A Dirichlet problem for a coupled system of two singularly perturbed convection-diffusion ordinary differential equations

Bellew, Seamus
(2003)
A Dirichlet problem for a coupled system of two singularly perturbed convection-diffusion ordinary differential equations.
Master of Science thesis, Dublin City University.

A Dirichlet problem for a system of two singularly perturbed convection-diffusion ordinary differential equations is examined where th e two singular perturbation parameters can be of a different order. A finite difference numerical method whose solutions converge pointwise independently of the singular perturbation parameters is constructed. A full theoretical analysis is provided which shows that the numerical method is robust. This is done over a piecewise uniform fitted mesh involving two transition points.
The first differential equation has only one dependent variable while the second equation has two dependent variables. The solution to the first differential equation is present in the second differential equation and th is introduces coupling which is examined in this thesis.
The solution of the first differential equation is decomposed into regular and singular components. The numerical solution is decomposed in an analogous manner. The convergence of the numerical method is analysed separately over each component. Sharp weighted derivative estimates for each of these components are examined as these are necessary for the analysis of the second differential equation.
The solution of the second differential equation is decomposed into regular, singular and coupling components. Again the numerical solution is decomposed analogously and the convergence of the numerical method is analysed separately over each component.