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Recent Advances in Computational Physics with Fractional Application, 2nd Edition

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A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 12519

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Dr. Lanre Akinyemi

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Department of Mathematics, Hampton University, Hampton, VA 23669, USA
Interests: fractional differential equations and their applications; methods and applications of nonlinear equations; iteration methods for differential equations; numerical and analytical methods for differential equations; mathematical modeling of flow in porous media
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Dr. Mehmet Senol

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Department of Mathematics, Nevşehir Haci Bektaş Veli Üniversitesi, Nevsehir 50300, Turkey
Interests: numerical and analytical methods for differential equations; numerical analysis; fractional differential equations and their applications
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Dr. Solomon Manukure

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Department of Mathematics, Florida A & M University, Tallahassee, FL 32307, USA
Interests: partial differential equations; nonlinear wave theory; integrable systems; soliton theory; Hamiltonian systems; algebro-geometric solutions; Riemann–Hilbert problem
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Special Issue Information

Dear Colleagues,

This Special Issue is devoted to “Recent Advances in Computational Physics with Fractional Application”.

Fractional calculus, a generalization of integer-order differentiation and integration, has applications in diverse and widespread fields of applied sciences and engineering, such as in image processing, financial modeling, control theory for dynamical systems, disease modeling, nanotechnology, random walks, anomalous transport and anomalous diffusion, viscoelasticity, as well as many others.

Nonlinear partial differential equations are used to explain a wide range of physical phenomena that arise in applied physics, including fluid dynamics, plasma physics, solid mechanics, and quantum field theory. Many of these equations are nonlinear and are thus frequently difficult to solve explicitly. Some direct and systematic methods have been developed to study nonlinear partial differential equations, such as the extended Tanh-function method, sub-equation method, inverse scattering method, G′/G expansion method, simplest method, Painlevé analysis, Cole–Hopf transformation, inverse scattering method, the Bäcklund transformation method, Hirota bilinear method, sine-Gordon expansion method, generalized auxiliary equation method, Kudryashov method, and many more.

As a result of recent developments in fractional calculus applications, many researchers have become interested in this field. This Special Issue on “Recent Advances in Computational Physics with Fractional Application” is devoted to uncovering leading researchers’ recent work in the above fields of fractional calculus.

Dr. Lanre Akinyemi
Dr. Mehmet Senol
Dr. Solomon Manukure
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

stability analysis
integral equations
semi-analytical method
mathematical modeling
traveling wave solutions
analytical and numerical methods
soliton theory and its applications
fractional calculus and its applications
ordinary and partial differential equations
symmetry analysis and conservation laws
mathematical modeling of flow in porous media
high-order numerical differential formulas for fractional derivatives
numerical and computational methods in fractional differential equations

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Recent Advances in Computational Physics with Fractional Application in Fractal and Fractional (29 articles)

Published Papers (11 papers)

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Research

15 pages, 5823 KiB

Open AccessArticle

Analytical Scheme for Time Fractional Kawahara and Modified Kawahara Problems in Shallow Water Waves

by Muhammad Nadeem, Asad Khan, Muhammad Awais Javeed and Zhong Yubin

Fractal Fract. 2024, 8(7), 395; https://doi.org/10.3390/fractalfract8070395 - 2 Jul 2024

Viewed by 423

Abstract

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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32 pages, 2597 KiB

Open AccessArticle

Kink, Dark, Bright, and Singular Optical Solitons to the Space–Time Nonlinear Fractional (41)-Dimensional Davey–Stewartson–Kadomtsev–Petviashvili Model⁺

by Abdulaziz Khalid Alsharidi and Moin-ud-Din Junjua

Fractal Fract. 2024, 8(7), 388; https://doi.org/10.3390/fractalfract8070388 (registering DOI) - 29 Jun 2024

Viewed by 401

Abstract

The new types of exact solitons of the space–time fractional nonlinear (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili (DSKP) model are achieved by applying the unified technique and modified extended tanh-expansion function technique. A novel definition of the fractional derivative known as the truncated M-fractional derivative is also used. This model describes both the non-elastic and elastic interactions between internal waves. This model is used to represent intricate nonlinear phenomena like shallow-water waves, plasma physics, and others. The obtained results are in the form of kink, singular, bright, periodic, and dark solitons. The observed results are verified and represented by 2D and 3D graphs. The observed results are not present in the literature due to the use of fractional derivatives. The impact of the truncated M-fractional derivative on the observed results is also represented by graphs. Hence, our observed results are fruitful for the future study of these models. The applied techniques are simple, fruitful, and reliable in solving the other models in applied mathematics. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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19 pages, 5495 KiB

Open AccessArticle

Exploring Novel Soliton Solutions to the Time-Fractional Coupled Drinfel’d–Sokolov–Wilson Equation in Industrial Engineering Using Two Efficient Techniques

by Md Nur Hossain, M. Mamun Miah, Moataz Alosaimi, Faisal Alsharif and Mohammad Kanan

Fractal Fract. 2024, 8(6), 352; https://doi.org/10.3390/fractalfract8060352 - 13 Jun 2024

Viewed by 1161

Abstract

The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the

(\frac{Ω^{'}}{Ω}, \frac{1}{Ω})

method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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19 pages, 7404 KiB

Open AccessArticle

Abundant Closed-Form Soliton Solutions to the Fractional Stochastic Kraenkel–Manna–Merle System with Bifurcation, Chaotic, Sensitivity, and Modulation Instability Analysis

by J. R. M. Borhan, M. Mamun Miah, Faisal Alsharif and Mohammad Kanan

Fractal Fract. 2024, 8(6), 327; https://doi.org/10.3390/fractalfract8060327 - 29 May 2024

Viewed by 463

Abstract

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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16 pages, 3262 KiB

Open AccessArticle

Extracting the Ultimate New Soliton Solutions of Some Nonlinear Time Fractional PDEs via the Conformable Fractional Derivative

by Md Ashik Iqbal, Abdul Hamid Ganie, Md Mamun Miah and Mohamed S. Osman

Fractal Fract. 2024, 8(4), 210; https://doi.org/10.3390/fractalfract8040210 - 3 Apr 2024

Cited by 2 | Viewed by 1642

Abstract

Nonlinear fractional-order differential equations have an important role in various branches of applied science and fractional engineering. This research paper shows the practical application of three such fractional mathematical models, which are the time-fractional Klein–Gordon equation (KGE), the time-fractional Sharma–Tasso–Olever equation (STOE), and the time-fractional Clannish Random Walker’s Parabolic equation (CRWPE). These models were investigated by using an expansion method for extracting new soliton solutions. Two types of results were found: one was trigonometric and the other one was an exponential form. For a profound explanation of the physical phenomena of the studied fractional models, some results were graphed in 2D, 3D, and contour plots by imposing the distinctive results for some parameters under the oblige conditions. From the numerical investigation, it was noticed that the obtained results referred smooth kink-shaped soliton, ant-kink-shaped soliton, bright kink-shaped soliton, singular periodic solution, and multiple singular periodic solutions. The results also showed that the amplitude of the wave augmented with the pulsation in time, which derived the order of time fractional coefficient, remarkably enhanced the wave propagation, and influenced the nonlinearity impacts. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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19 pages, 1196 KiB

Open AccessArticle

Fractional Optimal Control Model and Bifurcation Analysis of Human Syncytial Respiratory Virus Transmission Dynamics

by Muath Awadalla, Jihan Alahmadi, Kumama Regassa Cheneke and Sania Qureshi

Fractal Fract. 2024, 8(1), 44; https://doi.org/10.3390/fractalfract8010044 - 11 Jan 2024

Cited by 6 | Viewed by 1402

Abstract

In this paper, the Caputo-based fractional derivative optimal control model is looked at to learn more about how the human respiratory syncytial virus (RSV) spreads. Model solution properties such as boundedness and non-negativity are checked and found to be true. The fundamental reproduction number is calculated by using the next-generation matrix’s spectral radius. The fractional optimal control model includes the control functions of vaccination and treatment to illustrate the impact of these interventions on the dynamics of virus transmission. In addition, the order of the derivative in the fractional optimal control problem indicates that encouraging vaccination and treatment early on can slow the spread of RSV. The overall analysis and the simulated behavior of the fractional optimum control model are in good agreement, and this is due in large part to the use of the MATLAB platform. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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25 pages, 6122 KiB

Open AccessArticle

Finite Difference Modeling of Time Fractal Impact on Unsteady Magneto-hydrodynamic Darcy–Forchheimer Flow in Non-Newtonian Nanofluids with the q-Derivative

by Amani S. Baazeem, Yasir Nawaz and Muhammad Shoaib Arif

Fractal Fract. 2024, 8(1), 8; https://doi.org/10.3390/fractalfract8010008 - 20 Dec 2023

Viewed by 1173

Abstract

This contribution addresses a fractal numerical scheme that can be employed for handling fractal time-dependent parabolic equations. The numerical scheme presented in this contribution can be used to discretize integer order and fractal derivatives in a given differential equation. Therefore, the scheme and results can be used for both cases. The proposed finite difference scheme is based on two stages. Fractal time derivatives are discretized by employing the proposed approach. For the scalar convection–diffusion equation, we derive the stability condition of the proposed fractal scheme. Using a nonlinear chemical reaction, the approach is also used to solve the Quantum Calculus model of a Williamson nanofluid’s unsteady Darcy–Forchheimer flow over flat and oscillatory sheets. The findings indicate a negative correlation between the velocity profile and the porosity parameter and inertia coefficient, with an increase in these factors resulting in a drop in the velocity profile. Additionally, the fractal scheme under consideration is being compared to the fractal Crank–Nicolson method, revealing that the proposed scheme exhibits a superior convergence speed compared to the fractal Crank–Nicolson method. Several problems involving the motion of non-Newtonian nanofluids through magnetic fields and porous media can be investigated with the help of the proposed numerical scheme. This research has implications for developing more efficient heat transfer and energy conversion devices based on nanofluids. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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19 pages, 3782 KiB

Open AccessArticle

Multi-Peak and Propagation Behavior of M-Shape Solitons in (2 + 1)-Dimensional Integrable Schwarz-Korteweg-de Vries Problem

by Sarfaraz Ahmed, Aly R. Seadawy, Syed T. R. Rizvi and Umar Raza

Fractal Fract. 2023, 7(10), 709; https://doi.org/10.3390/fractalfract7100709 - 26 Sep 2023

Cited by 8 | Viewed by 912

Abstract

This paper examines the propagation of M-shape solitons and their interactions with kink waves to the (2 + 1)-dimensional integrable Schwarz-Korteweg-de Vries (ISKdV) problem by applying the symbolic computation with ansatz functions technique and logarithmic transformation. The governing model usually appears in the nonlinear shallow water waves and fluid mechanics. We discuss various nonlinear waves like multiwave solutions (MSs), homoclinic breather (HB), M-shape solitons, single exponential form (one-kink), and double exponential form (two-kink). These waves have lot of applications in fluid dynamics, nonlinear optics, chemical reaction networks, biological systems, climate science, and material science. We also study interaction among M-shape solitons with kink wave. At the end, we discuss the stability characteristics of all solutions. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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11 pages, 958 KiB

Open AccessArticle

An Accurate Approach to Simulate the Fractional Delay Differential Equations

by Mohamed Adel, Mohamed M. Khader, Salman Algelany and Khaled Aldwoah

Fractal Fract. 2023, 7(9), 671; https://doi.org/10.3390/fractalfract7090671 - 5 Sep 2023

Cited by 2 | Viewed by 1227

Abstract

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. The proposed method converts FDDEs into a set of algebraic equations. We lay out a study of the convergence analysis and figure out the upper bound on error for the approximate solution. Examples are provided to demonstrate the precision of the suggested approach. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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18 pages, 2433 KiB

Open AccessArticle

The Effects of Thermal Memory on a Transient MHD Buoyancy-Driven Flow in a Rectangular Channel with Permeable Walls: A Free Convection Flow with a Fractional Thermal Flux

by Nehad Ali Shah, Bander Almutairi, Dumitru Vieru and Ahmed A. El-Deeb

Fractal Fract. 2023, 7(9), 664; https://doi.org/10.3390/fractalfract7090664 - 1 Sep 2023

Viewed by 769

Abstract

This study investigates the effects of magnetic induction, ion slip and Hall current on the flow of linear viscous fluids in a rectangular buoyant channel. In a hydro-magnetic flow scenario with permeable and conducting walls, one wall has a temperature variation that changes over time, while the other wall keeps a constant temperature; the research focuses on this situation. Asymmetric wall heating and suction/injection effects are also examined in the study. Using the Laplace transform, analytical solutions in the Laplace domain for temperature, velocity and induced magnetic field have been determined. The Stehfest approach has been used to find numerical solutions in the real domain by reversing Laplace transforms. The generalized thermal process makes use of an original fractional constitutive equation, in which the thermal flux is influenced by the history of temperature gradients, which has an impact on both the thermal process and the fluid’s hydro-magnetic behavior. The influence of thermal memory on heat transfer, fluid movement and magnetic induction was highlighted by comparing the solutions of the fractional model with the classic one based on Fourier’s law. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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17 pages, 997 KiB

Open AccessArticle

Analytical Solutions of the Predator–Prey Model with Fractional Derivative Order via Applications of Three Modified Mathematical Methods

by Mounirah Areshi, Aly R. Seadawy, Asghar Ali, Amal F. Alharbi and Abdulrahman F. Aljohani

Fractal Fract. 2023, 7(2), 128; https://doi.org/10.3390/fractalfract7020128 - 30 Jan 2023

Cited by 6 | Viewed by 1566

Abstract

We have investigated wave solutions of the Predator–Prey (PP) model with fractional derivative order by novel three modified mathematical methods with the help of the Mathematica platform. The derived solutions are in the form of distinct functions such as trigonometric, hyperbolic, exponential and rational functional. For the physical phenomena of fractional model, some solutions are plotted in 2-dimensional and 3-dimensional by inserting specific values to attached parameters under sufficient condition on each solution. Hence, proposed schemes are enormously superbly mathematical tools to review wave solutions of several fractional models in nonlinear science. Full article

(This article belongs to the Special Issue Recent Advances in Computational Physics with Fractional Application, 2nd Edition)

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