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

Abstract

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.

An exponentially fitted spline method for singularly perturbed parabolic convection-diffusion problems with large time delay

“…They have shown that the proposed scheme is ε-uniform convergent of second-order accurate in the temporal direction and the first-order (up to a logarithmic factor) accurate in the spatial direction. Gelu and Duressa [21] proposed an implicit Euler method for time derivative with uniform mesh and extended cubic B-spline collocation method for space derivative on Shishkin mesh to solve singularly perturbed delay parabolic reaction-diffusion problem subject to mixed boundary conditions. The result obtained is shown to be accurate of O ∆t + N −2 ln 2 N by preserving an -uniform convergence.…”

“…Regarding the study of uniform convergence solution, a large amount of literature exists for singularly perturbed non-delay differential equations and many assumptions have been made on the stability [9,12,16]. There have been extensive development studies of singular perturbation problems in the classes of singularly perturbed systems [10,17,18,19,30], singular perturbation problems with two small parameters [4,6,11,27], singular perturbation problems with non-local boundary condition [20], singular perturbation problems with Robin type boundary condition [14,21], singularly perturbed system with Robin type boundary condition [15]. However, in recent years there has been a growing interest in the numerical study of singularly perturbed partial differential equations with delay both in space and time.…”

On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

“…In some instances, the employment of numerical methods is an appropriate alternative. Some of the well-known numerical and analytical techniques used for nonlinear PDEs include the spectral method [13], the finite element method [14], the collocation method [15][16][17][18], Adomian's decomposition method (ADM) [19,20], the variational iteration method (VIM) [21][22][23], and the homotopy perturbation method (HPM) [24,25]. ADM has been applied to solve non-linear equations in [19,20] by separating the equation into linear and non-linear components.…”

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

“…Singh et al [13] developed an overlapping Schwarz domain decomposition method for parabolic reaction diffusion problems with time delay. An extended cubic B-spline collocation method on Shishkin mesh has been developed for singularly perturbed delay parabolic reaction diffusion problem with mixed-type boundary conditions in [14]. A numerical method is developed for solving a singularly perturbed parabolic delay differential equation whose solution exhibits a boundary layer behaviour in [15] and singularly perturbed parabolic differential difference equations arising in computational neuroscience in [16].…”