Uniformly convergent difference schemes for singularly perturbed differential equations

Aditya Kaushik

Jan. 26, 2010, 4:15 p.m. P 215

We consider the difference schemes for singularly perturbed ordinary dif-
ferential equations(SPODE). When the perturbation parameter is very small,
the solution of the problem exhibits boundary layer behaviour. In the boun-
dary layer region the solution changes rapidly, while away from this region the
change in the solution is moderate. This simultaneous presence of two different
scales phenomena makes the problem stiff. In this talk we present parameter
uniform difference schemes for SPODE of two types, namely:

• Convection Reaction Diffusion:
εy''(x) + a(x)y'(x − δ) + b(x)y(x) = f (x), and

• Reaction Diffusion:
ε²y'' + α(x)y'(x − δ) + w(x)y(x) + β(x)y(x + η) = f (x),

with delay (δ) as well as advance (η). The main emphasis will be on the de-
velopment of parameter uniform error estimates on piecewise mesh. Although
the analysis is restricted to the ordinary differential equations, the proposed
technique is found useful in assessing the merits of numerical solution of other
nonlinear models, as well as partial differential equations.