A coupled system of singularly perturbed parabolic reaction-diffusion equations

Abstract

In this paper systems with an arbitrary number of singularly perturbed parabolic reaction-diffusion equations are examined. A numerical method is constructed for these systems which involves an appropriate layeradapted piecewise-uniform mesh. The numerical approximations generated from this method are shown to be uniformly convergent with respect to the singular perturbation parameters. Numerical experiments supporting the theoretical results are given.

An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems

“…The values of r 1,1 (0), r 1,1 (1) are calculated from (14). Then, using the estimates given in Lemma 2, it holds…”

“…Both extensions are not trivial because the definition of the method must be careful to obtain an inverse monotone scheme giving a high order approximation. Besides in [14,13] central differences are used to solve singularly perturbed systems with an arbitrary number of equations, and it is shown the difficulties in the study of the convergence with respect to the case of two equations. Finally, we find another important issue in the analysis of the asymptotic behavior of the derivatives up to sixth order of the regular and singular part of the solution which will be crucial to prove the uniform convergence of the numerical method.…”

“…In [13], for reaction-diffusion systems with an arbitrary number of equations, second order of uniform convergence of central differences scheme was proved. In [14] the analysis was extended to a parabolic reaction-diffusion system also with an arbitrary number of equations, proving first order of uniform convergence for a scheme combining the Euler method in time and central differences in space.…”

An asymptotic-numerical hybrid method for singularly perturbed system of two-point reaction-diffusion boundary-value problems

“…In recent years various methods were employed to obtain approximations to the solution of this kind of problems. In [2,11], finite difference methods (FDM) for singularly perturbed convection-diffusion systems, in [18,19] finite element method (FEM) and in [6,13,21,23] finite difference * Correspondence: suleyman.cengizci@antalya.edu.tr 2010 AMS Mathematics Subject Classification: 65L10, 65L11 methods for singularly perturbed reaction-diffusion systems are examined. For singularly perturbed system of reaction-diffusion BVPs, in [22], the authors provided a robust computational technique , whereas in [9], optimal order error estimates are obtained on equidistributed grids.…”

Parameter uniform numerical method for a system of two coupled singularly perturbed parabolic convection-diffusion equations

“…If A ≡ 0, the above problem (1) becomes a system of singularly perturbed parabolic reaction-diffusion equations. For these problems, some robust convergence numerical approaches on layer-adapted meshes are available in the literature, e.g., Shishkina and Shishkin [1,2], Gracia et al [3][4][5][6], Franklin et al [7].…”