The Numerical Solution of Singular Singulary-Perturbed Initial Value Problems.

We consider the vector initial value problem epsilon Y-dot = f(y,t,epsilon), Y(0) = Y superscript 0 (epsilon) in the situation when the m x m matrix fY(Y,t,0) is singular with constant rank k m and has k stable eigenvalues. We show how to determine the unique limiting solution Y sub 0 of the reduced problem f(Y sub 0, t,0) = 0 and how to obtain a uniform asymptotic expansion of the solution which is valid for small values of epsilon on finite t intervals. A numerical technique is developed to calculate the limiting solution and the results of some examples are compared with an existing code for stiff differential equations.