Advances in Fractional Order Derivatives and Their Applications, 2nd Edition

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: closed (30 April 2024) | Viewed by 6588

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Guest Editor
School of Mathematics, University of the Witwatersrand, Johannesburg 2001, South Africa
Interests: differential equations; symmetries; conservation laws; exact solutions; cosmology
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Special Issue Information

Dear Colleagues,

Fractional order derivatives have had a revolutionary impact on the scientific community. This area of study has grown in leaps and bounds, from analytical methods to numerical techniques. Consequently, applications of fractional order differential equations are now widespread across every possible area of research. 

The focus of this Special Issue is on the advancement of research on fractional order derivatives and their multi-faceted applications. Topics that are invited for submission include (but are not limited to) the following:

  • Mathematical modeling with fractional order derivatives;
  • Symmetry analysis of fractional order equations;
  • Conserved quantities related to fractional order models;
  • The various solution techniques for fractional order equations;
  • Special functions that are linked to the solution of fractional order equations;
  • Software to aid computations and analysis for fractional order derivatives and equations.

Please also feel free to read and download all published articles in our 1st volume:

https://www.mdpi.com/journal/fractalfract/special_issues/fractional_derivative

Dr. Sameerah Jamal
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 order derivatives
  • fractional calculus
  • Caputo derivatives
  • Riemann–Liouville derivatives
  • numerical analysis
  • modeling
  • application

Published Papers (9 papers)

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Research

27 pages, 987 KiB  
Article
On Numerical Simulations of Variable-Order Fractional Cable Equation Arising in Neuronal Dynamics
by Fouad Mohammad Salama
Fractal Fract. 2024, 8(5), 282; https://doi.org/10.3390/fractalfract8050282 - 8 May 2024
Viewed by 477
Abstract
In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling [...] Read more.
In recent years, various complex systems and real-world phenomena have been shown to include memory and hereditary properties that change with respect to time, space, or other variables. Consequently, fractional partial differential equations containing variable-order fractional operators have been extensively resorted for modeling such phenomena accurately. In this paper, we consider the two-dimensional fractional cable equation with the Caputo variable-order fractional derivative in the time direction, which is preferable for describing neuronal dynamics in biological systems. A point-wise scheme, namely, the Crank–Nicolson finite difference method, along with a group-wise scheme referred to as the explicit decoupled group method are proposed to solve the problem under consideration. The stability and convergence analyses of the numerical schemes are provided with complete details. To demonstrate the validity of the proposed methods, numerical simulations with results represented in tabular and graphical forms are given. A quantitative analysis based on the CPU timing, iteration counting, and maximum absolute error indicates that the explicit decoupled group method is more efficient than the Crank–Nicolson finite difference scheme for solving the variable-order fractional equation. Full article
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16 pages, 1578 KiB  
Article
Lie Symmetries and the Invariant Solutions of the Fractional Black–Scholes Equation under Time-Dependent Parameters
by Sameerah Jamal, Reginald Champala and Suhail Khan
Fractal Fract. 2024, 8(5), 269; https://doi.org/10.3390/fractalfract8050269 - 29 Apr 2024
Viewed by 507
Abstract
In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the [...] Read more.
In this paper, we consider the time-fractional Black–Scholes model with deterministic, time-varying coefficients. These time parametric constituents produce a model with greater flexibility that may capture empirical results from financial markets and their time-series datasets. We make use of transformations to reduce the underlying model to the classical heat transfer equation. We show that this transformation procedure is possible for a specific risk-free interest rate and volatility of stock function. Furthermore, we reverse these transformations and apply one-dimensional optimal subalgebras of the infinitesimal symmetry generators to establish invariant solutions. Full article
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21 pages, 467 KiB  
Article
Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types
by Ishtiaq Ali, Muhammad Yaseen and Iqra Akram
Fractal Fract. 2024, 8(5), 268; https://doi.org/10.3390/fractalfract8050268 - 29 Apr 2024
Viewed by 614
Abstract
Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method [...] Read more.
Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems. Full article
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20 pages, 707 KiB  
Article
Existence and Uniqueness Result for Fuzzy Fractional Order Goursat Partial Differential Equations
by Muhammad Sarwar, Noor Jamal, Kamaleldin Abodayeh, Chanon Promsakon and Thanin Sitthiwirattham
Fractal Fract. 2024, 8(5), 250; https://doi.org/10.3390/fractalfract8050250 - 25 Apr 2024
Viewed by 521
Abstract
In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s gH-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s gH-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert [...] Read more.
In this manuscript, we discuss fractional fuzzy Goursat problems with Caputo’s gH-differentiability. The second-order mixed derivative term in Goursat problems and two types of Caputo’s gH-differentiability pose challenges to dealing with Goursat problems. Therefore, in this study, we convert Goursat problems to equivalent systems fuzzy integral equations to deal properly with the mixed derivative term and two types of Caputo’s gH-differentiability. In this study, we utilize the concept of metric fixed point theory to discuss the existence of a unique solution of fractional fuzzy Goursat problems. For the useability of established theoretical work, we provide some numerical problems. We also discuss the solutions to numerical problems by conformable double Laplace transform. To show the validity of the solutions we provide 3D plots. We discuss, as an application, why fractional partial fuzzy differential equations are the generalization of usual partial fuzzy differential equations by providing a suitable reason. Moreover, we show the advantages of the proposed fractional transform over the usual Laplace transform. Full article
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19 pages, 1775 KiB  
Article
The Four-Dimensional Natural Transform Adomian Decomposition Method and (3+1)-Dimensional Fractional Coupled Burgers’ Equation
by Huda Alsaud and Hassan Eltayeb
Fractal Fract. 2024, 8(4), 227; https://doi.org/10.3390/fractalfract8040227 - 15 Apr 2024
Viewed by 616
Abstract
This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In [...] Read more.
This research article introduces the four-dimensional natural transform Adomian decomposition method (FNADM) for solving the (3+1)-dimensional time-singular fractional coupled Burgers’ equation, along with its associated initial conditions. The FNADM approach represents a fusion of four-dimensional natural transform techniques and Adomian decomposition methodologies. In order to observe the influence of time-Caputo fractional derivatives on the outcomes of the aforementioned models, two examples are illustrated along with their three-dimensional figures. The effectiveness and reliability of this approach are validated through the analysis of these examples related to the (3+1)-dimensional time-singular fractional coupled Burgers’ equations. This study underscores the method’s applicability and effectiveness in addressing the complex mathematical models encountered in various scientific and engineering domains. Full article
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14 pages, 661 KiB  
Article
Solitary and Periodic Wave Solutions of Fractional Zoomeron Equation
by Mohammad Alshammari, Khaled Moaddy, Muhammad Naeem, Zainab Alsheekhhussain, Saleh Alshammari and M. Mossa Al-Sawalha
Fractal Fract. 2024, 8(4), 222; https://doi.org/10.3390/fractalfract8040222 - 11 Apr 2024
Cited by 1 | Viewed by 636
Abstract
The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach [...] Read more.
The Zoomeron equation plays a significant role in many fields of physics, especially in soliton theory, such as helping to reveal new distinctive properties in different physical phenomena such as fluid dynamics, laser physics, and nonlinear optics. By using the Riccati–Bernoulli sub-ODE approach and the Backlund transformation, we search for soliton solutions of the fractional Zoomeron nonlinear equation. A number of solutions have been put forth, such as kink, anti-kink, cuspon kink, lump-type kink solitons, single solitons, and others defined in terms of pseudo almost periodic functions. The (2 + 1)-dimensional fractional Zoomeron equation given in a form undergoes precise dynamics. We use the computational software, Matlab 19, to express these solutions graphically by changing the value of various parameters involved. A detailed analysis of their dynamics allows us to obtain completely better insights necessarily with the elementary physical phenomena controlled by the fractional Zoomeron equation. Full article
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14 pages, 1172 KiB  
Article
Dengue Transmission Dynamics: A Fractional-Order Approach with Compartmental Modeling
by Mutum Zico Meetei, Shahbaz Zafar, Abdullah A. Zaagan, Ali M. Mahnashi and Muhammad Idrees
Fractal Fract. 2024, 8(4), 207; https://doi.org/10.3390/fractalfract8040207 - 2 Apr 2024
Viewed by 792
Abstract
This work presents a quantitative analysis of the transmission dynamics of dengue using the Caputo–Fabrizio fractional-order derivative. It presents an extensive framework for modeling a dengue epidemic, including the various stages of infection and encompassing a wide range of transmission pathways. The proposed [...] Read more.
This work presents a quantitative analysis of the transmission dynamics of dengue using the Caputo–Fabrizio fractional-order derivative. It presents an extensive framework for modeling a dengue epidemic, including the various stages of infection and encompassing a wide range of transmission pathways. The proposed model is subjected to a rigorous qualitative study, including the determination of a non-negative solution, the assessment of the basic reproduction number, and an evaluation of local stability. Numerical solutions are obtained using the Newton method. The fractional-order operator, developed using the Caputo–Fabrizio approach, provides a refined perspective on the transmission dynamics of dengue. This study contributes to a deeper understanding of the disease’s transmission mechanisms, considering both fractional-order dynamics and diverse transmission routes, thus offering insights for enhanced disease management and control. Full article
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25 pages, 370 KiB  
Article
The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions
by Renhao Ma, Yuan Meng and Huihui Pang
Fractal Fract. 2024, 8(4), 194; https://doi.org/10.3390/fractalfract8040194 - 28 Mar 2024
Viewed by 658
Abstract
This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder [...] Read more.
This paper is dedicated to studying the existence results of solutions to the nonlinear coupled system of Hilfer fractional differential equations and inclusions, with multi-strip and multi-point mixed boundary conditions. Through tools such as the Leray-Schauder alternative and the nonlinear alternative of Leray-Schauder type, continuous and measurable selection theorems, along with Leray-Schauder degree theory, the main results can be obtained. The Hilfer fractional differential system has practical implications for specific physical phenomena. Examples are provided to clarify the application of our main results. Full article
12 pages, 3552 KiB  
Article
Transient Dynamics of a Fractional Fisher Equation
by Enrique C. Gabrick, Paulo R. Protachevicz, Diogo L. M. Souza, José Trobia, Elaheh Sayari, Fernando S. Borges, Marcelo K. Lenzi, Iberê L. Caldas, Antonio M. Batista and Ervin K. Lenzi
Fractal Fract. 2024, 8(3), 143; https://doi.org/10.3390/fractalfract8030143 - 29 Feb 2024
Viewed by 970
Abstract
We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of [...] Read more.
We investigate the transient dynamics of the Fisher equation under nonlinear diffusion and fractional operators. Firstly, we investigate the effects of the nonlinear diffusivity parameter in the integer-order Fisher equation, by considering a Gaussian distribution as the initial condition. Measuring the spread of the Gaussian distribution by u(0,t)2, our results show that the solution reaches a steady state governed by the parameters present in the logistic function in Fisher’s equation. The initial transient is an anomalous diffusion process, but a power law cannot describe the whole transient. In this sense, the main novelty of this work is to show that a q-exponential function gives a better description of the transient dynamics. In addition to this result, we extend the Fisher equation via non-integer operators. As a fractional definition, we employ the Caputo fractional derivative and use a discretized system for the numerical approach according to finite difference schemes. We consider the numerical solutions in three scenarios: fractional differential operators acting in time, space, and in both variables. Our results show that the time to reach the steady solution strongly depends on the fractional order of the differential operator, with more influence by the time operator. Our main finding shows that a generalized q-exponential, present in the Tsallis formalism, describes the transient dynamics. The adjustment parameters of the q-exponential depend on the fractional order, connecting the generalized thermostatistics with the anomalous relaxation promoted by the fractional operators in time and space. Full article
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