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Advanced Research in Numerical Analysis of Partial Differential Equations
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Advanced Research in Numerical Analysis of Partial Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 1 May 2025 | Viewed by 3710

Special Issue Editors

Dr. Zhen Chao

E-Mail Website
Guest Editor
Department of Mathematics, Western Washington University, Bellingham, WA 98225, USA
Interests: numerical solution of partial differential equations; numerical linear algebra and scientific computing; computational biology
Dr. Xiang-Sheng Wang

E-Mail Website
Guest Editor
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70503, USA
Interests: asymptotic analysis and computational mathematics; differential equations and dynamical systems; mathematical biology and mathematical statistics; orthogonal polynomials and special functions

Special Issue Information

Dear Colleagues,

The numerical analysis of PDEs is a rich and active field of modern computational and applied mathematics. The steady growth of this field is stimulated by the ever-increasing demands of natural sciences, engineering, and economics, with the aim of providing accurate and reliable approximations to mathematical models involving PDEs, whose exact solutions are either too complicated to determine or are not known to exist. The purpose of this Special Issue is to contribute to this area by providing a collection of articles that showcase cutting-edge research in the numerical analysis of PDEs, focusing on advanced methodologies, algorithms, and applications.

Topics of interest including, but are not limited to, the following:

Advanced finite difference, finite element, spectral method, and boundary element methods for solving PDEs;
Parallel and distributed computing techniques for large-scale PDE problems;
Numerical optimization and inverse problems in PDEs;
Computational fluid dynamics (CFD) and computational electromagnetics (CEM);
Applications of PDEs in geophysics, materials science, and mathematical biology, etc.;
Software development and implementation for PDE solvers;
Fractional differential equations.

Dr. Zhen Chao
Dr. Xiang-Sheng Wang
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

  • finite element method
  • boundary element method
  • finite difference method
  • parallel computing techniques
  • spectral method
  • inverse problem
  • numerical optimization
  • computational fluid dynamics
  • multiscale methods
  • fractional differential equations

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Published Papers (4 papers)

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23 pages, 902 KiB  
Article
Fractional Dynamics: Applications of the Caputo Operator in Solving the Sawada–Kotera and Rosenau–Hyman Equations
by Khudhayr A. Rashedi, Musawa Yahya Almusawa, Hassan Almusawa, Tariq S. Alshammari and Adel Almarashi
Mathematics 2025, 13(2), 193; https://doi.org/10.3390/math13020193 - 8 Jan 2025
Viewed by 603
Abstract
This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional [...] Read more.
This study investigates the fractional-order Sawada–Kotera and Rosenau–Hyman equations, which significantly model non-linear wave phenomena in various scientific fields. We employ two advanced methodologies to obtain analytical solutions: the q-homotopy Mohand transform method (q-HMTM) and the Mohand variational iteration method (MVIM). The fractional derivatives in the equations are expressed using the Caputo operator, which provides a flexible framework for analyzing the dynamics of fractional systems. By leveraging these methods, we derive diverse types of solutions, including hyperbolic, trigonometric, and rational forms, illustrating the effectiveness of the techniques in addressing complex fractional models. Numerical simulations and graphical representations are provided to validate the accuracy and applicability of derived solutions. Special attention is given to the influence of the fractional parameter on behavior of the solution behavior, highlighting its role in controlling diffusion and wave propagation. The findings underscore the potential of q-HMTM and MVIM as robust tools for solving non-linear fractional differential equations. They offer insights into their practical implications in fluid dynamics, wave mechanics, and other applications governed by fractional-order models. Full article
(This article belongs to the Special Issue Advanced Research in Numerical Analysis of Partial Differential Equations)
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16 pages, 706 KiB  
Article
Finite-Time Partial Component Consensus for Nonlinear Leader-Following Multi-Agent Systems
by Zhaolei Yan, Baibin Yang, Manman Luo and Manfeng Hu
Mathematics 2024, 12(22), 3552; https://doi.org/10.3390/math12223552 - 13 Nov 2024
Viewed by 742
Abstract
The problem of finite-time partial component consensus (FTPCC) for first-order nonlinear multi-agent systems (MASs) is investigated in this paper for the first time. By incorporating the permutation matrix approach, we derive a novel error system for identical components, which facilitates stability analysis. Leveraging [...] Read more.
The problem of finite-time partial component consensus (FTPCC) for first-order nonlinear multi-agent systems (MASs) is investigated in this paper for the first time. By incorporating the permutation matrix approach, we derive a novel error system for identical components, which facilitates stability analysis. Leveraging partial variable stability theory and related foundational knowledge, we devise two adaptable protocols. These protocols are tailored to achieve FTPCC in nonlinear MASs, one for systems without disturbances and another for those with bounded disturbances. To validate our findings, numerical examples are provided, demonstrating the effectiveness of the proposed results. Full article
(This article belongs to the Special Issue Advanced Research in Numerical Analysis of Partial Differential Equations)
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15 pages, 905 KiB  
Article
Several Characterizations of the Generalized 1-Parameter 3-Variable Hermite Polynomials
by Shahid Ahmad Wani, Khalil Hadi Hakami and Hamad Zogan
Mathematics 2024, 12(16), 2459; https://doi.org/10.3390/math12162459 - 8 Aug 2024
Viewed by 820
Abstract
This paper presents a novel framework for introducing generalized 1-parameter 3-variable Hermite polynomials. These polynomials are characterized through generating functions and series definitions, elucidating their fundamental properties. Moreover, utilising a factorisation method, this study establishes recurrence relations, shift operators, and various differential equations, [...] Read more.
This paper presents a novel framework for introducing generalized 1-parameter 3-variable Hermite polynomials. These polynomials are characterized through generating functions and series definitions, elucidating their fundamental properties. Moreover, utilising a factorisation method, this study establishes recurrence relations, shift operators, and various differential equations, including differential, integro-differential, and partial differential equations. Full article
(This article belongs to the Special Issue Advanced Research in Numerical Analysis of Partial Differential Equations)
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22 pages, 400 KiB  
Article
Two Block Splitting Iteration Methods for Solving Complex Symmetric Linear Systems from Complex Helmholtz Equation
by Yanan Zhu, Naimin Zhang and Zhen Chao
Mathematics 2024, 12(12), 1888; https://doi.org/10.3390/math12121888 - 18 Jun 2024
Cited by 1 | Viewed by 960
Abstract
In this paper, we study the improved block splitting (IBS) iteration method and its accelerated variant, the accelerated improved block splitting (AIBS) iteration method, for solving linear systems of equations stemming from the discretization of the complex Helmholtz equation. We conduct a comprehensive [...] Read more.
In this paper, we study the improved block splitting (IBS) iteration method and its accelerated variant, the accelerated improved block splitting (AIBS) iteration method, for solving linear systems of equations stemming from the discretization of the complex Helmholtz equation. We conduct a comprehensive convergence analysis and derive optimal iteration parameters aimed at minimizing the spectral radius of the iteration matrix. Through numerical experiments, we validate the efficiency of both iteration methods. Full article
(This article belongs to the Special Issue Advanced Research in Numerical Analysis of Partial Differential Equations)
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