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Fractal Fract
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Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models
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Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models

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 (31 December 2023) | Viewed by 16811

Special Issue Editors

Prof. Dr. Ghazala Akram

E-Mail Website
Guest Editor
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
Interests: differential equations; computer-aided geometric design; fractional calculus; analytical and numerical techniques
Dr. Muhammad Abbas

E-Mail Website
Guest Editor
Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan
Interests: computer-aided geometric design; computer graphics; numerical methods; fractional calculus; applied mathematics
Prof. Dr. Ali Akgül

E-Mail Website
Guest Editor
1. Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon
2. Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey
3. Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, Mersin 99138, Turkey
Interests: applied mathematics; fractional calculus; numerical methods
Special Issues, Collections and Topics in MDPI journals
Dr. Maasoomah Sadaf

E-Mail Website
Guest Editor
Department of Mathematics, University of the Punjab, Lahore 54590, Pakistan
Interests: computational mathematics; differential equations; fractional evolution equations; analytical and numerical techniques

Special Issue Information

Dear Colleagues,

Nonlinearity occurs in all complex real-life phenomena. The construction and investigation of nonlinear mathematical models that arise in physics, bio-engineering, optics, fluid dynamics and other fields of science and engineering is necessary to understand the physical framework of the related real-life phenomena. Recently, fractional order models have become popular, due to their memory and hereditary properties, which classic integer order models have failed to offer. However, this requires the development and application of robust and effective techniques to solve nonlinear fractional order mathematical problems.
The aim of this Special Issue is to bring forward the development and applications of innovative mathematical techniques and novel results to provide a deeper insight into complex mechanisms that involve nonlinear fractional order evolution equations. Potential topics include, but are not limited to, the following:

  • Solutions of initial and boundary value problems using spline techniques;
  • Analytical and semi-analytical methods for solving fractional evolution models;
  • Development of nonlinear fractional order models;
  • Fractional effects of and solutions to fluid flow problems;
  • Optical solutions for fractional order optical systems;
  • Dynamical behavior of fractional order models in plasma theory;
  • Fractional evolution equations in bio-engineering;
  • Fractional effects on solitary wave dynamics;
  • Mathematical techniques to solve nonlinear fractional order systems of equations;
  • Numerical simulations for fractional order nonlinear evolution equations;
  • Approximate solution methods;
  • Fractional Bezier models;
  • Stability analysis;
  • Fractional Difference equations;
  • Fractional Integral Inequalities.

Prof. Dr. Ghazala Akram
Dr. Muhammad Abbas
Prof. Dr. Ali Akgul
Dr. Maasoomah Sadaf
Guest Editors

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

  • nonlinear fractional order models
  • fractional initial and boundary value problems
  • fractional evolution models
  • fractional operators
  • fluid flow problems
  • optical solutions
  • fractional wave equations
  • fractional dynamics
  • stability analysis
  • fractional difference problems

Published Papers (12 papers)

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Research

19 pages, 9255 KiB  
Article
An Efficient Cubic B-Spline Technique for Solving the Time Fractional Coupled Viscous Burgers Equation
by Usama Ghafoor, Muhammad Abbas, Tayyaba Akram, Emad K. El-Shewy, Mahmoud A. E. Abdelrahman and Noura F. Abdo
Fractal Fract. 2024, 8(2), 93; https://doi.org/10.3390/fractalfract8020093 - 31 Jan 2024
Viewed by 1050
Abstract
The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the [...] Read more.
The second order Burger’s equation model is used to study the turbulent fluids, suspensions, shock waves, and the propagation of shallow water waves. In the present research, we investigate a numerical solution to the time fractional coupled-Burgers equation (TFCBE) using Crank–Nicolson and the cubic B-spline (CBS) approaches. The time derivative is addressed using Caputo’s formula, while the CBS technique with the help of a θ-weighted scheme is utilized to discretize the first- and second-order spatial derivatives. The quasi-linearization technique is used to linearize the non-linear terms. The suggested scheme demonstrates unconditionally stable. Some numerical tests are utilized to evaluate the accuracy and feasibility of the current technique. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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18 pages, 920 KiB  
Article
Solutions to Fractional q-Kinetic Equations Involving Quantum Extensions of Generalized Hyper Mittag-Leffler Functions
by Mohammed Z. Alqarni, Mohamed Akel and Mohamed Abdalla
Fractal Fract. 2024, 8(1), 58; https://doi.org/10.3390/fractalfract8010058 - 16 Jan 2024
Viewed by 956
Abstract
This manuscript focuses on new generalizations of q-Mittag-Leffler functions, called generalized hyper q-Mittag-Leffler functions, and discusses their regions of convergence and various fractional q operators. Moreover, the solutions to the q-fractional kinetic equations in terms of the investigated generalized hyper [...] Read more.
This manuscript focuses on new generalizations of q-Mittag-Leffler functions, called generalized hyper q-Mittag-Leffler functions, and discusses their regions of convergence and various fractional q operators. Moreover, the solutions to the q-fractional kinetic equations in terms of the investigated generalized hyper q-Mittag-Leffler functions are obtained by applying the q-Sumudu integral transform. Furthermore, we present solutions obtained as numerical graphs using the MATLAB 2018 program. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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16 pages, 2668 KiB  
Article
A Comparative Study of Time Fractional Nonlinear Drinfeld–Sokolov–Wilson System via Modified Auxiliary Equation Method
by Ghazala Akram, Maasoomah Sadaf, Iqra Zainab, Muhammad Abbas and Ali Akgül
Fractal Fract. 2023, 7(9), 665; https://doi.org/10.3390/fractalfract7090665 - 03 Sep 2023
Cited by 5 | Viewed by 749
Abstract
The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The [...] Read more.
The time-fractional nonlinear Drinfeld–Sokolov–Wilson system, which has significance in the study of traveling waves, shallow water waves, water dispersion, and fluid mechanics, is examined in the presented work. Analytic exact solutions of the system are produced using the modified auxiliary equation method. The fractional implications on the model are examined under β-fractional derivative and a new fractional local derivative. Extracted solutions include rational, trigonometric, and hyperbolic functions with dark, periodic, and kink solitons. Additionally, by specifying values for fractional parameters, graphs are utilized to comprehend the fractional effects on the obtained solutions. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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27 pages, 3366 KiB  
Article
Mathematical Analysis of Fractal-Fractional Mathematical Model of COVID-19
by Muhammad Sinan and Nadiyah Hussain Alharthi
Fractal Fract. 2023, 7(5), 358; https://doi.org/10.3390/fractalfract7050358 - 27 Apr 2023
Cited by 2 | Viewed by 1209
Abstract
In this work, we modified a dynamical system that addresses COVID-19 infection under a fractal-fractional-order derivative. The model investigates the psychological effects of the disease on humans. We establish global and local stability results for the model under the aforementioned derivative. Additionally, we [...] Read more.
In this work, we modified a dynamical system that addresses COVID-19 infection under a fractal-fractional-order derivative. The model investigates the psychological effects of the disease on humans. We establish global and local stability results for the model under the aforementioned derivative. Additionally, we compute the fundamental reproduction number, which helps predict the transmission of the disease in the community. Using the Carlos Castillo-Chavez method, we derive some adequate results about the bifurcation analysis of the proposed model. We also investigate sensitivity analysis to the given model using the criteria of Chitnis and his co-authors. Furthermore, we formulate the characterization of optimal control strategies by utilizing Pontryagin’s maximum principle. We simulate the model for different fractal-fractional orders subject to various parameter values using Adam Bashforth’s numerical method. All numerical findings are presented graphically. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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23 pages, 735 KiB  
Article
The Novel Mittag-Leffler–Galerkin Method: Application to a Riccati Differential Equation of Fractional Order
by Lakhlifa Sadek, Ahmad Sami Bataineh, Hamad Talibi Alaoui and Ishak Hashim
Fractal Fract. 2023, 7(4), 302; https://doi.org/10.3390/fractalfract7040302 - 30 Mar 2023
Cited by 14 | Viewed by 1317
Abstract
We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by [...] Read more.
We present a new numerical approach to solving the fractional differential Riccati equations numerically. The approach—called the Mittag-Leffler–Galerkin method—comprises the finite Mittag-Leffler function and the Galerkin method. The error analysis of the method was studied. As a result, we present two theorems by which the error can be bounded. In addition to error analysis, the residual correction method, which allows us to estimate the error and obtain new approximate solutions, is also presented. To show how the method is applied, and the efficiency of the proposed method, some test examples were considered. When the numerical results obtained were examined, it was found that while the method achieves better results than some of the known methods in the literature, it also achieves results that are similar to those of others of the known methods. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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12 pages, 1050 KiB  
Article
The Stochastic Structural Modulations in Collapsing Maccari’s Model Solitons
by H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy and Mahmoud A. E. Abdelrahman
Fractal Fract. 2023, 7(4), 290; https://doi.org/10.3390/fractalfract7040290 - 28 Mar 2023
Cited by 3 | Viewed by 732
Abstract
The two-dimensional Maccari nonlinear system performs the energy and wave dynamical features in fiber communications and modern physical science as hydrodynamic and space plasma. Several new forms of solutions for the Maccari’s model are constructed by a unified solver method that mainly depends [...] Read more.
The two-dimensional Maccari nonlinear system performs the energy and wave dynamical features in fiber communications and modern physical science as hydrodynamic and space plasma. Several new forms of solutions for the Maccari’s model are constructed by a unified solver method that mainly depends on He’s variations method. The obtained solutions identify new wave stochastic structures with important features in energy physics such as rational explosive, breather, dispersive, explosive dissipated, dark solitons and blow-up (shock structure). It was elucidated that the random effects amend the energy wave strength or the collapsing due to model medium turbulence. Finally, the produced stochastic structures may be vital in some of these relationships between dispersions, nonlinearity and dissipative effects. The predominant energy waves that are collapsing or being forced may be applied to electrostatic auroral Langmuir structures and energy-generating ocean waves. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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25 pages, 1516 KiB  
Article
The Propagating Exact Solitary Waves Formation of Generalized Calogero–Bogoyavlenskii–Schiff Equation with Robust Computational Approaches
by Basem Al Alwan, Muhammad Abu Bakar, Waqas Ali Faridi, Antoniu-Claudiu Turcu, Ali Akgül and Mohammed Sallah
Fractal Fract. 2023, 7(2), 191; https://doi.org/10.3390/fractalfract7020191 - 14 Feb 2023
Cited by 15 | Viewed by 1403
Abstract
The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is [...] Read more.
The generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) is examined and analyzed in this paper. It has several applications in plasma physics and soliton theory, where it forecasts the soliton wave propagation profiles. In order to obtain the analytically exact solitons, the model under consideration is a nonlinear partial differential equation that is turned into an ordinary differential equation by using the next traveling wave transformation. The new extended direct algebraic technique and the modified auxiliary equation method are applied to the generalized Calogero–Bogoyavlenskii–Schiff equation to get new solitary wave profiles. As a result, novel and generalized analytical wave solutions are acquired in which singular solutions, mixed singular solutions, mixed complex solitary shock solutions, mixed shock singular solutions, mixed periodic solutions, mixed trigonometric solutions, mixed hyperbolic solutions, and periodic solutions are included with numerous soliton families. The propagation of the acquired soliton solution is graphically presented in contour, two- and three-dimensional visualization by selecting appropriate parametric values. It is graphically demonstrated how wave number impacts the obtained traveling wave structures. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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16 pages, 344 KiB  
Article
On Caputo Fractional Derivatives and Caputo–Fabrizio Integral Operators via (s, m)-Convex Functions
by Ammara Nosheen, Maria Tariq, Khuram Ali Khan, Nehad Ali Shah and Jae Dong Chung
Fractal Fract. 2023, 7(2), 187; https://doi.org/10.3390/fractalfract7020187 - 13 Feb 2023
Cited by 7 | Viewed by 1985
Abstract
This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are [...] Read more.
This paper contains a variety of new integral inequalities for (s,m)-convex functions using Caputo fractional derivatives and Caputo–Fabrizio integral operators. Various generalizations of Hermite–Hadamard-type inequalities containing Caputo–Fabrizio integral operators are derived for those functions whose derivatives are (s,m)-convex. Inequalities involving the digamma function and special means are deduced as applications. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
19 pages, 2171 KiB  
Article
Dynamics of Nonlinear Optics with Different Analytical Approaches
by Naeem Ullah, Muhammad Imran Asjad, Musawa Yahya Almusawa and Sayed M. Eldin
Fractal Fract. 2023, 7(2), 138; https://doi.org/10.3390/fractalfract7020138 - 02 Feb 2023
Cited by 5 | Viewed by 1103
Abstract
In this article, we investigate novel optical solitons solutions for the Lakshmanan–Porsezian–Daniel (LPD) equation, along with group velocity dispersion and spatio-temporal dispersion, via three altered analytical techniques. A variety of bright, singular, dark, periodic singular, and kink solitons solutions are constructed via the [...] Read more.
In this article, we investigate novel optical solitons solutions for the Lakshmanan–Porsezian–Daniel (LPD) equation, along with group velocity dispersion and spatio-temporal dispersion, via three altered analytical techniques. A variety of bright, singular, dark, periodic singular, and kink solitons solutions are constructed via the Kudryashov method, the generalized tanh method and the Sardar-subequation method. The dynamical behavior of the extracted solutions is demonstrated in graphical form such as 3D plots, 2D plots, and contour plots. The originality of the obtained solutions is recognized by comparison with each other and solutions previously stated in the literature for the LPD model, which displays the efficiency of the methods under consideration. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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20 pages, 3999 KiB  
Article
The Sensitive Visualization and Generalized Fractional Solitons’ Construction for Regularized Long-Wave Governing Model
by Riaz Ur Rahman, Waqas Ali Faridi, Magda Abd El-Rahman, Aigul Taishiyeva, Ratbay Myrzakulov and Emad Ahmad Az-Zo’bi
Fractal Fract. 2023, 7(2), 136; https://doi.org/10.3390/fractalfract7020136 - 01 Feb 2023
Cited by 23 | Viewed by 1493
Abstract
The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite [...] Read more.
The solution of partial differential equations has generally been one of the most-vital mathematical tools for describing physical phenomena in the different scientific disciplines. The previous studies performed with the classical derivative on this model cannot express the propagating behavior at heavy infinite tails. In order to address this problem, this study addressed the fractional regularized long-wave Burgers problem by using two different fractional operators, Beta and M-truncated, which are capable of predicting the behavior where the classical derivative is unable to show dynamical characteristics. This fractional equation is first transformed into an ordinary differential equation using the fractional traveling wave transformation. A new auxiliary equation approach was employed in order to discover new soliton solutions. As a result, bright, periodic, singular, mixed periodic, rational, combined dark–bright, and dark soliton solutions were found based on the constraint relation imposed on the auxiliary equation parameters. The graphical visualization of the obtained results is displayed by taking the suitable parametric values and predicting that the fractional order parameter is responsible for controlling the behavior of propagating solitary waves and also providing the comparison between fractional operators and the classical derivative. We are confident about the vital applications of this study in many scientific fields. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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23 pages, 5525 KiB  
Article
Explicit Soliton Structure Formation for the Riemann Wave Equation and a Sensitive Demonstration
by Sheikh Zain Majid, Waqas Ali Faridi, Muhammad Imran Asjad, Magda Abd El-Rahman and Sayed M. Eldin
Fractal Fract. 2023, 7(2), 102; https://doi.org/10.3390/fractalfract7020102 - 17 Jan 2023
Cited by 35 | Viewed by 1581
Abstract
The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using [...] Read more.
The motive of the study was to explore the nonlinear Riemann wave equation, which describes the tsunami and tidal waves in the sea and homogeneous and stationary media. This study establishes the framework for the analytical solutions to the Riemann wave equation using the new extended direct algebraic method. As a result, the soliton patterns of the Riemann wave equation have been successfully illustrated, with exact solutions offered by the plane solution, trigonometry solution, mixed hyperbolic solution, mixed periodic and periodic solutions, shock solution, mixed singular solution, mixed trigonometric solution, mixed shock single solution, complex soliton shock solution, singular solution, and shock wave solutions. Graphical visualization is provided of the results with suitable values of the involved parameters by Mathematica. It was visualized that the velocity of the soliton and the wave number controls the behavior of the soliton. We are confident that our research will assist physicists in predicting new notions in mathematical physics. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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23 pages, 3270 KiB  
Article
New Explicit Propagating Solitary Waves Formation and Sensitive Visualization of the Dynamical System
by Rana Muhammad Zulqarnain, Wen-Xiu Ma, Sayed M. Eldin, Khush Bukht Mehdi and Waqas Ali Faridi
Fractal Fract. 2023, 7(1), 71; https://doi.org/10.3390/fractalfract7010071 - 09 Jan 2023
Cited by 6 | Viewed by 1234
Abstract
This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities [...] Read more.
This work discusses the soliton solutions for the fractional complex Ginzburg–Landau equation in Kerr law media. It is a particularly fascinating model in this context as it is a dissipative variant of the Hamiltonian nonlinear Schrödinger equation with solutions that create localized singularities in finite time. The ϕ6-model technique is one of the generalized methodologies exerted on the fractional complex Ginzburg–Landau equation to find the new solitary wave profiles. As a result, solitonic wave patterns develop, including Jacobi elliptic function, periodic, dark, bright, single, dark-bright, exponential, trigonometric, and rational solitonic structures, among others. The assurance of the practicality of the solitary wave results is provided by the constraint condition corresponding to each achieved solution. The graphical 3D and contour depiction of the attained outcomes is shown to define the pulse propagation behaviors while imagining the pertinent data for the involved parameters. The sensitive analysis predicts the dependence of the considered model on initial conditions. It is a reliable and efficient technique used to generate generalized solitonic wave profiles with diverse soliton families. Furthermore, we ensure that all results are innovative and mark remarkable impacts on the prevailing solitary wave theory literature. Full article
(This article belongs to the Special Issue Numerical Simulations and Advanced Techniques for Nonlinear Fractional Evolution Models)
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