Article
Mathematics, Applied
Jin Zhang, Xiaoqi Ma, Yanhui Lv
Summary: In this paper, a singularly perturbed convection diffusion problem is discussed, and a finite element method on Shishkin mesh is constructed to address the issue of uniform convergence. The paper proves the minimum principle and stability result, and derives asymptotic expansion of the solution to establish a priori estimates. Uniform convergence of almost order k in the energy norm is proven, with k being the order of piecewise polynomials.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Abhay Kumar Chaturvedi, S. Chandra Sekhara Rao
Summary: This article investigates a two-dimensional singularly perturbed convection-reaction-diffusion interface problem with discontinuities in the coefficients and source term. A Local Discontinuous Galerkin method is constructed on a Shishkin mesh, and the error in the computed solution converges at a rate of O((N-1ln N)r+12). Numerical results are provided to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
S. Chandra Sekhara Rao, Abhay Kumar Chaturvedi
Summary: This article investigates a two-dimensional singularly perturbed convection-reaction-diffusion problem with discontinuities, proposing a decomposition of the solution and constructing a finite difference scheme on an appropriate Shishkin mesh. The numerical results support the theoretical conclusions.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this paper, a singularly perturbed convection-diffusion problem with a discontinuous convection is discussed. The interior layer appearing in the solution due to this discontinuity is solved using a streamline diffusion finite element method on Shishkin mesh, and the optimal order of convergence in a modified streamline diffusion norm is derived. Numerical results are presented to validate the theoretical conclusion.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Swati Yadav, Pratima Rai
Summary: This article investigates a one-dimensional singularly perturbed parabolic convection-diffusion problem with an interior turning point, where the convection coefficient vanishes inside the spatial domain and exhibits an interior layer. A higher order numerical method is developed and rigorously analyzed for the numerical solution, showing almost second order ε-uniform convergence. The method's higher accuracy and convergence rate are confirmed through numerical experiments on two test problems, with comparisons to previous schemes for similar problems.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Article
Computer Science, Interdisciplinary Applications
Ram Shiromani, Vembu Shanthi, Higinio Ramos
Summary: In this article, a two-dimensional singularly perturbed convection-reaction-diffusion elliptic type problem is investigated, where the diffusion and convection terms are multiplied by two parameters epsilon and mu, respectively. The source term in the problem exhibits jump discontinuities along the x- and y-axis. The solutions to such problems exhibit boundary and corner layers due to the presence of perturbation parameters. The suitable numerical approach for solving this problem, which includes interior layers due to the discontinuity in the source term, is the main focus of this article. The proposed method utilizes an upwind finite-difference technique with an appropriate layer-adapted piecewise uniform Shishkin mesh, and examples are presented to demonstrate its effectiveness and agreement with theoretical analysis.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Mustafa Kudu, Ilhame Amirali, Gabil M. Amiraliyev
Summary: This paper investigates a class of parameterized singularly perturbed problems with integral boundary condition and proposes a finite difference scheme of hybrid type with an appropriate Shishkin mesh. It is proven that the method converges almost second order in the discrete maximum norm, which is illustrated by numerical results supporting the theoretical findings.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Xiaoqi Ma, Jin Zhang
Summary: In this study, we investigate the supercloseness property of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh for a singularly perturbed convection diffusion problem. We propose a new composite interpolation method that combines Gaul3 Radau projection outside the layer and Gaul3 Lobatto projection inside the layer. By selecting appropriate penalty parameters at different mesh points, we obtain the supercloseness of k + 21th order (k >= 1) in an energy norm.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Pramod Chakravarthy Podila, Vishwas Sundrani, Higinio Ramos
Summary: This paper presents an efficient wavelet-based numerical scheme for solving fourth-order singularly perturbed boundary value problems with discontinuous data. The method is computationally fast, cheap, and reliable, and can be easily extended to similar classes of problems.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
Wen Lian, Zhanbing Bai
Summary: This article considers a class of singularly perturbed fourth order nonlinear differential equation, and investigates the existence of solutions to boundary value problem (BVP) without singularly perturbed using nonlinear analysis method and upper and lower solution theory. In addition, it examines the existence, uniqueness, and asymptotic estimation of the solution to singularly perturbed boundary value problem (SPBVP) based on differential inequality techniques and appropriate upper-lower solutions.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Computer Science, Artificial Intelligence
T. E. Simos, Ioannis Th Famelis
Summary: The training algorithm adapts the training points grid to be more tense in areas of interest, leading to very accurate Neural Network solutions that produce smaller errors compared to competitors in various test problems.
NEURAL COMPUTING & APPLICATIONS
(2022)
Article
Mathematics
Mirzakulova Aziza Erkomekovna, Dauylbayev Muratkhan Kudaibergenovich
Summary: This paper investigates the two-point integral boundary value problem with boundary jumps for a third order linear integro-differential equation with a small parameter at the two highest derivatives. An asymptotic expansion of the solution is constructed, which provides accuracy with respect to the small parameter. A justification of the asymptotic is also provided.
MISKOLC MATHEMATICAL NOTES
(2023)
Article
Mathematics, Applied
Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke
Summary: This study focuses on weak boundary layer solutions to singularly perturbed ODE systems, including systems with spatial non-smoothness. Unlike smooth problems, the asymptotic convergence rates do not hold true in general for non-smooth cases, and specific local uniqueness and stability conditions differ in the vectorial case. Adjustments are needed for the monotonicity condition to ensure stability in vectorial cases compared to the scalar case.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics, Applied
Kumar Rajeev Ranjan, S. Gowrisankar
Summary: This paper introduces numerical methods for singularly perturbed convection-diffusion problems with a turning point, developing a non-symmetric discontinuous Galerkin finite element method for both boundary layers and cusp-type interior layers. The uniform error estimates in L-2-norm and DG-norm are obtained and confirmed through numerical experiments.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Hongliang Li, Pingbing Ming
Summary: This paper proposes an asymptotic-preserving finite element method for solving a fourth order singular perturbation problem, which preserves the asymptotic transition of the underlying partial differential equation. The NZT element is analyzed as a representative, and a linear convergence rate is proved for the solution with sharp boundary layer. Numerical examples in two and three dimensions are consistent with the theoretical prediction.
APPLIED MATHEMATICS LETTERS
(2024)
Article
Mathematics, Applied
Thai Anh Nhan, Scott MacLachlan, Niall Madden
NUMERICAL ALGORITHMS
(2018)
Article
Mathematics, Applied
Thai Anh Nhan, Martin Stynes, Relja Vulanovic
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Thai Anh Nhan, Relja Vulanovic
Summary: In this study, a linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is discretized using the original Bakhvalov mesh. The analysis of the error in the numerical solution shows first-order pointwise accuracy, which is consistent across perturbation parameters. This is the first comprehensive analysis for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh, and numerical experiments confirm the theoretical findings.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Relja Vulanovic, Thai Anh Nhan
Summary: This study considers the Kellogg-Tsan decomposition of the solution to the linear one-dimensional singularly perturbed convection-diffusion problem and improves it by including the solution of the corresponding reduced problem. The upwind scheme on a modified Shishkin-type mesh is used to approximate the unknown component of the decomposition. It is proved that the error is O(epsilon(ln epsilon)N-2(-1)), where epsilon is the perturbation parameter and N is the number of mesh steps, demonstrating high accuracy of the method through numerical examples.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Physics, Multidisciplinary
Vinh Quang Mai, Thai Anh Nhan, Zakia Hammouch
Summary: Enzymes are biological catalysts that accelerate biochemical reactions in living organisms, and cells use regulatory mechanisms like enzymatic inhibition to control metabolite concentrations. A novel mathematical model describing enzymatic non-competitive inhibition by product was developed, providing insights into reaction velocity and Michaelis-Menten constant. The model's asymptotic solutions were found to align well with numerical solutions, and a global sensitivity analysis identified key mechanisms affecting enzyme activities.
Article
Mathematics, Applied
Thai Anh Nhan, Relja Vulanovic
Summary: This paper discusses a finite-difference method for solving linear singularly perturbed convection-diffusion problems in one dimension. The previous proof of parameter-uniform convergence using the truncation-error and barrier-function approach on Bakhvalov-type meshes is extended to a hybrid second-order scheme. A new representation of the meshes is also proposed.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Le Phuong Quan, Thai Anh Nhan
ARABIAN JOURNAL OF MATHEMATICS
(2020)
Article
Education & Educational Research
Hien Nhan, Thai Anh Nhan
EDUCATION SCIENCES
(2019)
Article
Mathematics, Interdisciplinary Applications
Le Phuong Quan, Thai Anh Nhan
MATHEMATICAL AND COMPUTATIONAL APPLICATIONS
(2018)
