scholarly journals A Uniformly Convergent Collocation Method for Singularly Perturbed Delay Parabolic Reaction-Diffusion Problem

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Fasika Wondimu Gelu ◽  
Gemechis File Duressa
Keyword(s):  
Collocation Method ◽  
Reaction Diffusion ◽  
Diffusion Problem ◽  
Euler Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
Step Size ◽  
Uniformly Convergent ◽  
Type Boundary

In this article, a numerical solution is proposed for singularly perturbed delay parabolic reaction-diffusion problem with mixed-type boundary conditions. The problem is discretized by the implicit Euler method on uniform mesh in time and extended cubic B-spline collocation method on a Shishkin mesh in space. The parameter-uniform convergence of the method is given, and it is shown to be ε -uniformly convergent of O Δ t + N − 2 ln 2 N , where Δ t and N denote the step size in time and number of mesh intervals in space, respectively. The proposed method gives accurate results by choosing suitable value of the free parameter λ . Some numerical results are carried out to support the theory.

B-SPLINE COLLOCATION METHOD FOR A SINGULARLY PERTURBED REACTION–DIFFUSION PROBLEM USING ARTIFICIAL VISCOSITY

2009 ◽  
Vol 06 (01) ◽  
pp. 23-41 ◽  
Author(s):  
MOHAN K. KADALBAJOO ◽  
PUNEET ARORA
Keyword(s):  
Exact Solution ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Artificial Viscosity ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method

In this paper, we develop a B-spline collocation method using artificial viscosity for solving a class of singularly perturbed reaction–diffusion equations. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh, and use the B-spline collocation method, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method, with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

scholarly journals An Efficient Computational Method for Singularly Perturbed Delay Parabolic Partial Differential Equations

2021 ◽  
Vol 15 ◽  
pp. 105-117
Author(s):  
Imiru Takele Daba ◽  
Gemechis File Duressa
Keyword(s):  
Numerical Scheme ◽  
Euler Method ◽  
Computational Method ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
Fine Mesh ◽  
Theoretical Predictions ◽  
Uniformly Convergent

In this communication, a parameter uniform numerical scheme is proposed to solve singularly perturbed delay parabolic convection-diffusion equations. Taylor’s series expansion is applied to approximate the shift term. Then the resulting singularly perturbed parabolic convection-diffusion equation is solved by utilizing the implicit Euler method for temporal discretization on uniform mesh and hybrid numerical scheme based on a midpoint upwind scheme in the coarse mesh regions and a cubic spline method in the fine mesh regions on a piecewise uniform Shishkin mesh for the spatial discretization. The proposed numerical scheme is shown to be an ε−uniformly convergent accuracy of first-order in time and almost second-order in space directions. Some test examples are considered to testify the theoretical predictions.

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