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Mathematics
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Advances in Numerical Analysis of Partial Differential Equations
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Advances in Numerical Analysis of 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: 28 February 2026 | Viewed by 2995

Special Issue Editor

Prof. Dr. Gang Li

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China
Interests: numerical analysis and partial differential equations

Special Issue Information

Dear Colleagues,

‘Advances in Numerical Analysis of Partial Differential Equations’ provides a suitable platform for the development of original research in the fields of numerical solutions for partial differential equations from science and engineering. The purpose of this Special Issue is to gather contributions from experts on numerical analysis and scientific computing with application domains, including, but not limited to, computational fluid dynamics, hyperbolic conservation laws, shallow water equations, magnetohydrodynamics, relativistic fluid mechanics, and actuators. Contributions should have a main emphasis on numerical methods for the innovation of the method and their applications.

Prof. Dr. Gang Li
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. 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

  • high-order accuracy
  • partial differential equations
  • physical-constraint-preserving property
  • science and engineering

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Published Papers (4 papers)

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Research

29 pages, 592 KB  
Article
Stability Analysis and Finite Difference Approximations for a Damped Wave Equation with Distributed Delay
by Manal Alotaibi
Mathematics 2025, 13(17), 2714; https://doi.org/10.3390/math13172714 - 23 Aug 2025
Viewed by 357
Abstract
This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure [...] Read more.
This paper presents a fully implicit finite difference scheme for the numerical approximation of a wave equation featuring strong damping and a distributed delay term. The discretization employs second-order accurate approximations in both time and space. Although implicit, the scheme does not ensure unconditional stability due to the nonlocal nature of the delayed damping. To address this, we perform a stability analysis based on Rouché’s theorem from complex analysis and derive a sufficient condition for asymptotic stability of the discrete system. The resulting criterion highlights the interplay among the discretization parameters, the damping coefficient, and the delay kernel. Two quadrature techniques, the composite trapezoidal rule (CTR) and the Gaussian quadrature rule (GQR), are employed to approximate the convolution integral. Numerical experiments validate the theoretical predictions and illustrate both stable and unstable dynamics across different parameter regimes. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)
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30 pages, 1523 KB  
Article
Modeling and Simulation of Attraction–Repulsion Chemotaxis Mechanism System with Competing Signal
by Anandan P. Aswathi, Amar Debbouche, Yadhavan Karuppusamy and Lingeshwaran Shangerganesh
Mathematics 2025, 13(15), 2486; https://doi.org/10.3390/math13152486 - 1 Aug 2025
Viewed by 363
Abstract
This paper addresses an attraction–repulsion chemotaxis system governed by Neumann boundary conditions within a bounded domain Ω⊂R3 that has a smooth boundary. The primary focus of the study is the chemotactic response of a species (cell population) to two competing [...] Read more.
This paper addresses an attraction–repulsion chemotaxis system governed by Neumann boundary conditions within a bounded domain Ω⊂R3 that has a smooth boundary. The primary focus of the study is the chemotactic response of a species (cell population) to two competing signals. We establish the existence and uniqueness of a weak solution to the system by analyzing the solvability of an approximate problem and utilizing the Leray–Schauder fixed-point theorem. By deriving appropriate a priori estimates, we demonstrate that the solution of the approximate problem converges to a weak solution of the original system. Additionally, we conduct computational studies of the model using the finite element method. The accuracy of our numerical implementation is evaluated through error analysis and numerical convergence, followed by various numerical simulations in a two-dimensional domain to illustrate the dynamics of the system and validate the theoretical findings. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)
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13 pages, 386 KB  
Article
Reachable Set Estimation of Discrete Singular Systems with Time-Varying Delays and Bounded Peak Inputs
by Hongli Yang, Lijuan Yang and Ivan Ganchev Ivanov
Mathematics 2025, 13(1), 79; https://doi.org/10.3390/math13010079 - 28 Dec 2024
Cited by 1 | Viewed by 748
Abstract
This paper studies the estimation of reachable sets for discrete-time singular systems with time-varying delays and bounded peak inputs. A novel linear matrix inequality condition for the reachable set estimation of the time-varying time-delay discrete singular system is derived using an inverse convex [...] Read more.
This paper studies the estimation of reachable sets for discrete-time singular systems with time-varying delays and bounded peak inputs. A novel linear matrix inequality condition for the reachable set estimation of the time-varying time-delay discrete singular system is derived using an inverse convex combination and the discrete form of the Wirtinger inequality. Furthermore, the symmetric matrix involved in the obtained results does not need to be positively definite. Compared to decomposing the time-delay discrete singular system under consideration into fast and slow subsystems, the method presented in this paper is simpler and involves fewer variables. Two numerical examples are provided to illustrate the proposed method. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)
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19 pages, 1385 KB  
Article
A Solution-Structure B-Spline-Based Framework for Hybrid Boundary Problems on Implicit Domains
by Ammar Qarariyah, Tianhui Yang and Fang Deng
Mathematics 2024, 12(24), 3973; https://doi.org/10.3390/math12243973 - 18 Dec 2024
Cited by 2 | Viewed by 1052
Abstract
Solving partial differential equations (PDEs) on complex domains with hybrid boundary conditions presents significant challenges in numerical analysis. In this paper, we introduce a solution-structure-based framework that transforms non-homogeneous hybrid boundary problems into homogeneous ones, allowing exact conformity to the boundary conditions. By [...] Read more.
Solving partial differential equations (PDEs) on complex domains with hybrid boundary conditions presents significant challenges in numerical analysis. In this paper, we introduce a solution-structure-based framework that transforms non-homogeneous hybrid boundary problems into homogeneous ones, allowing exact conformity to the boundary conditions. By leveraging B-splines within the R-function method structure and adopting the stability principles of the WEB method, we construct a well-conditioned basis for numerical analysis. The framework is validated through a number of numerical examples of Poisson equations with hybrid boundary conditions on different implicit domains in two and three dimensions. The results reflect that the approach can achieve the optimal approximation order in solving hybrid problems. Full article
(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)
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