CUBIC SPLINE ADAPTIVE WAVELET SCHEME TO SOLVE SINGULARLY PERTURBED REACTION DIFFUSION PROBLEMS

In this paper, the collocation method proposed by Cai and Wang1 has been reviewed in detail to solve singularly perturbed reaction diffusion equation of elliptic and parabolic types. The method is based on an interpolating wavelet transform using cubic spline on dyadic points. Adaptive feature is performed automatically by thresholding the wavelet coefficients. Numerical examples are presented for elliptic and parabolic problems. The purposed method comes up as a powerful tool for studying singular perturbation problems in term of effective grid generation and CPU time.

Download Full-text

An Almost Sixth-Order Finite-Difference Method for Semilinear Singular Perturbation Problems

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2004-0020 ◽

2004 ◽

Vol 4 (3) ◽

pp. 368-383 ◽

Cited By ~ 5

Author(s):

Relja Vulanović

Keyword(s):

Finite Difference ◽

Reaction Diffusion ◽

Perturbation Parameter ◽

Singularly Perturbed ◽

Sixth Order ◽

Finite Difference Schemes ◽

Hybrid Scheme ◽

Perturbation Problems ◽

High Order Finite Difference ◽

Diffusion Problems

AbstractThe discretization meshes of the Shishkin type are more suitable for high- order finite-difference schemes than Bakhvalov-type meshes. This point is illustrated by the construction of a hybrid scheme for a class of semilinear singularly perturbed reaction-diffusion problems. A sixth-order five-point equidistant scheme is used at most of the mesh points inside the boundary layers, whereas lower-order three-point schemes are used elsewhere. It is proved under certain conditions that this combined scheme is almost sixth-order accurate and that its error does not increase when the perturbation parameter tends to zero.

Download Full-text

FINITE DIFFERENCE SCHEME FOR A SINGULARLY PERTURBED PARABOLIC EQUATIONS IN THE PRESENCE OF INITIAL AND BOUNDARY LAYERS

Mathematical Modelling and Analysis ◽

10.3846/1392-6292.2008.13.483-492 ◽

2008 ◽

Vol 13 (4) ◽

pp. 483-492

Author(s):

Nicolas Cordero ◽

Kevin Cronin ◽

Grigorii Shishkin ◽

Lida Shishkina ◽

Martin Stynes

Keyword(s):

Difference Scheme ◽

Boundary Layers ◽

A Priori Estimates ◽

A Priori ◽

Reaction Diffusion ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Singularly Perturbed ◽

Reaction Diffusion Equation ◽

Parabolic Problems

The grid approximation of an initial‐boundary value problem is considered for a singularly perturbed parabolic reaction‐diffusion equation. The second‐order spatial derivative and the temporal derivative in the differential equation are multiplied by parameters å 2 1 and å 2 2, respectively, that take arbitrary values in the open‐closed interval (0,1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial‐boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor‐product grid, piecewise‐uniform in xand t, a difference scheme is constructed that converges å‐uniformly at the rate O(N−2 ln2 N + N0 −1 ln N0 ), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.

Download Full-text