Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics

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

Deadline for manuscript submissions: 25 July 2024 | Viewed by 15069

Special Issue Editor


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Guest Editor
Department of Mathematics, Faculty of Science, Mersin University, Mersin 33110, Turkey
Interests: numerical analysis; mathematical physics; partial differential equations; fractional calculus
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear differential equations are generally used to create mathematical models of real-life problems and to obtain their solutions. Therefore, many researchers have achieved important results by developing new methods in terms of finding analytical, numerical and exact solutions to nonlinear differential equations. In these studies, the nonlinear differential equations generally discussed include integer and fractional derivatives.

The aim of this Special Issue is to construct and apply analytical, numerical and exact methods for approaching nonlinear differential equations which have applications in the field of physics. In addition, this Special Issue will focus particularly on examining the physical behavior of the obtained results and analyzing them in detail.

Researchers are encouraged to introduce and discuss their new original papers on the solutions to nonlinear differential equations in engineering and applied science. Potential research topics include, but are not limited to, the following themes:

  • Recent advances in fractional calculus
  • Fractional calculus models in engineering and applied science
  • Fractional differential and difference equations
  • Functional fractional differential equations
  • Computational methods for integer or fractional order PDEs in applied science
  • Exact solutions to nonlinear physical problems
  • Numerical methods for initial and boundary value problems
  • Multiplicative differential equations and their applications
  • Fuzzy differential equations and their applications
  • Stochastic differential equations and their applications

Dr. Yusuf Gürefe
Guest Editor

Manuscript Submission Information

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Keywords

  • fractional calculus
  • special functions in fractional calculus
  • mathematical modelling in physics
  • nonlinear models in mathematical physics
  • dynamics of physical systems
  • numerical solutions
  • exact solutions
  • soliton theory
  • computational physics
  • multiplicative calculus
  • fuzzy differential calculus
  • stochastic differential equations

Published Papers (12 papers)

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Research

17 pages, 4005 KiB  
Article
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
Viewed by 446
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
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22 pages, 2651 KiB  
Article
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
Viewed by 737
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
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13 pages, 1850 KiB  
Article
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 812
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
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18 pages, 345 KiB  
Article
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 936
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 H1 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+h2). Finally, some numerical examples are provided to further demonstrate the superiority of this method through comparative analysis with other algorithms. Full article
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33 pages, 7511 KiB  
Article
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
Viewed by 971
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
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19 pages, 979 KiB  
Article
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 8 | Viewed by 983
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
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14 pages, 1429 KiB  
Article
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 1 | Viewed by 889
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
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22 pages, 2894 KiB  
Article
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 43 | Viewed by 1126
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
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16 pages, 4456 KiB  
Article
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 1727
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
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14 pages, 1247 KiB  
Article
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 5 | Viewed by 1238
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
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14 pages, 1249 KiB  
Article
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 4 | Viewed by 1564
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
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17 pages, 1370 KiB  
Article
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 8 | Viewed by 1242
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
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