Fractal and Fractional

Research

19 pages, 3626 KiB

Open AccessArticle

Exact Fractional Solution by Nucci’s Reduction Approach and New Analytical Propagating Optical Soliton Structures in Fiber-Optics

by Waqas Ali Faridi, Muhammad Imran Asjad and Sayed M. Eldin

Fractal Fract. 2022, 6(11), 654; https://doi.org/10.3390/fractalfract6110654 - 5 Nov 2022

Cited by 17 | Viewed by 1525

Abstract

This study examines the Chen–Lee–Liu dynamical equation, which represents the propagation of optical pulses in optical fibers and plasma. A new extended direct algebraic technique and Nucci’s scheme are used to find new solitary wave profiles. The method covers thirty-seven solitonic wave profiles, [...] Read more.

This study examines the Chen–Lee–Liu dynamical equation, which represents the propagation of optical pulses in optical fibers and plasma. A new extended direct algebraic technique and Nucci’s scheme are used to find new solitary wave profiles. The method covers thirty-seven solitonic wave profiles, including approximately all soliton families, in an efficient and generic manner. New solitonic wave patterns are obtained, including a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, trigonometric solution, shock solution, mixed shock singular solution, mixed singular solution, complex solitary shock solution, singular solution and shock wave solutions. The exact fractional solution is obtained using Nucci’s reduction approach. The impact of the fractional order parameter on the solution is considered using both mathematical expressions and graphical visualization. The fractional order parameter is responsible for controlling the singularity of the solution which is graphically displayed. A sensitivity analysis was used to predict the sensitivity of equations with respect to initial conditions. To demonstrate the pulse propagation characteristics, while taking suitable values for the parameters involved, 2-D, 3-D, and contour graphics of the outcomes achieved are presented. The influence of the fractional order

ζ

is shown graphically. A periodic-singular wave with lower amplitude and dark-singular behaviour is inferred from the graphical behaviour of the trigonometric function solution

H_{1}

and the rational function solution

H_{34}

from the obtained solutions, respectively. Full article

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

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10 pages, 605 KiB

Open AccessArticle

Exact Traveling Wave Solutions of the Local Fractional Bidirectional Propagation System Equations

by Xue Sang, Zongguo Zhang, Hongwei Yang and Xiaofeng Han

Fractal Fract. 2022, 6(11), 653; https://doi.org/10.3390/fractalfract6110653 - 5 Nov 2022

Cited by 3 | Viewed by 1253

Abstract

In this paper, within the scope of the local fractional derivative theory, bidirectional propagation system local fractional equations are researched. Compared with the unidirectional propagation of nonlinear waves in a pipeline, the bidirectional propagation system equations studied in this paper can better describe [...] Read more.

In this paper, within the scope of the local fractional derivative theory, bidirectional propagation system local fractional equations are researched. Compared with the unidirectional propagation of nonlinear waves in a pipeline, the bidirectional propagation system equations studied in this paper can better describe the propagation of nonlinear waves in a channel. This study is groundbreaking and offers more possibilities for the bidirectional propagation of nonlinear waves in the simulation pipeline. The exact traveling wave solutions of the non-differentiable type defined on the Cantor sets are obtained. The characteristics of the particular solutions of a fixed fractal dimension are discussed. It is proven that the local fractional nonlinear bidirectional wave equations can describe the interaction of fractal waves. It is also shown that the study of traveling wave solutions of nonlinear local fractional equations has important significance in mathematical physics. Full article

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

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15 pages, 478 KiB

Open AccessArticle

Fractals Flow Simulation for Groundwater Flow with Varying Apertures by Using Analytic Element Method

by Maryam Atta, Sardar Muhammad Hussain, Farooq Hussain, Hasrat Hussain Shah, Hassan Shah and Jong-Suk Ro

Fractal Fract. 2022, 6(10), 573; https://doi.org/10.3390/fractalfract6100573 - 9 Oct 2022

Cited by 3 | Viewed by 1394

Abstract

The work presented in this article is composed of 2-dimensional groundwater flow simulations for fractured porous media with different aperture of fractures by using the Analytic Element Method. In order to investigate the flow behavior and its effect on fractures, we considered different [...] Read more.

The work presented in this article is composed of 2-dimensional groundwater flow simulations for fractured porous media with different aperture of fractures by using the Analytic Element Method. In order to investigate the flow behavior and its effect on fractures, we considered different systems of fractures with varying apertures, hydraulic conductivities and orientations in the presence of uniform flow field and a well. We also introduced the matrix method to solve the problems for which the unknown coefficients are obtained from the discharge potential of all the elements present in the systems. The numerical solution of the prescribed problem is based on a series expansion, while the influence of each fracture is expressed in a series that satisfy Laplace’s equation. Full article

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

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21 pages, 1706 KiB

Open AccessArticle

Fractional View Analysis of Swift–Hohenberg Equations by an Analytical Method and Some Physical Applications

by Salemah A. Almutlak, Rasool Shah, Wajaree Weera, Samir A. El-Tantawy and Lamiaa S. El-Sherif

Fractal Fract. 2022, 6(9), 524; https://doi.org/10.3390/fractalfract6090524 - 16 Sep 2022

Cited by 1 | Viewed by 1656

Abstract

This study investigates the fractional-order Swift–Hohenberg equations using the natural decomposition method with non-singular kernel derivatives. The fractional derivative in the sense of Caputo–Fabrizio is considered. The Adomian decomposition technique (ADT) is a great deal to the overall natural transformation to create closed-form [...] Read more.

This study investigates the fractional-order Swift–Hohenberg equations using the natural decomposition method with non-singular kernel derivatives. The fractional derivative in the sense of Caputo–Fabrizio is considered. The Adomian decomposition technique (ADT) is a great deal to the overall natural transformation to create closed-form results of the given models. This technique provides a closed-form result for the suggested models. In addition, this technique is attractive, simple, and preferred over other techniques. The graphs of the solution in fractional and integer-order show that the achieved solutions are very close to the actual result of the examples. It is also investigated that the result of fractional-order models converges to the integer-order model’s solution. Furthermore, the proposed method validity is examined using numerical examples. The obtained results for the given problems fully support the theory of the proposed method. The present method is a straightforward and accurate analytical method to analyze other fractional-order partial differential equations, such as many evolution equations that govern the dynamics of nonlinear waves in plasma physics. Full article

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

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11 pages, 4631 KiB

Open AccessArticle

A New L²-Gradient Flow-Based Fractional-in-Space Modified Phase-Field Crystal Equation and Its Mass Conservative and Energy Stable Method

by Hyun Geun Lee

Fractal Fract. 2022, 6(9), 472; https://doi.org/10.3390/fractalfract6090472 - 27 Aug 2022

Cited by 3 | Viewed by 1488

Abstract

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the

L^{2}

-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the

L^{2}

-gradient flow-based fractional-in-space modified phase-field [...] Read more.

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the

L^{2}

-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the

L^{2}

-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical

H^{- 1}

-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework. Full article

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

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11 pages, 3057 KiB

Open AccessArticle

Exact Solutions of the Nonlinear Modified Benjamin-Bona-Mahony Equation by an Analytical Method

by Trad Alotaibi and Ali Althobaiti

Fractal Fract. 2022, 6(7), 399; https://doi.org/10.3390/fractalfract6070399 - 20 Jul 2022

Cited by 8 | Viewed by 1646

Abstract

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of [...] Read more.

The current manuscript investigates the exact solutions of the modified Benjamin-Bona-Mahony (BBM) equation. Due to its efficiency and simplicity, the modified auxiliary equation method is adopted to solve the problem under consideration. As a result, a variety of the exact wave solutions of the modified BBM equation are obtained. Furthermore, the findings of the current study remain strong since Jacobi function solutions generate hyperbolic function solutions and trigonometric function solutions, as liming cases of interest. Some of the obtained solutions are illustrated graphically using appropriate values for the parameters. Full article

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

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22 pages, 7116 KiB

Open AccessArticle

Effectiveness of Radiation on Magneto-Combined Convective Boundary Layer Flow in Polar Nanofluid around a Spherical Shape

by Mohammed Z. Swalmeh, Feras Shatat, Firas A. Alwawi, Mohd Asrul Hery Ibrahim, Ibrahim Mohammed Sulaiman, Nusayba Yaseen and Mohammad F. M. Naser

Fractal Fract. 2022, 6(7), 383; https://doi.org/10.3390/fractalfract6070383 - 6 Jul 2022

Cited by 13 | Viewed by 1779

Abstract

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including polymeric solutions, liquid [...] Read more.

Many physical aspects emerging from the local structure and micromotions of liquid particles can be studied by utilizing the governing model of micropolar liquid. It has the ability to explain the behavior of a wide range of real fluids, including polymeric solutions, liquid crystals, lubricants, and animal blood. This earned it a major role in the treatment of many industrial and engineering applications. Radiative heat transmission induced by a combined convection flow of micropolar fluid over a solid sphere, and its enhancement via nanoparticle oxides, are investigated in this study. An applied magnetic field and a constant wall temperature are also considered. The Tiwari–Das model is used to construct the mathematical model. An approximate numerical solution is included using the Keller box method, in which its numerical calculations are performed via MATLAB software, to obtain numerical results and graphic outputs reflecting the effects of critical parameters on the physical quantities associated with heat transfer. The investigation results point out that a weakness in the intensity of the magnetic field, or an increment in the nanoparticle volume fraction, causes an increment in velocity. Raising the radiation parameter promotes energy transport, angular velocity, and velocity. Full article

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

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19 pages, 1626 KiB

Open AccessArticle

Modeling and Numerical Simulation for Covering the Fractional COVID-19 Model Using Spectral Collocation-Optimization Algorithm

by Mohamed M. Khader and Mohamed Adel

Fractal Fract. 2022, 6(7), 363; https://doi.org/10.3390/fractalfract6070363 - 29 Jun 2022

Cited by 12 | Viewed by 1552

Abstract

A primary aim of this study is to examine and simulate a fractional Coronavirus disease model by providing an efficient method for solving numerically this important model. In the Liouville-Caputo sense, the examined model consists of five fractional-order differential equations. With the Vieta-Lucas [...] Read more.

A primary aim of this study is to examine and simulate a fractional Coronavirus disease model by providing an efficient method for solving numerically this important model. In the Liouville-Caputo sense, the examined model consists of five fractional-order differential equations. With the Vieta-Lucas spectral collocation method, the unknown functions can be discretized and fractional derivatives can be obtained. With the system of nonlinear algebraic equations obtained, we can simplify the examined problem. In this system, the unknown coefficients are discovered by constructing and solving it as a restricted optimization problem. Some theoretical investigations are stated to examine the convergence analysis and stability analysis of the proposed approach and model. The results produced using the fractional finite difference technique (FDM), where the fractional differentiation operator was discretized using the Grünwald-Letnikov approach, are compared. The FDM relies heavily upon accurately turning the proposed model into a system of algebraic equations. To assess the algorithm’s correctness and usefulness, a numerical simulation is included. Full article

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

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22 pages, 27412 KiB

Open AccessArticle

A Study of the Soliton Solutions with an Intrinsic Fractional Discrete Nonlinear Electrical Transmission Line

by Hassan Almusawa and Adil Jhangeer

Fractal Fract. 2022, 6(6), 334; https://doi.org/10.3390/fractalfract6060334 - 16 Jun 2022

Cited by 24 | Viewed by 1713

Abstract

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct [...] Read more.

This study aims to identify soliton structures as an inherent fractional discrete nonlinear electrical transmission lattice. Here, the analysis is founded on the idea that the electrical properties of a capacitor typically contain a non-integer-order time derivative in a realistic system. We construct a non-integer order nonlinear partial differential equation of such voltage dynamics using Kirchhoff’s principles for the model under study. It was discovered that the behavior for newly generated soliton solutions is impacted by both the non-integer-order time derivative and connected parameters. Regardless of structure, the fractional-order alters the propagation velocity of such a voltage wave, thus bringing up a localized framework under low coupling coefficient values. The generalized auxiliary equation method drove us to these solitary structures while employing the modified Riemann–Liouville derivatives and the non-integer order complex transform. As well as addressing sensitivity testing, we also investigate how our model’s altered dynamical framework shows quasi-periodic properties. Some randomly selected solutions are shown graphically for physical interpretation, and conclusions are held at the end. Full article

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

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12 pages, 391 KiB

Open AccessArticle

Fractional View Analysis of Cahn–Allen Equations by New Iterative Transform Method

by Liaqat Ali, Rasool Shah and Wajaree Weera

Fractal Fract. 2022, 6(6), 293; https://doi.org/10.3390/fractalfract6060293 - 27 May 2022

Cited by 4 | Viewed by 1467

Abstract

In this article, the new iterative transform method is applied to evaluate the time-fractional Cahn–Allen model solution. In this technique, Elzaki transformation is a mixture of the new iteration technique. Two problems are studied to demonstrate and confirm the accuracy of the proposed [...] Read more.

In this article, the new iterative transform method is applied to evaluate the time-fractional Cahn–Allen model solution. In this technique, Elzaki transformation is a mixture of the new iteration technique. Two problems are studied to demonstrate and confirm the accuracy of the proposed technique. The current technique’s mathematical analysis showed that the method is simple to understand and reliable. These solutions indicate that the proposed technique is advantageous and simple to apply in science and engineering problems. Full article

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

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17 pages, 5589 KiB

Open AccessArticle

Fractional Variation Network for THz Spectrum Denoising without Clean Data

by Qingliang Jiao, Jing Xu, Ming Liu, Fengfeng Zhao, Liquan Dong, Mei Hui, Lingqin Kong and Yuejin Zhao

Fractal Fract. 2022, 6(5), 246; https://doi.org/10.3390/fractalfract6050246 - 29 Apr 2022

Cited by 3 | Viewed by 2079

Abstract

Deep learning can remove the noise of the terahertz (THz) spectrum via its powerful feature extraction ability. However, this technology suffers from several limitations, including clean training data being difficult to obtain, the amount of training data being small, and the restored effect [...] Read more.

Deep learning can remove the noise of the terahertz (THz) spectrum via its powerful feature extraction ability. However, this technology suffers from several limitations, including clean training data being difficult to obtain, the amount of training data being small, and the restored effect being unsatisfactory. In this paper, a novel THz spectrum denoising method is proposed. Low-quality underwater images and transfer learning are used to alleviate the limitation of the training data amount. Then, the principle of Noise2Noise is applied to further reduce the limitations of clean training data. Moreover, a THz denoising network based on Transformer is proposed, and fractional variation is introduced in the loss function to improve the denoising effect. Experimental results demonstrate that the proposed method estimates the high-quality THz spectrum in simulation and measured data experiments, and it also has a satisfactory result in THz imaging. Full article

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

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13 pages, 11440 KiB

Open AccessArticle

Novel Patterns in Fractional-in-Space Nonlinear Coupled FitzHugh–Nagumo Models with Riesz Fractional Derivative

by Xiaoyu Li, Che Han and Yulan Wang

Fractal Fract. 2022, 6(3), 136; https://doi.org/10.3390/fractalfract6030136 - 28 Feb 2022

Cited by 19 | Viewed by 2449

Abstract

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent [...] Read more.

In this paper, the Fourier spectral method is used to solve the fractional-in-space nonlinear coupled FitzHugh–Nagumo model.Numerical simulation is carried out to elucidate the diffusion behavior of patterns for the fractional 2D and 3D FitzHugh–Nagumo model. The results of numerical experiments are consistent with the theoretical results of other scholars, which verifies the accuracy of the method. We show that stable spatio-temporal patterns can be sustained for a long time; these patterns are different from any previously obtained in numerical studies. Here, we show that behavior patterns can be described well by the fractional FitzHugh–Nagumo and Gray–Scott models, which have unique properties that integer models do not have. Results show that the Fourier spectral method has strong competitiveness, reliability, and solving ability for solving 2D and 3D fractional-in-space nonlinear reaction-diffusion models. Full article

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

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22 pages, 2273 KiB

Open AccessArticle

Analysis of Lie Symmetries with Conservation Laws and Solutions of Generalized (4 + 1)-Dimensional Time-Fractional Fokas Equation

by Zhuo Jiang, Zong-Guo Zhang, Jing-Jing Li and Hong-Wei Yang

Fractal Fract. 2022, 6(2), 108; https://doi.org/10.3390/fractalfract6020108 - 13 Feb 2022

Cited by 11 | Viewed by 1986

Abstract

High-dimensional fractional equations research is a cutting-edge field with significant practical and theoretical implications in mathematics, physics, biological fluid mechanics, and other fields. Firstly, in this paper, the (4 + 1)-dimensional time-fractional Fokas equation in a higher-dimensional integrable system is studied by using [...] Read more.

High-dimensional fractional equations research is a cutting-edge field with significant practical and theoretical implications in mathematics, physics, biological fluid mechanics, and other fields. Firstly, in this paper, the (4 + 1)-dimensional time-fractional Fokas equation in a higher-dimensional integrable system is studied by using semi-inverse and fractional variational theory. Then, the Lie symmetry analysis and conservation law analysis are carried out for the higher dimensional fractional order model with the symmetry of fractional order. Finally, the fractional-order equation is solved using the bilinear approach to produce the rogue wave and multi-soliton solutions, and the fractional equation is numerically solved using the Radial Basis Functions (RBFs) method. Full article

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

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24 pages, 4750 KiB

Open AccessArticle

Impact of Non-Uniform Periodic Magnetic Field on Unsteady Natural Convection Flow of Nanofluids in Square Enclosure

by Tarikul Islam, Mehmet Yavuz, Nazma Parveen and Md. Fayz-Al-Asad

Fractal Fract. 2022, 6(2), 101; https://doi.org/10.3390/fractalfract6020101 - 11 Feb 2022

Cited by 25 | Viewed by 2603

Abstract

In this article, unsteady free convective heat transport of copper-water nanofluid within a square-shaped enclosure with the dominance of non-uniform horizontal periodic magnetic effect is investigated numerically. Various nanofluids are also used to investigate temperature performance. The Brownian movement of nano-sized particles is [...] Read more.

In this article, unsteady free convective heat transport of copper-water nanofluid within a square-shaped enclosure with the dominance of non-uniform horizontal periodic magnetic effect is investigated numerically. Various nanofluids are also used to investigate temperature performance. The Brownian movement of nano-sized particles is included in the present model. A sinusoidal function of the y coordinate is considered for the magnetic effect, which works as a non-uniform magnetic field. The left sidewall is warmed at a higher heat, whereas the right sidewall is cooled at a lower heat. The upper and bottom walls are insulated. For solving the governing non-linear partial differential equation, Galerkin weighted residual finite element method is devoted. Comparisons are made with previously published articles, and we found there to be excellent compliance. The influence of various physical parameters, namely, the volume fraction of nanoparticles, period of the non-uniform magnetic field, Rayleigh number, the shape and diameter of nanoparticles, and Hartmann number on the temperature transport and fluid flow are researched. The local and average Nusselt number is also calculated to investigate the impact of different parameters on the flow field. The results show the best performance of heat transport for the Fe₃O₄-water nanofluid than for other types of nanofluids. The heat transport rate increases 20.14% for Fe₃O₄-water nanofluid and 8.94% for TiO₂-water nanofluid with 1% nanoparticles volume. The heat transportation rate enhances with additional nanoparticles into the base fluid whereas it decreases with the increase of Hartmann number and diameter of particles. A comparison study of uniform and non-uniform magnetic effects is performed, and a higher heat transfer rate is observed for a non-uniform magnetic effect compared to a uniform magnetic effect. Moreover, periods of magnetic effect and a nanoparticle’s Brownian movement significantly impacts the temperature transport and fluid flow. The solution reaches unsteady state to steady state within a very short time. Full article

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

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14 pages, 1002 KiB

Open AccessArticle

Analytical Study on Sodium Alginate Based Hybrid Nanofluid Flow through a Shrinking/Stretching Sheet with Radiation, Heat Source and Inclined Lorentz Force Effects

by P. Hammachukiattikul, M. Govindaraju, Muhammad Sohail, R. Vadivel, Nallappan Gunasekaran and Sameh Askar

Fractal Fract. 2022, 6(2), 68; https://doi.org/10.3390/fractalfract6020068 - 27 Jan 2022

Cited by 14 | Viewed by 2417

Abstract

This study investigated the flow and heat transfer of sodium alginate-based hybrid nanofluids with a stretching/shrinking surface. The heat source/sink, Joule heating, inclined magnetic field, and thermal radiation influences are also examined in the designed model. The mixers of non-magnetic and magnetic nanoparticles [...] Read more.

This study investigated the flow and heat transfer of sodium alginate-based hybrid nanofluids with a stretching/shrinking surface. The heat source/sink, Joule heating, inclined magnetic field, and thermal radiation influences are also examined in the designed model. The mixers of non-magnetic and magnetic nanoparticles are utilized, such as

C u

and

F e_{3} O_{4}

. The Casson fluid model is applied to determine the viscoplastic characteristics of sodium alginate (SA). The necessary governing SA-based hybrid nanofluid flow equations are solved analytically by hypergeometric function. SA-based hybrid nanofluid velocity, temperature, skin friction, and Nusselt number results are discussed in detail with various pertinent parameters, such as radiation, heat source/sink, inclined angle, magnetic field, Eckert number, and Casson parameters. It is noted that the dimensions of both

C u

and

F e_{3} O_{4}

hybrid nanoparticles and Casson parameters are minimized by the momentum surface layer thickness. The magnetic field, radiation, heat source and Casson parameters serve to enhance the thermal boundary layer thickness. Finally, the current result was verified with previously published works. Full article

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

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19 pages, 1405 KiB

Open AccessArticle

Fractional Order Modeling the Gemini Virus in Capsicum annuum with Optimal Control

by Kottakkaran Sooppy Nisar, Kumararaju Logeswari, Veliappan Vijayaraj, Haci Mehmet Baskonus and Chokkalingam Ravichandran

Fractal Fract. 2022, 6(2), 61; https://doi.org/10.3390/fractalfract6020061 - 25 Jan 2022

Cited by 66 | Viewed by 3338

Abstract

In this article, a fractional model of the Capsicum annuum (C. annuum) affected by the yellow virus through whiteflies (Bemisia tabaci) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal [...] Read more.

In this article, a fractional model of the Capsicum annuum (C. annuum) affected by the yellow virus through whiteflies (Bemisia tabaci) is examined. We analyzed the model by equilibrium points, reproductive number, and local and global stability. The optimal control methods are discussed to decrease the infectious B. tabaci and C. annuum by applying the Verticillium lecanii (V. lecanii) with the Atangana–Baleanu derivative. Numerical results described the population of plants and comparison values of using V. lecanni. The results show that using 60% of V. lecanni will control the spread of the yellow virus in infected B. tabaci and C. annuum in 10 days, which helps farmers to afford the costs of cultivating chili plants. Full article

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

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15 pages, 2250 KiB

Open AccessArticle

Effectiveness of Newtonian Heating on Magneto-Free Convective Flow of Polar Nanofluid across a Solid Sphere

by Hossam A. Nabwey, Ahmed M. Rashad, Amal M. A. EL-Hakiem and Sumayyah I. Alshber

Fractal Fract. 2022, 6(2), 57; https://doi.org/10.3390/fractalfract6020057 - 23 Jan 2022

Cited by 2 | Viewed by 2060

Abstract

This paper explains the free convective flowing of micropolar nanofluid through a solid sphere with Newtonian heating and the magnetic field influence. Sets of partial differential equations are converted by using convenient transformations to ordinary differential equations. The system of similar and nonsimilar [...] Read more.

This paper explains the free convective flowing of micropolar nanofluid through a solid sphere with Newtonian heating and the magnetic field influence. Sets of partial differential equations are converted by using convenient transformations to ordinary differential equations. The system of similar and nonsimilar equations is solved numerically using the Runge–Kutta–Fehlberg method (RKF45) using MAPLE software (version 20).The numerical results are validated by comparison with previously published works, and excellent agreement is found between them. The influence of the magnetic field parameter, solid volume fraction, and micropolar parameter on velocity, temperature, and angular velocity profiles are shown graphically. In addition, both the skin friction coefficient and Nusselt number are also discussed. It is found that the skin friction increases with an increase in the solid volume fraction of both nanoparticles and Newtonian heating and micropolar parameters. In addition, the magnetic field reduces both the skin friction and the Nusselt number. Moreover, the solid volume fraction and Newtonian heating parameter enhance the Nusselt number. Full article

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

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12 pages, 1793 KiB

Open AccessArticle

Study on Date–Jimbo–Kashiwara–Miwa Equation with Conformable Derivative Dependent on Time Parameter to Find the Exact Dynamic Wave Solutions

by Md Ashik Iqbal, Ye Wang, Md Mamun Miah and Mohamed S. Osman

Fractal Fract. 2022, 6(1), 4; https://doi.org/10.3390/fractalfract6010004 - 23 Dec 2021

Cited by 181 | Viewed by 4150

Abstract

In this article, we construct the exact dynamical wave solutions to the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative by using an efficient and well-established approach, namely: the two-variable

(G ’ / G, 1 / G)

-expansion method. The solutions of the [...] Read more.

In this article, we construct the exact dynamical wave solutions to the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative by using an efficient and well-established approach, namely: the two-variable

(G ’ / G, 1 / G)

-expansion method. The solutions of the Date–Jimbo–Kashiwara–Miwa equation with conformable derivative play a vital role in many scientific occurrences. The regular dynamical wave solutions of the abovementioned equation imply three different fundamental functions, which are the trigonometric function, the hyperbolic function, and the rational function. These solutions are classified graphically into three categories, such as singular periodic solitary, kink soliton, and anti-kink soliton wave solutions. Furthermore, the effect of the fractional parameter on these solutions is discussed through 2D plots. Full article

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

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23 pages, 675 KiB

Open AccessArticle

Designing a Matrix Collocation Method for Fractional Delay Integro-Differential Equations with Weakly Singular Kernels Based on Vieta–Fibonacci Polynomials

by Khadijeh Sadri, Kamyar Hosseini, Dumitru Baleanu, Soheil Salahshour and Choonkil Park

Fractal Fract. 2022, 6(1), 2; https://doi.org/10.3390/fractalfract6010002 - 22 Dec 2021

Cited by 11 | Viewed by 3072

Abstract

In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientific fields, such as astrophysics, economy, control, biology, and electro-dynamics. The emerged [...] Read more.

In the present work, the numerical solution of fractional delay integro-differential equations (FDIDEs) with weakly singular kernels is addressed by designing a Vieta–Fibonacci collocation method. These equations play immense roles in scientific fields, such as astrophysics, economy, control, biology, and electro-dynamics. The emerged fractional derivative is in the Caputo sense. By resultant operational matrices related to the Vieta–Fibonacci polynomials (VFPs) for the first time accompanied by the collocation method, the problem taken into consideration is converted into a system of algebraic equations, the solving of which leads to an approximate solution to the main problem. The existence and uniqueness of the solution of this category of fractional delay singular integro-differential equations (FDSIDEs) are investigated and proved using Krasnoselskii’s fixed-point theorem. A new formula for extracting the VFPs and their derivatives is given, and the orthogonality of the derivatives of VFPs is easily proved via it. An error bound of the residual function is estimated in a Vieta–Fibonacci-weighted Sobolev space, which shows that by properly choosing the number of terms of the series solution, the approximation error tends to zero. Ultimately, the designed algorithm is examined on four FDIDEs, whose results display the simple implementation and accuracy of the proposed scheme, compared to ones obtained from previous methods. Furthermore, the orthogonality of the VFPs leads to having sparse operational matrices, which makes the execution of the presented method easy. Full article

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

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26 pages, 17550 KiB

Open AccessArticle

Dynamics of Fractional Model of Biological Pest Control in Tea Plants with Beddington–DeAngelis Functional Response

by Sindhu J. Achar, Chandrali Baishya, Pundikala Veeresha and Lanre Akinyemi

Fractal Fract. 2022, 6(1), 1; https://doi.org/10.3390/fractalfract6010001 - 21 Dec 2021

Cited by 36 | Viewed by 3232

Abstract

In this study, we depicted the spread of pests in tea plants and their control by biological enemies in the frame of a fractional-order model, and its dynamics are surveyed in terms of boundedness, uniqueness, and the existence of the solutions. To reduce [...] Read more.

In this study, we depicted the spread of pests in tea plants and their control by biological enemies in the frame of a fractional-order model, and its dynamics are surveyed in terms of boundedness, uniqueness, and the existence of the solutions. To reduce the harm to the tea plant, a harvesting term is introduced into the equation that estimates the growth of tea leaves. We analyzed various points of equilibrium of the projected model and derived the conditions for the stability of these equilibrium points. The complex nature is examined by changing the values of various parameters and fractional derivatives. Numerical computations are conducted to strengthen the theoretical findings. Full article

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

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23 pages, 9260 KiB

Open AccessArticle

Dual Penta-Compound Combination Anti-Synchronization with Analysis and Application to a Novel Fractional Chaotic System

by Lone Seth Jahanzaib, Pushali Trikha, Rajaa T. Matoog, Shabbir Muhammad, Ahmed Al-Ghamdi and Mahmoud Higazy

Fractal Fract. 2021, 5(4), 264; https://doi.org/10.3390/fractalfract5040264 - 7 Dec 2021

Cited by 5 | Viewed by 2321

Abstract

This paper studies a fractional-order chaotic system with sine non-linearities and highlights its dynamics using the Lyapunov spectrum, bifurcation analysis, stagnation points, the solution of the system, the impact of the fractional order on the system, etc. The system considering uncertainties and disturbances [...] Read more.

This paper studies a fractional-order chaotic system with sine non-linearities and highlights its dynamics using the Lyapunov spectrum, bifurcation analysis, stagnation points, the solution of the system, the impact of the fractional order on the system, etc. The system considering uncertainties and disturbances was synchronized using dual penta-compound combination anti-synchronization among four master systems and twenty slave systems by non-linear control and the adaptive sliding mode technique. The estimates of the disturbances and uncertainties were also obtained using the sliding mode technique. The application of the achieved synchronization in secure communication is illustrated with the help of an example. Full article

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

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9 pages, 1133 KiB

Open AccessArticle

The Influence of Noise on the Exact Solutions of the Stochastic Fractional-Space Chiral Nonlinear Schrödinger Equation

by Wael W. Mohammed, Omar Bazighifan, Mohammed M. Al-Sawalha, A. Othman Almatroud and Elkhateeb S. Aly

Fractal Fract. 2021, 5(4), 262; https://doi.org/10.3390/fractalfract5040262 - 7 Dec 2021

Cited by 19 | Viewed by 2648

Abstract

In this paper, we consider the stochastic fractional-space Chiral nonlinear Schrödinger equation (S-FS-CNSE) derived via multiplicative noise. We obtain the exact solutions of the S-FS-CNSE by using the Riccati equation method. The obtained solutions are extremely important in the development of nuclear medicine, [...] Read more.

In this paper, we consider the stochastic fractional-space Chiral nonlinear Schrödinger equation (S-FS-CNSE) derived via multiplicative noise. We obtain the exact solutions of the S-FS-CNSE by using the Riccati equation method. The obtained solutions are extremely important in the development of nuclear medicine, the entire computer industry and quantum mechanics, especially in the quantum hall effect. Moreover, we discuss how the multiplicative noise affects the exact solutions of the S-FS-CNSE. This equation has never previously been studied using a combination of multiplicative noise and fractional space. Full article

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

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16 pages, 637 KiB

Open AccessArticle

Propagation of Surface Waves in a Rotating Coated Viscoelastic Half-Space under the Influence of Magnetic Field and Gravitational Forces

by Ali Mubaraki, Saad Althobaiti and Rahmatullah Ibrahim Nuruddeen

Fractal Fract. 2021, 5(4), 250; https://doi.org/10.3390/fractalfract5040250 - 2 Dec 2021

Cited by 7 | Viewed by 2340

Abstract

The present manuscript focuses on the study of surface wave propagation in a rotating coated viscoelastic half-space and its response to external forces comprised of the magnetic field and gravitational forces. A celebrated normal mode analysis procedure is adopted as the methodology of [...] Read more.

The present manuscript focuses on the study of surface wave propagation in a rotating coated viscoelastic half-space and its response to external forces comprised of the magnetic field and gravitational forces. A celebrated normal mode analysis procedure is adopted as the methodology of interest for its high level of efficiency in the literature. The analytically obtained frequency equation is analyzed for certain scenarios of curiosity, in addition to the determination of the resulting displacements and stresses. Moreover, certain physical data of relevance with the viscoelasticity index of unity are considered for the numerical simulations. As for the findings, the presented graphical illustrations showed that both the magnetic field and rotation positively accelerated the dispersion of surface waves in the coated half-space, while the obtained approximate fields in the half-space are found to be oscillatory as they steadily move towards the limiting point. Full article

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

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29 pages, 4101 KiB

Open AccessArticle

Dynamics of Different Nonlinearities to the Perturbed Nonlinear Schrödinger Equation via Solitary Wave Solutions with Numerical Simulation

by Asim Zafar, Muhammad Raheel, Muhammad Qasim Zafar, Kottakkaran Sooppy Nisar, Mohamed S. Osman, Roshan Noor Mohamed and Ashraf Elfasakhany

Fractal Fract. 2021, 5(4), 213; https://doi.org/10.3390/fractalfract5040213 - 12 Nov 2021

Cited by 28 | Viewed by 2055

Abstract

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned nonlinearities are known as the Kerr law, power, dual [...] Read more.

This paper investigates the solitary wave solutions for the perturbed nonlinear Schrödinger equation with six different nonlinearities with the essence of the generalized classical derivative, which is known as the beta derivative. The aforementioned nonlinearities are known as the Kerr law, power, dual power law, triple power law, quadratic–cubic law and anti-cubic law. The dark, bright, singular and combinations of these solutions are retrieved using an efficient, simple integration scheme. These solutions suggest that this method is more simple, straightforward and reliable compared to existing methods in the literature. The novelty of this paper is that the perturbed nonlinear Schrödinger equation is investigated in different nonlinear media using a novel derivative operator. Furthermore, the numerical simulation for certain solutions is also presented. Full article

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

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21 pages, 6937 KiB

Open AccessArticle

A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis

by Muhammad Bilal Riaz, Jan Awrejcewicz, Adil Jhangeer and Muhammad Junaid-U-Rehman

Fractal Fract. 2021, 5(4), 170; https://doi.org/10.3390/fractalfract5040170 - 18 Oct 2021

Cited by 13 | Viewed by 1891

Abstract

This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional [...] Read more.

This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some

3 D

,

2 D

, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method. Full article

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

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19 pages, 29954 KiB

Open AccessArticle

Solving a Higher-Dimensional Time-Fractional Diffusion Equation via the Fractional Reduced Differential Transform Method

by Salah Abuasad, Saleh Alshammari, Adil Al-rabtah and Ishak Hashim

Fractal Fract. 2021, 5(4), 168; https://doi.org/10.3390/fractalfract5040168 - 15 Oct 2021

Cited by 5 | Viewed by 2003

Abstract

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo [...] Read more.

In this study, exact and approximate solutions of higher-dimensional time-fractional diffusion equations were obtained using a relatively new method, the fractional reduced differential transform method (FRDTM). The exact solutions can be found with the benefit of a special function, and we applied Caputo fractional derivatives in this method. The numerical results and graphical representations specified that the proposed method is very effective for solving fractional diffusion equations in higher dimensions. Full article

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

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15 pages, 8261 KiB

Open AccessArticle

Numerical Exploration via Least Squares Estimation on Three Dimensional MHD Yield Exhibiting Nanofluid Model with Porous Stretching Boundaries

by Tamour Zubair, Muhammad Usman, Umar Nazir, Poom Kumam and Muhammad Sohail

Fractal Fract. 2021, 5(4), 167; https://doi.org/10.3390/fractalfract5040167 - 14 Oct 2021

Cited by 2 | Viewed by 1775

Abstract

The numerical study of a three-dimensional magneto-hydrodynamic (MHD) Casson nano-fluid with porous and stretchy boundaries is the focus of this paper. Radiation impacts are also supposed. A feasible similarity variable may convert a verbalized set of nonlinear “partial” differential equations (PDEs) into a [...] Read more.

The numerical study of a three-dimensional magneto-hydrodynamic (MHD) Casson nano-fluid with porous and stretchy boundaries is the focus of this paper. Radiation impacts are also supposed. A feasible similarity variable may convert a verbalized set of nonlinear “partial” differential equations (PDEs) into a system of nonlinear “ordinary” differential equations (ODEs). To investigate the solutions of the resulting dimensionless model, the least-square method is suggested and extended. Maple code is created for the expanded technique of determining model behaviour. Several simulations were run, and graphs were used to provide a thorough explanation of the important parameters on velocities, skin friction, local Nusselt number, and temperature. The comparison study attests that the suggested method is well-matched, trustworthy, and accurate for investigating the governing model’s answers. This method may be expanded to solve additional physical issues with complicated geometry. Full article

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

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22 pages, 630 KiB

Open AccessArticle

Effect of Magnetic Field with Parabolic Motion on Fractional Second Grade Fluid

by Nazish Iftikhar, Muhammad Bilal Riaz, Jan Awrejcewicz and Ali Akgül

Fractal Fract. 2021, 5(4), 163; https://doi.org/10.3390/fractalfract5040163 - 11 Oct 2021

Cited by 7 | Viewed by 1877

Abstract

This paper is an analysis of the flow of magnetohydrodynamics (MHD) second grade fluid (SGF) under the influence of chemical reaction, heat generation/absorption, ramped temperature and concentration and thermodiffusion. The fluid was made to flow through a porous medium. It has been proven [...] Read more.

This paper is an analysis of the flow of magnetohydrodynamics (MHD) second grade fluid (SGF) under the influence of chemical reaction, heat generation/absorption, ramped temperature and concentration and thermodiffusion. The fluid was made to flow through a porous medium. It has been proven in many already-published articles that heat and mass transfer do not always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such a dynamical process very accurately; thus, a different concept of differentiation is needed to capture such a process. Very recently, new classes of differential operators were introduced and have been recognized to be efficient in capturing processes following the power law, the decay law and the crossover behaviors. For the study of heat and mass transfer, we applied the newly introduced differential operators to model such flow. The equations for heat, mass and momentum are established in the terms of Caputo (C), Caputo–Fabrizio (CF) and Atangana–Baleanu in Caputo sense (ABC) fractional derivatives. The Laplace transform, inversion algorithm and convolution theorem were used to derive the exact and semi-analytical solutions for all cases. The obtained analytical solutions were plotted for different values of existing parameters. It is concluded that the fluid velocity shows increasing behavior for

κ

,

G_{r}

and

G_{m}

, while velocity decreases for

P_{r}

and M. For

K_{r}

, both velocity and concentration curves show decreasing behavior. Fluid flow accelerates under the influence of

S_{r}

and R. Temperature and concentration profiles increase for

S_{r}

and R. Moreover, the ABC fractional operator presents a larger memory effect than C and CF fractional operators. Full article

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

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21 pages, 545 KiB

Open AccessEditor’s ChoiceArticle

Thermophysical Investigation of Oldroyd-B Fluid with Functional Effects of Permeability: Memory Effect Study Using Non-Singular Kernel Derivative Approach

by Muhammad Bilal Riaz, Jan Awrejcewicz, Aziz-Ur Rehman and Ali Akgül

Fractal Fract. 2021, 5(3), 124; https://doi.org/10.3390/fractalfract5030124 - 15 Sep 2021

Cited by 24 | Viewed by 2771

Abstract

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage [...] Read more.

It is well established fact that the functional effects, such as relaxation and retardation of materials, can be measured for magnetized permeability based on relative increase or decrease during magnetization. In this context, a mathematical model is formulated based on slippage and non-slippage assumptions for Oldroyd-B fluid with magnetized permeability. An innovative definition of Caputo-Fabrizio time fractional derivative is implemented to hypothesize the constitutive energy and momentum equations. The exact solutions of presented problem, are determined by using mathematical techniques, namely Laplace transform with slipping boundary conditions have been invoked to tackle governing equations of velocity and temperature. The Nusselt number and limiting solutions have also been persuaded to estimate the heat emission rate through physical interpretation. In order to provide the validation of the problem, the absence of retardation time parameter led the investigated solutions with good agreement in literature. Additionally, comprehensively scrutinize the dynamics of the considered problem with parametric analysis is accomplished, the graphical illustration is depicted for slipping and non-slipping solutions for temperature and velocity. A comparative studies between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. Full article

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

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