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

Deadline for manuscript submissions: closed (28 February 2026) | Viewed by 7128

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Prof. Dr. Gang Li

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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

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Keywords

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

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Published Papers (7 papers)

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Research

24 pages, 8750 KB

Open AccessArticle

Finite Element Analysis for the Stationary Navier–Stokes Equations with Mixed Boundary Conditions

by Ya Cui, Qingfang Liu and Jia Liu

Mathematics 2026, 14(2), 333; https://doi.org/10.3390/math14020333 - 19 Jan 2026

Viewed by 484

Abstract

This paper studies the stationary incompressible Navier-Stokes equations with mixed boundary conditions using a velocity-pressure finite element formulation. We first establish a variational framework and prove existence of solutions under suitable regularity assumptions, followed by a Galerkin discretization with error estimates. Three iterative algorithms (the Stokes, Newton, and Oseen schemes) are then analyzed, with stability conditions and error bounds derived for each. Numerical experiments confirm the theoretical results: all methods achieve second-order convergence for velocity and pressure. Among the three schemes, the Newton iteration is the most efficient in terms of computational time, while the Oseen iteration exhibits the strongest robustness with respect to decreasing viscosity coefficients. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)

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21 pages, 918 KB

Open AccessArticle

A Numerical Approach for the Simultaneous Identification of a Source Term and a Robin Boundary Coefficient in Time-Fractional Reaction–Diffusion Equations

by Miglena N. Koleva

Mathematics 2026, 14(2), 324; https://doi.org/10.3390/math14020324 - 18 Jan 2026

Viewed by 516

Abstract

In the present study, we develop numerical approaches for the simultaneous determination of a time-dependent right-hand side and a Robin boundary coefficient in linear and quasilinear Caputo time-fractional reaction–diffusion problems based on boundary and interior observations. The well-posedness of the corresponding direct problems [...] Read more.

L 2 - 1_{σ}

scheme, and the solution is decomposed with respect to the unknown functions. The correctness of the proposed method is proved. For the nonlinear diffusion problem, a quasilinearization technique is employed, and the spatial discretization is carried out using finite difference schemes. An iterative procedure is developed to solve the resulting inverse problem. Numerical simulations with noisy data are presented and discussed to demonstrate the efficiency of the method. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)

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35 pages, 5897 KB

Open AccessArticle

An Extrinsic Enriched Finite Element Method Based on RBFs for the Helmholtz Equation

by Qingliang Liu, Zhihong Zou, Yingbin Chai, Wei Li and Wei Chu

Mathematics 2026, 14(1), 200; https://doi.org/10.3390/math14010200 - 5 Jan 2026

Cited by 1 | Viewed by 703

Abstract

The traditional finite element method (FEM) usually exhibits significant numerical dispersion error for solving the Helmholtz equation in relatively high-frequency range, resulting in insufficiently accurate solutions. To address this problem, this paper proposes a novel enriched finite element method (EFEM) based on radial basis functions (RBFs) which are frequently used in meshless numerical techniques. In the proposed method, the partition of unity (PU) framework is retained, and nodal interpolation functions are formed using the RBFs. Furthermore, the linear dependence (LD) problem commonly encountered in many of the PU-based methods using polynomial basis functions (PBFs) is effectively avoided by using the present RBFs. To enrich the approximation space generated by the RBFs, the PBFs are introduced to construct the local enrichment functions. Several typical numerical experiments are conducted in this work. The results indicate that the proposed method can significantly reduce the dispersion error and yield accurate solutions even for relatively high-frequency Helmholtz problems. More importantly, the proposed method can be directly implemented with standard quadrilateral meshes as in FEM. Therefore, the proposed method represents a promising numerical scheme for solving relatively high-frequency Helmholtz problems. Full article

(This article belongs to the Special Issue Advances in Numerical Analysis of Partial Differential Equations)

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29 pages, 592 KB

Open AccessArticle

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 966

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 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

Open AccessArticle

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 960

Abstract

This paper addresses an attraction–repulsion chemotaxis system governed by Neumann boundary conditions within a bounded domain

Ω \subset R^{3}

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 \subset R^{3}

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

Open AccessArticle

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 2 | Viewed by 1159

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 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

Open AccessArticle

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 3 | Viewed by 1567

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 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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