Article
Mathematics, Applied
Ram Shiromani, Vembu Shanthi, J. Vigo-Aguiar
Summary: This paper investigates a class of singularly perturbed 2-D elliptic convection-diffusion partial differential equations with non-smooth convection and source terms. An efficient numerical method is developed to approximate the linear problem, with spatial discretization based on a finite difference scheme. The theoretical outcomes are supported by extensive numerical experiments, including a comparison of accuracy and computational cost of the proposed numerical method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Ram Shiromani, Vembu Shanthi, Pratibhamoy Das
Summary: In this article, a higher order convergent approximation is proposed for a class of singularly perturbed 2-D convection-diffusion-reaction elliptic problems with discontinuous convection and source terms. The proposed numerical approach utilizes a hybrid-difference technique on a layer-adaptive piecewise-uniform Shishkin mesh. Numerical experiments with various types of discontinuities are conducted to verify the sharpness of the proposed results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Sekar Elango, Bundit Unyong
Summary: This article investigates the use of two non-uniform meshes in the finite difference method for solving singularly perturbed mixed-delay differential equations. The multiplication of the second-order derivative by a small parameter creates boundary layers at x=0 and x=3, as well as strong interior layers at x=1 and x=2 due to the delay terms. It is proven that the method has almost first-order convergence on the Shishkin mesh and first-order convergence on the Bakhvalov-Shishkin mesh. Error estimates are derived in the discrete maximum norm, and practical examples are provided for validation.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Applied
J. L. Gracia, E. O'Riordan
Summary: In this study, a singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified to match the discontinuity in the initial condition and satisfy the homogenous parabolic differential equation associated with the problem. By using an upwind finite difference operator with a layer-adapted mesh, the numerical approximation of the difference between the analytical function and the solution of the parabolic problem is shown to be parameter-uniform, with numerical results illustrating the theoretical error bounds established in the paper.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Ajay Singh Rathore, Vembu Shanthi, Higinio Ramos
Summary: This article examines a singularly perturbed second-order Fredholm integro-differential equation with a discontinuous source term. An exponentially-fitted numerical method on a Shishkin mesh is applied to solve the problem. The method is proven to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
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
Sanjay Ku Sahoo, Vikas Gupta
Summary: This article investigates a singularly perturbed convection-diffusion equation with discontinuous convective and source terms. Through the use of specially designed meshes and extrapolation schemes, the problem is successfully solved and the theoretical results are verified.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2022)
Article
Mathematics, Applied
K. Aarthika, Ram Shiromani, V. Shanthi
Summary: This paper investigates a two-dimensional singularly perturbed reaction-diffusion equation with a discontinuous source term, and proposes a numerical approach using a hybrid finite difference method and a layer adapted piece-wise uniform Shishkin mesh. The convergence of this method with respect to the perturbation parameter is proven. The numerical results validate the theoretical findings.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Carmelo Clavero, Ram Shiromani, Vembu Shanthi
Summary: In this paper, we study a singularly perturbed two-dimensional convection-diffusion elliptic interface problem with Robin boundary conditions, where the source term is discontinuous. The parameter epsilon, which represents the coefficient of the highest-order terms in the differential equation and in the boundary conditions, can be arbitrarily small. Due to the presence of the discontinuous source term and the diffusion parameter, the solutions to these problems generally exhibit boundary, corner, and weak-interior layers. We apply a numerical approach using a finite-difference technique on a layer-adapted piecewise uniform Shishkin mesh to provide an accurate estimation of the error. Numerical results are presented to demonstrate the sharpness of these results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
Zhongdi Cen, Jian Huang, Aimin Xu
Summary: In this paper, a quadratic B-spline collocation method is proposed to solve a singularly perturbed semilinear reaction-diffusion problem with a discontinuous source term. The method discretizes the problem on a Shishkin-type mesh using a quadratic B-spline collocation on both sides of the discontinuous point. The obtained scheme is shown to be stable and almost second-order uniformly convergent, which is supported by numerical experiments.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2023)
Article
Mathematics, Applied
J. L. Gracia, E. O'Riordan
Summary: This article investigates a singular perturbed parabolic problem of convection-diffusion type with incompatible inflow boundary and initial conditions. When the coefficients are constant, a set of singular functions is identified to match the incompatibilities in the data and satisfy the associated homogeneous differential equation. In the case of variable coefficients and continuous boundary/initial data, a numerical method is developed with its convergence rate depending on the level of compatibility satisfied by the data. Numerical results are provided to validate the theoretical error bounds for both approaches.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Swati Yadav, Pratima Rai
Summary: In this article, a higher order numerical method is constructed and analyzed for a class of two dimensional parabolic singularly perturbed problem of convection-diffusion type with vanishing convection coefficient. The proposed scheme is proven to be uniformly convergent with respect to parameter epsilon, and exhibits high accuracy according to the numerical results.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Xiaowei Liu, Min Yang, Jin Zhang
Summary: In this paper, the weak Galerkin method on a Shishkin rectangular mesh is analyzed for a singularly perturbed convection-diffusion problem in two dimensions. The method achieves supercloseness through a specially constructed interpolant, which consists of vertices-edges-element interpolant inside the layers and modified Gauss-Radau interpolant outside the layers in the interior of each element, and vertices-edges-element interpolant inside the layers and weighted L2 projection outside the layers on the boundary of each element. Additionally, over-penalization technique is used inside the layers, and supercloseness of order k + 1/2 is proved, even up to almost k + 1 under appropriate assumptions. Numerical experiments validate the theoretical result.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
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
Meenakshi Shivhare, Pramod Chakravarthy Podila, Higinio Ramos, Jesus Vigo-Aguiar
Summary: In this paper, we studied a time-dependent singularly perturbed differential-difference equation with small shifts in the field of neuroscience. We approximated the terms containing delay and advance parameters using Taylor's series expansion. The continuous problem was semi-discretized using the Crank-Nicolson finite difference method in the time direction and quadratic B-spline collocation method in the space direction. The method was proven to have second-order uniform convergence in both space and time directions, and theoretical estimates were carried out to support the obtained numerical experiments.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Musa Ahmed Demba, Higinio Ramos, Poom Kumam, Wiboonsak Watthayu, Norazak Senu, Idris Ahmed
Summary: In this study, a trigonometrically adapted 6(4) explicit Runge-Kutta-Nystrom pair with six stages is formulated, which can integrate the usual test equation exactly. The local truncation error and the periodicity interval of the new method are calculated, demonstrating its maintained algebraic order and almost P-stability.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Sunil Kumar, Kuldeep, Higinio Ramos, Joginder Singh
Summary: An efficient hybrid numerical method is developed for solving coupled systems of singularly perturbed linear parabolic problems, with first order convergence in time and third order convergence in space. The additive scheme helps decouple the vector approximate solution components for increased computational efficiency. Numerical results confirm theoretical convergence and demonstrate the efficiency of the proposed strategy.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Mufutau Ajani Rufai, Francesca Mazzia, Higinio Ramos
Summary: This research presents an adaptive optimized one-step Nystrom method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The method employs a new approach for selecting collocation points and utilizes an embedding-like procedure to estimate the error. Numerical experiments demonstrate that the introduced error estimation and step-size control strategy outperform other existing numerical methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Computer Science, Interdisciplinary Applications
Saurabh Tomar, Mehakpreet Singh, Kuppalapalle Vajravelu, Higinio Ramos
Summary: This paper presents a novel method for calculating the Lagrange multiplier, which improves the efficiency of the variational iteration method in solving nonlinear problems.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Mathematics, Applied
Morteza Bisheh-Niasar, Higinio Ramos
Summary: This paper presents an efficient numerical approach for solving first-order delay differential equations with a piece-wise constant delay. The approach is based on a five-point hybrid block method designed for ordinary differential equations. Interpolation technique is utilized to evaluate delay terms at grid points. The paper investigates the method's characteristics such as zero stability, local truncation errors, convergence, and stability region. Numerical experiments demonstrate the efficiency and accuracy of the proposed method compared to other existing methods in the literature.
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
(2023)
Article
Mathematics, Applied
Ram Shiromani, Vembu Shanthi, Higinio Ramos
Summary: This article investigates a class of singularly perturbed two-dimensional steady-state convection-diffusion problems with Robin boundary conditions. The solutions to these problems exhibit regular boundary layers and corner layers. A numerical approach is used to provide a good approximation of the exact solutions using a finite-difference technique with a layer-adapted piecewise-uniform Shishkin mesh. Numerical examples demonstrate the accuracy of the approximations and their agreement with theoretical results.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Ajay Singh Rathore, Vembu Shanthi, Higinio Ramos
Summary: This article examines a singularly perturbed second-order Fredholm integro-differential equation with a discontinuous source term. An exponentially-fitted numerical method on a Shishkin mesh is applied to solve the problem. The method is proven to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented to validate the theoretical findings.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics
Manikandan Mariappan, Chandru Muthusamy, Higinio Ramos
Summary: This article develops and analyzes a numerical scheme to solve a singularly perturbed parabolic system of n reaction-diffusion equations. The scheme considers m equations with a perturbation parameter and the rest without it. It uses finite difference approximations on a uniform mesh in the temporal variable and a piecewise uniform Shishkin mesh in the spatial variable. Convergence properties and error analyses are derived, and numerical experiments are presented to support the theoretical results.
Article
Mathematics
Mufutau Ajani Rufai, Higinio Ramos
Summary: This research article introduces an efficient method for integrating Lane-Emden-Fowler equations of second-order singular initial value problems (SIVPs) using a pair of hybrid block methods with a variable step-size mode. The method pairs an optimized Nystrom technique with a set of formulas applied at the initial step to circumvent the singularity at the beginning of the interval. The variable step-size formulation is implemented using an embedded-type approach, resulting in an efficient technique that outperforms its counterpart methods that used fixed step-size implementation. The numerical simulations confirm the better performance of the variable step-size implementation.
Article
Mathematics, Applied
R. I. Abdulganiy, H. Ramos, J. A. Osilagun, S. A. Okunuga, Sania Qureshi
Summary: This research presents a fourth-order convergent functionally-fitted block hybrid Falkner method based on interpolation and collocation for solving the approximate solution of the Kepler equations and related problems. The proposed method achieves high stability and accuracy.
COMPUTATIONAL & APPLIED MATHEMATICS
(2023)
Article
Multidisciplinary Sciences
Higinio Ramos, Mufutau Ajani Rufai, Bruno Carpentieri
Summary: This paper introduces an efficient approach for solving Lane-Emden-Fowler problems, which utilizes two Nystrom schemes for integration and generates simultaneous approximations by solving an algebraic system of equations, outperforming existing numerical methods.
Article
Mathematics, Interdisciplinary Applications
Mufutau Ajani Rufai, Bruno Carpentieri, Higinio Ramos
Summary: This paper presents a new hybrid block method for solving initial value problems of ODEs and time-dependent partial differential equations in applied sciences and engineering. The proposed method uses an adaptive stepsize strategy to control the estimated error, and numerical simulations show that it is more efficient than other existing numerical methods.
FRACTAL AND FRACTIONAL
(2023)
Article
Mathematics, Interdisciplinary Applications
Ahmed Mohamed Hassan, Higinio Ramos, Osama Moaaz
Summary: The present article aims to study the oscillatory properties of a class of second-order dynamic equations on time scales. The noncanonical case, which received less attention compared to the canonical dynamic equations, is considered in this study. The approach adopted involves converting the noncanonical equation to a corresponding canonical equation. By using this transformation and employing various techniques, new, more effective, and sharp oscillation criteria are created. The effectiveness and importance of the results are explained by applying them to specific cases of the studied equation.
FRACTAL AND FRACTIONAL
(2023)
Article
Computer Science, Interdisciplinary Applications
Nikhil Sriwastav, Amit K. Barnwal, Higinio Ramos, Ravi P. Agarwal, Mehakpreet Singh
Summary: In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation is proposed for approximating a class of two-point singular boundary value problems (SBVPs) with Dirichlet and Neumann-Robin boundary conditions. The approach involves making an initial guess in contrast to the boundary conditions, solving the initial value problem using the Legendre wavelet operational matrix method, and iteratively improving the initial condition using a shooting projection method until the desired accuracy is achieved.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2024)