In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order
ε
-uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.
In this study, a parameter-uniform numerical scheme is built and analyzed to treat a singularly perturbed parabolic differential equation involving large spatial delay. The solution of the considered problem has two strong boundary layers due to the effect of the perturbation parameter, and the large delay causes a strong interior layer. The behavior of the layers makes it difficult to solve such problem analytically. To treat the problem, we developed a numerical scheme using the weighted average ($$\theta$$
θ
-method) difference approximation on a uniform time mesh and the central difference method on a piece-wise uniform spatial mesh. We established the Stability and convergence analysis for the proposed scheme and obtained that the method is uniformly convergent of order two in the temporal direction and almost second order in the spatial direction. To validate the applicability of the proposed numerical scheme, two model examples are treated and confirmed with the theoretical findings.
This paper presents numerical treatments for a class of singularly perturbed parabolic partial differential equations with nonlocal boundary conditions. The problem has strong boundary layers at x = 0 and x = 1. The nonstandard finite difference method was developed to solve the considered problem in the spatial direction, and the implicit Euler method was proposed to solve the resulting system of IVPs in the temporal direction. The nonlocal boundary condition is approximated by Simpsons 13 rule. The stability and uniform convergence analysis of the scheme are studied. The developed scheme is second-order uniformly convergent in the spatial direction and first-order in the temporal direction. Two test examples are carried out to validate the applicability of the developed numerical scheme. The obtained numerical results reflect the theoretical estimate.
Objectives: This study focuses on some iterative algorithms (Jacobi, Gauss-Seidel, SOR, and Multigrid) for solving Poisson's equations and investigating their performance based on the computational time complexity point of view. These iterative methods are derived to solve linear systems resulting from the discretization of Poisson's equations using finite difference methods.Results: For each method, the rate of convergence is analyzed. Our analysis shows that the Multigrid method is efficient among the other methods to solve Poisson's equations. Numerical experiments are carried out to validate the theoretical results.AMS Subject Classification: Primary 65F10; 65N06; secondary 65N55
Objectives: An accurate exponential fitted numerical method is developed to solve singularly perturbed time lag problem. Solution of the problem exhibits a boundary layer as the perturbation parameter approach to zero. A priori bounds and properties on the continuous solution is discussed. Result: The backward-Euler method is applied in time direction and higher order finite difference method is employed to the spatial derivative approximation. An exponential fitting factor is induced on the difference scheme for stabilizing the computed solution. Using comparison principle, stability of the method is examined and analyzed. It is proved that the method converges uniformly with linear order of convergence both in space and time. To validate theoretical findings of the scheme, two test examples are given. Comparison is employed with the result available in the literature and it indicates that the proposed method has better accuracy than the schemes in literature.
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