Uniformly convergent diﬀerence schemes for singularly perturbed diﬀerential equations
Aditya KaushikJan. 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.