Solving third-order singularly perturbed problems using rational pseudospectral methods

For problems whose solutions have several pairs of singularities close to the computationalinterval, usually characterised by layers, an improvement on the exponentialconvergence rate of rational pseudospectral methods is presented. For this propose,a class of linear singularly perturbed two-point boundary value problems (BVPs) forthird-order ordinary di erential equations (ODEs) with a small positive parametermultiplying the highest derivative are considered. A numerical method is described inthis paper to solve such problems. In this method, at rst, the given third-order BVPtransformed into a system of two ODEs of rst and second-order subject to suitableinitial and boundary conditions. Then, the rational pseudospectral method is applied tosolve the system of ODEs, where the collocation points are the transformed Chebyshevpoints by using a new map, that proposed recently by the same author. This map causesan improvement on the exponential convergence rate of rational pseudospectral methodsthrough enlargement of the ellipse of analyticity of the underlying solutions of system.To obtain the parameters of the using map, a analysis on the zeroth-order asymptoticexpansion of the solution of the BVP is given to determine the location and width ofboundary layer of the given problem.

کلیدواژه ها:

Singularly perturbed problem ، Rational Pseudospectral method ، Boundary layers