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Fractal Fract
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Numerical and Exact Methods for Nonlinear Differential Equations and Applications...
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Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition

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

Deadline for manuscript submissions: 1 October 2025 | Viewed by 3200

Special Issue Editor

Dr. Yusuf Gürefe

E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Science, Mersin University, 33110 Mersin, 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 particularly focus 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

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

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

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • 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

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Related Special Issue

Published Papers (6 papers)

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Research

8 pages, 799 KiB  
Article
Optical Solutions of the Nonlinear Kodama Equation with the M-Truncated Derivative via the Extended (G′/G)-Expansion Method
by Zhao Li
Fractal Fract. 2025, 9(5), 300; https://doi.org/10.3390/fractalfract9050300 - 5 May 2025
Viewed by 183
Abstract
The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended (G′/G)-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is [...] Read more.
The main purpose of this article is to study the optical solutions of the nonlinear Kodama equation with the M-truncated derivative by using the extended (G′/G)-expansion method. Firstly, the nonlinear Kodama equation with the M-truncated derivative is transformed into a nonlinear ordinary differential equation based on the principle of homogeneous equilibrium and the traveling wave transformation. Secondly, the optical solutions of the nonlinear Kodama equation with the M-truncated derivative are constructed by using the extended (G′/G)-expansion method. Finally, three-dimensional, two-dimensional, and contour maps of partial solutions are obtained by using Matlab R2023b mathematical software. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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20 pages, 7934 KiB  
Article
Dynamical Behavior Analysis of Generalized Chen–Lee–Liu Equation via the Riemann–Hilbert Approach
by Wenxia Chen, Chaosheng Zhang and Lixin Tian
Fractal Fract. 2025, 9(5), 282; https://doi.org/10.3390/fractalfract9050282 - 26 Apr 2025
Viewed by 176
Abstract
In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its N-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on [...] Read more.
In this paper, we investigate the dynamics of the generalized Chen–Lee–Liu (gCLL) equation utilizing the Riemann–Hilbert method to derive its N-soliton solution. We incorporate a self-steepening term and a Kerr nonlinear term to characterize nonlinear light propagation in optical fibers based on the Chen–Lee–Liu (CLL) equation more accurately. The Riemann–Hilbert problem is addressed through spectral analysis derived from the Lax pair formulation, resulting in an N-soliton solution for a reflectance-less system. We also present explicit formulas for solutions involving one and two solitons, thereby providing theoretical support for stable long-distance signal transmission in optical fiber communication. Furthermore, by adjusting parameters and conducting comparative analyses, we generate three-dimensional soliton images that warrant further exploration. The stability of soliton solutions in optical fibers offers novel insights into the intricate propagation behavior of light pulses, and it is crucial for maintaining the integrity of communication signals. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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15 pages, 2815 KiB  
Article
Computational Study of Time-Fractional Kawahara and Modified Kawahara Equations with Caputo Derivatives Using Natural Homotopy Transform Method
by Muhammad Nadeem, Loredana Florentina Iambor, Ebraheem Alzahrani and Azeem Hafiz P. Ajmal
Fractal Fract. 2025, 9(4), 247; https://doi.org/10.3390/fractalfract9040247 - 15 Apr 2025
Viewed by 232
Abstract
This article presents a computational analysis of approximate solutions for the time-fractional nonlinear Kawahara problem (KP) and the modified Kawahara problem (modified KP). This study utilizes the natural homotopy transform scheme (NHTS), which integrates the natural transform (NT) with the homotopy perturbation scheme [...] Read more.
This article presents a computational analysis of approximate solutions for the time-fractional nonlinear Kawahara problem (KP) and the modified Kawahara problem (modified KP). This study utilizes the natural homotopy transform scheme (NHTS), which integrates the natural transform (NT) with the homotopy perturbation scheme (HPS). We derive the algebraic expression of nonlinear terms through the implementation of HPS. The fractional derivatives are considered in the Caputo form. Numerical results and visualizations present the practical interest and effectiveness of the fractional derivatives. The accuracy of the approximate results, coupled with their precise outcomes, emphasizes the reliability of the method. These findings demonstrate that NHTS is a robust and effective approach for solving time-fractional problems through series expansions. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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24 pages, 8587 KiB  
Article
Integrable Riesz Fractional-Order Generalized NLS Equation with Variable Coefficients: Inverse Scattering Transform and Analytical Solutions
by Hongwei Li, Sheng Zhang and Bo Xu
Fractal Fract. 2025, 9(4), 228; https://doi.org/10.3390/fractalfract9040228 - 3 Apr 2025
Viewed by 317
Abstract
Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations [...] Read more.
Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations with variable coefficients. Specifically, based on the analysis of anomalous dispersion relation (ADR), a novel variable-coefficient Riesz fractional-order generalized NLS (vcRfgNLS) equation is derived. By utilizing the relevant matrix spectral problems (MSPs), the vcRfgNLS equation is solved through the inverse scattering transform (IST), and analytical solutions including n-soliton solution as a special case are obtained. In addition, an explicit form of the vcRfgNLS equation depending on the completeness of squared eigenfunctions (SEFs) is presented. In particular, the 1-soliton solution and 2-soliton solution are taken as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. It turns out that both the variable coefficients and fractional order can influence the velocity of soliton propagation, but there is no energy dissipation throughout the entire motion process. Such soliton solutions may not only have important value for studying the super-dispersion transport of nonlinear waves in non-uniform media, but also for realizing a new generation of ultra-high-speed optical communication engineering. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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15 pages, 3454 KiB  
Article
Soliton Solutions and Chaotic Dynamics of the Ion-Acoustic Plasma Governed by a (3+1)-Dimensional Generalized Korteweg–de Vries–Zakharov–Kuznetsov Equation
by Amjad E. Hamza, Mohammed Nour A. Rabih, Amer Alsulami, Alaa Mustafa, Khaled Aldwoah and Hicham Saber
Fractal Fract. 2024, 8(11), 673; https://doi.org/10.3390/fractalfract8110673 - 19 Nov 2024
Cited by 2 | Viewed by 871
Abstract
This study explores the novel dynamics of the (3+1)-dimensional generalized Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. A Galilean transformation is employed to derive the associated system of equations. Perturbing this system allows us to investigate the presence and characteristics of chaotic behavior, including return maps, [...] Read more.
This study explores the novel dynamics of the (3+1)-dimensional generalized Korteweg–de Vries–Zakharov–Kuznetsov (KdV-ZK) equation. A Galilean transformation is employed to derive the associated system of equations. Perturbing this system allows us to investigate the presence and characteristics of chaotic behavior, including return maps, fractal dimension, power spectrum, recurrence plots, and strange attractors, supported by 2D and time-dependent phase portraits. A sensitivity analysis is demonstrated to show how the system behaves when there are small changes in initial values. Finally, the planar dynamical system method is used to derive anti-kink, dark soliton, and kink soliton solutions, advancing our understanding of the range of solutions admitted by the model. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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20 pages, 6607 KiB  
Article
Investigating the Dynamics of a Unidirectional Wave Model: Soliton Solutions, Bifurcation, and Chaos Analysis
by Tariq Alraqad, Muntasir Suhail, Hicham Saber, Khaled Aldwoah, Nidal Eljaneid, Amer Alsulami and Blgys Muflh
Fractal Fract. 2024, 8(11), 672; https://doi.org/10.3390/fractalfract8110672 - 18 Nov 2024
Cited by 2 | Viewed by 939
Abstract
The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to [...] Read more.
The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to derive new solutions for the governing model. Second, it analyzes the complex dynamical structure of the governing model using bifurcation, chaos, and sensitivity analyses. To provide a more accurate depiction of the underlying dynamics, they use quantum mechanics to explain the intricate behavior of the system. To illustrate the physical behavior of the obtained solutions, 2D and 3D plots, along with a phase plane analysis, are presented using appropriate parameter values. These results validate the effectiveness of the employed method, providing thorough and consistent solutions with significant computational efficiency. The investigated soliton solutions will be valuable in understanding complex physical structures in various scientific fields, including ferromagnetic dynamics, nonlinear optics, soliton wave theory, and fiber optics. This approach proves highly effective in handling the complexities inherent in engineering and mathematical problems, especially those involving fractional-order systems. Full article
(This article belongs to the Special Issue Numerical and Exact Methods for Nonlinear Differential Equations and Applications in Physics, 2nd Edition)
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