Analytical and Numerical Methods for Nonlinear Singular Singularly-Perturbed Initial Value Problems

This paper considers the vector problem ε y = f(y,t,ε ),y(0) prescribed, when the m × m matrix fy (y,t,0) is singular and of rank k,0 < _ = k m, with k stable eigenvalues. The reduced system f(Y₀ ,t,0) = 0 is, then, insufficient to determine the limiting solution Y₀ for $t > 0$. Instead, depending on the row-reduced form of fy, a differential equation is obtained for Y₀ and an initial vector Y₀ (0) is obtained as the steady-state solution of an inner problem. Alternative analytical and numerical possibilities are explored, and analytical and numerical solutions are obtained for representative problems.