bifurcation theory, demonstrating that, under specific parameter conditions, the continuous system exhibits a Hopf bifurcation, while the discrete version exhibits a Neimark–Sacker bifurcation and a period-doubling bifurcation. After providing numerical simulations to illustrate the theoretically derived conclusions and by summarizing the various analytical results obtained, we finally present four interesting conclusions that can contribute to better management and preservation of ecological systems. Full article
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33 pages, 678 KiB  
Article
Exploring Families of Solitary Wave Solutions for the Fractional Coupled Higgs System Using Modified Extended Direct Algebraic Method
by Muhammad Bilal, Javed Iqbal, Rashid Ali, Fuad A. Awwad and Emad A. A. Ismail
Fractal Fract. 2023, 7(9), 653; https://doi.org/10.3390/fractalfract7090653 - 30 Aug 2023
Cited by 1 | Viewed by 721
Abstract
In this paper, we suggest the modified Extended Direct Algebraic Method (mEDAM) to examine the existence and dynamics of solitary wave solutions in the context of the fractional coupled Higgs system, with Caputo’s fractional derivatives. The method begins with the formulation of nonlinear [...] Read more.
In this paper, we suggest the modified Extended Direct Algebraic Method (mEDAM) to examine the existence and dynamics of solitary wave solutions in the context of the fractional coupled Higgs system, with Caputo’s fractional derivatives. The method begins with the formulation of nonlinear differential equations using a fractional complex transformation, followed by the derivation of solitary wave solutions. Two-dimensional, Three-dimensional and contour graphs are used to investigate the behavior of traveling wave solutions. The research reveals many families of solitary wave solutions as well as their deep interrelationships and dynamics. These discoveries add to a better understanding of the dynamics of the fractionally coupled Higgs system and have potential applications in areas that use nonlinear Fractional Partial Differential Equations (FPDEs). Full article
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24 pages, 1506 KiB  
Article
Dynamic of Some Relapse in a Giving Up Smoking Model Described by Fractional Derivative
by Fawaz K. Alalhareth, Ahmed Boudaoui, Yacine El hadj Moussa, Noura Laksaci and Mohammed H. Alharbi
Fractal Fract. 2023, 7(7), 543; https://doi.org/10.3390/fractalfract7070543 - 14 Jul 2023
Cited by 1 | Viewed by 803
Abstract
Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model [...] Read more.
Smoking is associated with various detrimental health conditions, including cancer, heart disease, stroke, lung illnesses, diabetes, and fatal diseases. Motivated by the application of fractional calculus in epidemiological modeling and the exploration of memory and nonlocal effects, this paper introduces a mathematical model that captures the dynamics of relapse in a smoking cessation context and presents the dynamic behavior of the proposed model utilizing Caputo fractional derivatives. The model incorporates four compartments representing potential, persistent (heavy), temporally recovered, and permanently recovered smokers. The basic reproduction number R0 is computed, and the local and global dynamic behaviors of the free equilibrium smoking point (Y0) and the smoking-present equilibrium point (Y*) are analyzed. It is demonstrated that the free equilibrium smoking point (Y0) exhibits global asymptotic stability when R01, while the smoking-present equilibrium point (Y*) is globally asymptotically stable when R0>1. Additionally, analytical results are validated through a numerical simulation using the predictor–corrector PECE method for fractional differential equations in Matlab software. Full article
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