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Mathematics
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Research on Applied Partial Differential Equations
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Research on Applied Partial Differential Equations

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

Deadline for manuscript submissions: 31 December 2025 | Viewed by 2482

Special Issue Editors

Dr. Erwin Suazo

E-Mail Website
Guest Editor
School of Mathematical & Statistical Sciences, The University of Texas, Rio Grande Valley (UTRGV), Edinburg, TX, USA
Interests: evolution equations; nonlinear wave propagation; quantum mechanics; fractional calculus; partial differential equations; applications in applied mathematics
Dr. Jose M. Vega-Guzman

E-Mail Website
Guest Editor
Department of Mathematics, Lamar University, Beaumont, TX 77710, USA
Interests: evolution equations; nonlinear wave propagation; quantum mechanics; nonlinear optics; applications of ODEs and PDEs in natural sciences; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to advance the development of partial differential equation analysis, including theoretical and numerical analysis and applications in different areas of applied science. We invite researchers to contribute high-quality, original research articles as well as review articles on recent advances including, but not limited to, the following topics: modeling, theoretical analysis and numerical methods for ordinary and partial differential equations, exact solutions for nonlinear differential equations, Lie group analysis, integrable systems and solitons, fractional PDEs, and partial and ordinary stochastic differential equations.

Dr. Erwin Suazo
Dr. Jose M. Vega-Guzman
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

  • fractional calculus
  • ordinary differential equations
  • partial differential equations
  • stochastic differential equations
  • nonlinear wave propagation
  • applications of difference and differential equations in finance (or applications in finance)
  • differential learning methods

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Published Papers (3 papers)

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Research

15 pages, 2360 KiB  
Article
Analytic Investigation of a Generalized Variable-Coefficient KdV Equation with External-Force Term
by Gongxun Li, Zhiyan Wang, Ke Wang, Nianqin Jiang and Guangmei Wei
Mathematics 2025, 13(10), 1642; https://doi.org/10.3390/math13101642 - 17 May 2025
Viewed by 195
Abstract
This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair [...] Read more.
This paper investigates integrable properties of a generalized variable-coefficient Korteweg–de Vries (gvcKdV) equation incorporating dissipation, inhomogeneous media, and an external-force term. Based on Painlevé analysis, sufficient and necessary conditions for the equation’s Painlevé integrability are obtained. Under specific integrability conditions, the Lax pair for this equation is successfully constructed using the extended Ablowitz–Kaup–Newell–Segur system (AKNS system). Furthermore, the Riccati-type Bäcklund transformation (R-BT), Wahlquist–Estabrook-type Bäcklund transformation (WE-BT), and the nonlinear superposition formula are derived. In utilizing these transformations and the formula, explicit one-soliton-like and two-soliton-like solutions are constructed from a seed solution. Moreover, the infinite conservation laws of the equation are systematically derived. Finally, the influence of variable coefficients and the external-force term on the propagation characteristics of a solitory wave is discussed, and soliton interaction is illustrated graphically. Full article
(This article belongs to the Special Issue Research on Applied Partial Differential Equations)
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17 pages, 3762 KiB  
Article
A Time–Space Numerical Procedure for Solving the Sideways Heat Conduction Problem
by Ching-Chuan Tan, Chao-Feng Shih, Jian-Hung Shen and Yung-Wei Chen
Mathematics 2025, 13(5), 751; https://doi.org/10.3390/math13050751 - 25 Feb 2025
Viewed by 402
Abstract
This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated with measurement noise. These perturbations may lead [...] Read more.
This paper proposes a solution to the sideways heat conduction problem (SHCP) based on the time and space integration direction. Conventional inverse problems depend highly on the available data, particularly when the observed data are contaminated with measurement noise. These perturbations may lead to significant oscillations in the solution. The uniqueness of the solution in this SHCP requires revaluation when boundary conditions (BCs) or initial conditions (ICs) are missing. First, the spatial gradient between two points resolves the missing BCs in the computational domain by a one-step Lie group scheme. Further, the SHCP can be transformed into a backward-in-time heat conduction problem (BHCP). The second-order backward explicit integration can be applied to determine the ICs using the two-point solution at each time step. The performance of the suggested strategy is demonstrated with three numerical examples. The exact solution and the numerical results correspond well, despite the absence of some boundary and initial conditions. The only method of preventing numerical instability in this study is to alter the direction of numerical integration instead of relying on regularization techniques. Therefore, a numerical formula with two integration directions proves to be more accurate and stable compared to existing methods for the SHCP. Full article
(This article belongs to the Special Issue Research on Applied Partial Differential Equations)
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21 pages, 3084 KiB  
Article
On Blow-Up and Explicit Soliton Solutions for Coupled Variable Coefficient Nonlinear Schrödinger Equations
by José M. Escorcia and Erwin Suazo
Mathematics 2024, 12(17), 2694; https://doi.org/10.3390/math12172694 - 29 Aug 2024
Cited by 1 | Viewed by 1054
Abstract
This work is concerned with the study of explicit solutions for a generalized coupled nonlinear Schrödinger equations (NLS) system with variable coefficients. Indeed, by employing similarity transformations, we show the existence of rogue wave and dark–bright soliton-like solutions for such a generalized NLS [...] Read more.
This work is concerned with the study of explicit solutions for a generalized coupled nonlinear Schrödinger equations (NLS) system with variable coefficients. Indeed, by employing similarity transformations, we show the existence of rogue wave and dark–bright soliton-like solutions for such a generalized NLS system, provided the coefficients satisfy a Riccati system. As a result of the multiparameter solution of the Riccati system, the nonlinear dynamics of the solution can be controlled. Finite-time singular solutions in the L∞ norm for the generalized coupled NLS system are presented explicitly. Finally, an n-dimensional transformation between a variable coefficient NLS coupled system and a constant coupled system coefficient is presented. Soliton and rogue wave solutions for this high-dimensional system are presented as well. Full article
(This article belongs to the Special Issue Research on Applied Partial Differential Equations)
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