Uniformly convergent difference schemes for singularly perturbed differential equations

Aditya Kaushik

Jan. 26, 2010, 3:15 p.m. P 215

We consider the difference schemes for singularly perturbed ordinary differential equations(SPODE). When the perturbation parameter is very small, the solution of the problem exhibits boundary layer behaviour. In the boundary 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 development 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.