Fractal and Fractional

Research

20 pages, 14994 KiB

Open AccessArticle

The Extended Weierstrass Transformation Method for the Biswas–Arshed Equation with Beta Time Derivative

by Sertac Goktas, Aslı Öner and Yusuf Gurefe

Fractal Fract. 2024, 8(10), 593; https://doi.org/10.3390/fractalfract8100593 - 9 Oct 2024

Cited by 1 | Viewed by 1121

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, [...] Read more.

In this article, exact solutions of the Biswas–Arshed equation are obtained using the extended Weierstrass transformation method (EWTM). This method is widely used in solid-state physics, electrodynamics, and mathematical physics, and it yields exact solution functions involving trigonometric, rational trigonometric, Weierstrass elliptic, wave, and rational functions. The process involves expanding the solution functions of an elliptic differential equation into finite series by transforming them into Weierstrass functions. Furthermore, it generates parametric solutions for nonlinear algebraic equation systems, which are particularly useful in mathematical physics. These solutions are derived using the Mathematica package program. To analyze the behavior of these determined solution functions, the article employs separate two- and three-dimensional graphs showing the real and imaginary components, along with contour and density graphs. These visuals aid in comprehending the physical characteristics exhibited by these solution functions. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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20 pages, 6930 KiB

Open AccessArticle

Riemann–Hilbert Method Equipped with Mixed Spectrum for N-Soliton Solutions of New Three-Component Coupled Time-Varying Coefficient Complex mKdV Equations

by Sheng Zhang, Xianghui Wang and Bo Xu

Fractal Fract. 2024, 8(6), 355; https://doi.org/10.3390/fractalfract8060355 - 14 Jun 2024

Viewed by 1391

Abstract

This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating [...] Read more.

This article extends the celebrated Riemann–Hilbert (RH) method equipped with mixed spectrum to a new integrable system of three-component coupled time-varying coefficient complex mKdV equations (ccmKdVEs for short) generated by the mixed spectral equations (msEs). Firstly, the ccmKdVEs and the msEs for generating the ccmKdVEs are proposed. Then, based on the msEs, a solvable RH problem related to the ccmKdVEs is constructed. By using the constructed RH problem with mixed spectrum, scattering data for the recovery of potential formulae are further determined. In the case of reflectionless coefficients, explicit N-soliton solutions of the ccmKdVEs are ultimately obtained. Taking N equal to 1 and 2 as examples, this paper reveals that the spatiotemporal solution structures with time-varying nonlinear dynamic characteristics localized in the ccmKdVEs is attributed to the multiple selectivity of mixed spectrum and time-varying coefficients. In addition, to further highlight the application of our work in fractional calculus, by appropriately selecting these time-varying coefficients, the ccmKdVEs are transformed into a conformable time-fractional order system of three-component coupled complex mKdV equations. Based on the obtained one-soliton solutions, a set of initial values are assigned to the transformed fractional order system, and the N-th iteration formulae of approximate solutions for this fractional order system are derived through the variational iteration method (VIM). Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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17 pages, 4005 KiB

Open AccessArticle

Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics

by Muhammad Nadeem and Loredana Florentina Iambor

Fractal Fract. 2024, 8(5), 277; https://doi.org/10.3390/fractalfract8050277 - 7 May 2024

Cited by 2 | Viewed by 1043

Abstract

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the [...] Read more.

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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22 pages, 2651 KiB

Open AccessArticle

Impressive Exact Solitons to the Space-Time Fractional Mathematical Physics Model via an Effective Method

by Abdulaziz Khalid Alsharidi and Moin-ud-Din Junjua

Fractal Fract. 2024, 8(5), 248; https://doi.org/10.3390/fractalfract8050248 - 24 Apr 2024

Cited by 1 | Viewed by 1404

Abstract

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of [...] Read more.

A new class of truncated M-fractional exact soliton solutions for a mathematical physics model known as a truncated M-fractional (1+1)-dimensional nonlinear modified mixed-KdV model are achieved. We obtain these solutions by using a modified extended direct algebraic method. The obtained results consist of trigonometric, hyperbolic trigonometric and mixed functions. We also discuss the effect of fractional order derivative. To validate our results, we utilized the Mathematica software. Additionally, we depict some of the obtained kink, periodic, singular, and kink-singular wave solitons, using two and three dimensional graphs. The obtained results are useful in the fields of fluid dynamics, nonlinear optics, ocean engineering and others. Furthermore, these employed techniques are not only straightforward, but also highly effective when used to solve non-linear fractional partial differential equations (FPDEs). Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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13 pages, 1850 KiB

Open AccessArticle

Investigation of a Spatio-Temporal Fractal Fractional Coupled Hirota System

by Obaid J. Algahtani

Fractal Fract. 2024, 8(3), 178; https://doi.org/10.3390/fractalfract8030178 - 21 Mar 2024

Viewed by 1368

Abstract

This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that [...] Read more.

This article aims to examine the nonlinear excitations in a coupled Hirota system described by the fractal fractional order derivative. By using the Laplace transform with Adomian decomposition (LADM), the numerical solution for the considered system is derived. It has been shown that the suggested technique offers a systematic and effective method to solve complex nonlinear systems. Employing the Banach contraction theorem, it is confirmed that the LADM leads to a convergent solution. The numerical analysis of the solutions demonstrates the confinement of the carrier wave and the presence of confined wave packets. The dispersion nonlinear parameter reduction equally influences the wave amplitude and spatial width. The localized internal oscillations in the solitary waves decreased the wave collapsing effect at comparatively small dispersion. Furthermore, it is also shown that the amplitude of the solitary wave solution increases by reducing the fractal derivative. It is evident that decreasing the order

α

modifies the nature of the solitary wave solutions and marginally decreases the amplitude. The numerical and approximation solutions correspond effectively for specific values of time (t). However, when the fractal or fractional derivative is set to one by increasing time, the wave amplitude increases. The absolute error analysis between the obtained series solutions and the accurate solutions are also presented. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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18 pages, 345 KiB

Open AccessArticle

The Finite Difference Method and Analysis for Simulating the Unsteady Generalized Maxwell Fluid with a Multi-Term Time Fractional Derivative

by Yu Wang, Tianzeng Li and Yu Zhao

Fractal Fract. 2024, 8(3), 136; https://doi.org/10.3390/fractalfract8030136 - 26 Feb 2024

Viewed by 1592

Abstract

The finite difference method is used to solve a new class of unsteady generalized Maxwell fluid models with multi-term time-fractional derivatives. The fractional order range of the Maxwell model index is from 0 to 2, which is hard to approximate with general methods. [...] Read more.

The finite difference method is used to solve a new class of unsteady generalized Maxwell fluid models with multi-term time-fractional derivatives. The fractional order range of the Maxwell model index is from 0 to 2, which is hard to approximate with general methods. In this paper, we propose a new finite difference scheme to solve such problems. Based on the discrete

H^{1}

norm, the stability and convergence of the considered discrete scheme are discussed. We also prove that the accuracy of the method proposed in this paper is

O (τ + h^{2})

. Finally, some numerical examples are provided to further demonstrate the superiority of this method through comparative analysis with other algorithms. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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33 pages, 7511 KiB

Open AccessArticle

Fractional View Analysis System of Korteweg–de Vries Equations Using an Analytical Method

by Yousef Jawarneh, Zainab Alsheekhhussain and M. Mossa Al-Sawalha

Fractal Fract. 2024, 8(1), 40; https://doi.org/10.3390/fractalfract8010040 - 7 Jan 2024

Cited by 2 | Viewed by 1490

Abstract

This study introduces two innovative methods, the new transform iteration method and the residual power series transform method, to solve fractional nonlinear system Korteweg–de Vries (KdV) equations. These equations, fundamental in describing nonlinear wave phenomena, present complexities due to the involvement of fractional [...] Read more.

This study introduces two innovative methods, the new transform iteration method and the residual power series transform method, to solve fractional nonlinear system Korteweg–de Vries (KdV) equations. These equations, fundamental in describing nonlinear wave phenomena, present complexities due to the involvement of fractional derivatives. In demonstrating the application of the new transform iteration method and the residual power series transform method, computational analyses showcase their efficiency and accuracy in computing solutions for fractional nonlinear system KdV equations. Tables and figures accompanying this research present the obtained solutions, highlighting the superior performance of the new transform iteration method and the residual power series transform method compared to existing methods. The results underscore the efficacy of these novel methods in handling complex nonlinear equations involving fractional derivatives, suggesting their potential for broader applicability in similar mathematical problems. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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19 pages, 979 KiB

Open AccessArticle

Application of Analytical Techniques for Solving Fractional Physical Models Arising in Applied Sciences

by Mashael M. AlBaidani, Abdul Hamid Ganie, Fahad Aljuaydi and Adnan Khan

Fractal Fract. 2023, 7(8), 584; https://doi.org/10.3390/fractalfract7080584 - 28 Jul 2023

Cited by 16 | Viewed by 1596

Abstract

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation [...] Read more.

In this paper, we examined the approximations to the time-fractional Kawahara equation and modified Kawahara equation, which model the creation of nonlinear water waves in the long wavelength area and the transmission of signals. We implemented two novel techniques, namely the homotopy perturbation transform method and the Elzaki transform decomposition method. The derivative having fractional-order is taken in Caputo sense. The Adomian and He’s polynomials make it simple to handle the nonlinear terms. To illustrate the adaptability and effectiveness of derivatives with fractional order to represent the water waves in long wavelength regions, numerical data have been given graphically. A key component of the Kawahara equation is the symmetry pattern, and the symmetrical nature of the solution may be observed in the graphs. The importance of our suggested methods is illustrated by the convergence of analytical solutions to the precise solutions. The techniques currently in use are straightforward and effective for solving fractional-order issues. The offered methods reduced computational time is their main advantage. It will be possible to solve fractional partial differential equations using the study’s findings as a tool. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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14 pages, 1429 KiB

Open AccessArticle

Approximate Solution to Fractional Order Models Using a New Fractional Analytical Scheme

by Muhammad Nadeem and Loredana Florentina Iambor

Fractal Fract. 2023, 7(7), 530; https://doi.org/10.3390/fractalfract7070530 - 5 Jul 2023

Cited by 6 | Viewed by 1306

Abstract

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series [...] Read more.

In the present work, a new fractional analytical scheme (NFAS) is developed to obtain the approximate results of fourth-order parabolic fractional partial differential equations (FPDEs). The fractional derivatives are considered in the Caputo sense. In this scheme, we show that a Taylor series destructs the recurrence relation and minimizes the heavy computational work. This approach presents the results in the sense of convergent series. In addition, we provide the convergence theorem that shows the authenticity of this scheme. The proposed strategy is very simple and straightforward for obtaining the series solution of the fractional models. We take some differential problems of fractional orders to present the robustness and effectiveness of this developed scheme. The significance of NFAS is also shown by graphical and tabular expressions. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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22 pages, 2894 KiB

Open AccessArticle

Investigating Families of Soliton Solutions for the Complex Structured Coupled Fractional Biswas–Arshed Model in Birefringent Fibers Using a Novel Analytical Technique

by Humaira Yasmin, Noufe H. Aljahdaly, Abdulkafi Mohammed Saeed and Rasool Shah

Fractal Fract. 2023, 7(7), 491; https://doi.org/10.3390/fractalfract7070491 - 21 Jun 2023

Cited by 50 | Viewed by 1696

Abstract

This research uses a novel analytical method known as the modified Extended Direct Algebraic Method (mEDAM) to explore families of soliton solutions for the complex structured Coupled Fractional Biswas–Arshed Model (CFBAM) in Birefringent Fibers. The Direct Algebraic Method (DAM) is extended by the [...] Read more.

This research uses a novel analytical method known as the modified Extended Direct Algebraic Method (mEDAM) to explore families of soliton solutions for the complex structured Coupled Fractional Biswas–Arshed Model (CFBAM) in Birefringent Fibers. The Direct Algebraic Method (DAM) is extended by the mEDAM’s methodology to compute more analytical solutions that would otherwise be difficult to acquire. We use this method to derive several families of soliton solutions and examine their characteristics. We also look at how different model parameters, such as amplitude, width, and propagation speed, affect the dynamics of soliton. Our use of 2D and 3D graphics to illustrate the soliton solutions also makes it possible to see the soliton dynamics more clearly. The outcomes also demonstrate that the method suggested has proven successful in producing soliton solutions for intricate structures such as the CFBAM. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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16 pages, 4456 KiB

Open AccessArticle

Numerical Solutions of the (2+1)-Dimensional Nonlinear and Linear Time-Dependent Schrödinger Equations Using Three Efficient Approximate Schemes

by Neveen G. A. Farag, Ahmed H. Eltanboly, Magdi S. El-Azab and Salah S. A. Obayya

Fractal Fract. 2023, 7(2), 188; https://doi.org/10.3390/fractalfract7020188 - 13 Feb 2023

Cited by 1 | Viewed by 2633

Abstract

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, [...] Read more.

In this paper, the (2+1)-dimensional nonlinear Schrödinger equation (2D NLSE) abreast of the (2+1)-dimensional linear time-dependent Schrödinger equation (2D TDSE) are thoroughly investigated. For the first time, these two notable 2D equations are attempted to be solved using three compelling pseudo-spectral/finite difference approaches, namely the split-step Fourier transform (SSFT), Fourier pseudo-spectral method (FPSM), and the hopscotch method (HSM). A bright 1-soliton solution is considered for the 2D NLSE, whereas a Gaussian wave solution is determined for the 2D TDSE. Although the analytical solutions of these partial differential equations can sometimes be reached, they are either limited to a specific set of initial conditions or even perplexing to find. Therefore, our suggested approximate solutions are of tremendous significance, not only for our proposed equations, but also to apply to other equations. Finally, systematic comparisons of the three suggested approaches are conducted to corroborate the accuracy and reliability of these numerical techniques. In addition, each scheme’s error and convergence analysis is numerically exhibited. Based on the MATLAB findings, the novelty of this work is that the SSFT has proven to be an invaluable tool for the presented 2D simulations from the speed, accuracy, and convergence perspectives, especially when compared to the other suggested schemes. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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14 pages, 1247 KiB

Open AccessArticle

Higher-Order Dispersive and Nonlinearity Modulations on the Propagating Optical Solitary Breather and Super Huge Waves

by H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy and Mahmoud A. E. Abdelrahman

Fractal Fract. 2023, 7(2), 127; https://doi.org/10.3390/fractalfract7020127 - 30 Jan 2023

Cited by 6 | Viewed by 1730

Abstract

The nonlinearity form of the Schrödinger equation (NLSE) gives a sterling account for energy and solitary transmission properties in modern communications with optical-fiber energ- reinforcement actions. The solitary representation during fiber transmissions was regulated by NLSE coefficients such as nonlinear Kerr, evolutions, and [...] Read more.

The nonlinearity form of the Schrödinger equation (NLSE) gives a sterling account for energy and solitary transmission properties in modern communications with optical-fiber energ- reinforcement actions. The solitary representation during fiber transmissions was regulated by NLSE coefficients such as nonlinear Kerr, evolutions, and dispersions, which controlled the energy changes through the model. Sometimes, the energy values predicted from the NLSEs computations may diverge due to variations in the amplitude and width caused by scattering, dispersive, and dissipative features of fiber materials. Higher-order nonlinear Schrödinger equations (HONLSEs) should be explored to alleviate these implications in energy and wave features. The unified solver approach is employed in this work to evaluate the HONLSEs. Steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and solitary features were altered by higher-order actions. The unified solver approach is employed in this work to reform the HONLSE solutions and its energy properties. The steepness, HO dispersions, and nonlinearity self-frequency influences have been taken into consideration. The energy and soliton features in the investigated model were altered by the higher-order impacts. Furthermore, the new HONLSE solutions explain a wide range of important complex phenomena in wave energy and its applications. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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14 pages, 1249 KiB

Open AccessArticle

Fractional View Study of the Brusselator Reaction–Diffusion Model Occurring in Chemical Reactions

by Saleh Alshammari, M. Mossa Al-Sawalha and Jamal R. Humaidi

Fractal Fract. 2023, 7(2), 108; https://doi.org/10.3390/fractalfract7020108 - 20 Jan 2023

Cited by 6 | Viewed by 2226

Abstract

In this paper, we study a fractional Brusselator reaction–diffusion model with the help of the residual power series transform method. Specific reaction–diffusion chemical processes are modeled by applying the fractional Brusselator reaction–diffusion model. It should be mentioned that many problems in nonlinear science [...] Read more.

In this paper, we study a fractional Brusselator reaction–diffusion model with the help of the residual power series transform method. Specific reaction–diffusion chemical processes are modeled by applying the fractional Brusselator reaction–diffusion model. It should be mentioned that many problems in nonlinear science are characterized by fractional differential equations, where an unknown term occurs when a fractional-order derivative is operating on it. The analytic method of this problem is rarely discussed in the literature, despite numerous scholars having researched its application and usefulness. To validate our proposed method’s accuracy, we compare the numerical results of the residual power series transform method and the exact result with different fractional orders. The solution shows that the introduced approach is a good tool for solving linear and nonlinear fractional system differential equations. Finally, we provide two and three-dimensional graphical plots to support the impact of the fractional derivative on the behavior of the achieved profile results to the proposed equations. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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17 pages, 1370 KiB

Open AccessArticle

Fractional Study of the Non-Linear Burgers’ Equations via a Semi-Analytical Technique

by Naveed Iqbal, Muhammad Tajammal Chughtai and Roman Ullah

Fractal Fract. 2023, 7(2), 103; https://doi.org/10.3390/fractalfract7020103 - 18 Jan 2023

Cited by 12 | Viewed by 1961

Abstract

Most complex physical phenomena are described by non-linear Burgers’ equations, which help us understand them better. This article uses the transformation and the fractional Taylor’s formula to find approximate solutions for non-linear fractional-order partial differential equations. Solving non-linear Burgers’ equations with the right [...] Read more.

Most complex physical phenomena are described by non-linear Burgers’ equations, which help us understand them better. This article uses the transformation and the fractional Taylor’s formula to find approximate solutions for non-linear fractional-order partial differential equations. Solving non-linear Burgers’ equations with the right starting data shows that the method utilized is correct and can be utilized. Based on the limit of the idea, a rapid convergence McLaurin series is used to obtain close series solutions for both models with less work and more accuracy. To see how time-Caputo fractional derivatives affect how the results of the above models behave, in three dimension figures are drawn. The results showed that the proposed method is an easy, flexible, and helpful way to solve and understand a wide range of non-linear physical models. Full article

(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics)

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