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Mathematics
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Exact Solutions and Numerical Solutions of Differential Equations, 2nd Edition
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Exact Solutions and Numerical Solutions of Differential Equations, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C1: Difference and Differential Equations".

Deadline for manuscript submissions: 20 September 2025 | Viewed by 1068

Special Issue Editors

Dr. Ben Muatjetjeja

E-Mail Website
Guest Editor
1. Department of Mathematical Sciences, North-West University, Private Bag X 2046, Mmabatho 2735, South Africa
2. Department of Mathematics, Faculty of Science, University of Botswana, Private Bag 22, Gaborone, Botswana
Interests: symmetries of differentials equations; soliton theory
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Abdullahi Adem

E-Mail Website
Guest Editor
Department of Mathematical Sciences, University of South Africa, UNISA, Pretoria 0003, South Africa
Interests: conservation laws of partial differentials equations; mathematical physics
Special Issues, Collections and Topics in MDPI journals
Dr. P. Kaelo

E-Mail Website
Guest Editor
Department of Mathematics, University of Botswana, Private Bag UB00704, Gaborone, Botswana
Interests: numerical analysis; monotone nonlinear equations; optimization
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear differential equations play a significant role in many real-life phenomena, such as in fluid dynamics, optics, acoustics, plasma physics, engineering, and in many other areas of nonlinear science. Thus, it is vital to find solutions to these equations in order to understand and interpret the structure modeled by them.

Moreover, researchers have developed a variety of analytical and numerical techniques that can be employed to solve nonlinear differential equations. Some of the well-known techniques include the Lie symmetry method, the inverse scattering transformation approach, Ansatz methods, multistep methods, finite difference/element/volume methods, and many other techniques enumerated in the literature.

This Special Issue will be devoted to unveiling the most recent progress in obtaining analytical and numerical solutions to nonlinear differential equations via various methods and to stimulating collaborative research activities.

Dr. Ben Muatjetjeja
Prof. Dr. Abdullahi Adem
Dr. P. Kaelo
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • symmetries of differential equations
  • soliton theory conservation laws of partial differential equations
  • mathematical physics
  • numerical analysis
  • monotone nonlinear equations

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Published Papers (2 papers)

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Research

27 pages, 2813 KiB  
Article
Study of Optical Solitons and Quasi-Periodic Behaviour for the Fractional Cubic Quintic Nonlinear Pulse Propagation Model
by Lotfi Jlali, Syed T. R. Rizvi, Sana Shabbir and Aly R. Seadawy
Mathematics 2025, 13(13), 2117; https://doi.org/10.3390/math13132117 - 28 Jun 2025
Viewed by 161
Abstract
This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial [...] Read more.
This study explores analytical soliton solutions for the cubic–quintic time-fractional nonlinear non-paraxial pulse transmission model. This versatile model finds numerous uses in fiber optic communication, nonlinear optics, and optical signal processing. The strength of the quintic and cubic nonlinear components plays a crucial role in nonlinear processes, such as self-phase modulation, self-focusing, and wave combining. The fractional nonlinear Schrödinger equation (FNLSE) facilitates precise control over the dynamic properties of optical solitons. Exact and methodical solutions include those involving trigonometric functions, Jacobian elliptical functions (JEFs), and the transformation of JEFs into solitary wave (SW) solutions. This study reveals that various soliton solutions, such as periodic, rational, kink, and SW solitons, are identified using the complete discrimination polynomial methods (CDSPM). The concepts of chaos and bifurcation serve as the framework for investigating the system qualitatively. We explore various techniques for detecting chaos, including three-dimensional and two-dimensional graphs, time-series analysis, and Poincarè maps. A sensitivity analysis is performed utilizing a variety of initial conditions. Full article
(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations, 2nd Edition)
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25 pages, 3109 KiB  
Article
Generalized Modified Unstable Nonlinear Schrödinger’s Equation: Optical Solitons and Modulation Instability
by Jamilu Sabi’u, Ibrahim Sani Ibrahim, Khomsan Neamprem, Surattana Sungnul and Sekson Sirisubtawee
Mathematics 2025, 13(12), 2032; https://doi.org/10.3390/math13122032 - 19 Jun 2025
Viewed by 694
Abstract
This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of [...] Read more.
This paper proposes the generalized modified unstable nonlinear Schrödinger’s equation with applications in modulated wavetrain instabilities. The extended direct algebra and generalized Ricatti equation methods are applied to find innovative soliton solutions to the equation. The solutions are obtained in the form of elliptic, hyperbolic, and trigonometric functions. Moreover, a Galilean transformation is used to convert the problem into a dynamical system. We use the theory of planar dynamical systems to derive the equilibrium points of the dynamical system and analyze the Hamiltonian polynomial. We further investigate the bifurcation phase portrait of the system and study its chaotic behaviors when an external force is applied to the system. Graphical 2D and 3D plots are explored to support our mathematical analysis. A sensitivity analysis confirms that the variation in initial conditions has no substantial effect on the stability of the solutions. Furthermore, we give the modulation instability gain spectrum of the considered model and graphically indicate its dynamics using 2D plots. The reported results demonstrate not only the dynamics of the analyzed equation but are also conceptually relevant in establishing the temporal development of modest disturbances in stable or unstable media. These disturbances will be critical for anticipating, planning treatments, and creating novel mechanisms for modulated wavetrain instabilities. Full article
(This article belongs to the Special Issue Exact Solutions and Numerical Solutions of Differential Equations, 2nd Edition)
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