Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
2.8
3.577
Journals
Fractal Fract
Special Issues
Numerical and Exact Methods for Nonlinear Differential Equations and Applications...
share announcement

Special Issue "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: 15 August 2023 | Viewed by 3352

Special Issue Editors

Dr. Yusuf Gürefe

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Mersin University, Mersin, Turkey
Interests: numerical analysis; mathematical physics; partial differential equations; fractional calculus
Prof. Dr. Nguyen Huy Tuan

E-Mail Website
Guest Editor
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
Interests: partial differential equations; stochastic partial differential equations; regularity; ill-posed problem.

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
Prof. Dr. Nguyen Huy Tuan
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • 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 (4 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

get_app
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
Viewed by 676
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)
►▼ Show Figures

Figure 1

get_app
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
Viewed by 658
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)
►▼ Show Figures

Figure 1

get_app
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 2 | Viewed by 891
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)
►▼ Show Figures

Figure 1

get_app
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 1 | Viewed by 570
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)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-4
Back to TopTop