Numerical techniques for a class of singular perturbation problems
An efficient two part numerical method is derived for approximate solution of a singularly perturbed second order boundary value problem for an ordinary differential equation. The system considered has the form (i) \(\epsilon y''+a(x)y'+b(x)y=f(x)\), \(0<x<1\), (ii) \(y(0)=\alpha\), \(y(1)=\beta\). The procedure used is to replace the given equation with a corresponding differential equation of first order having a small retarding argument \(\delta\). Then the new equation is approximated for various \(\delta\) at mesh points by a tridiagonal matrix system which is solved by a discrete invariant embedding technique. Solutions of the retarded system are computed for a sequence of the parameter \(\delta_ n\) until \(| y(x,\delta_{n+1})-y(x,\delta_ n)| <\sigma\) where \(\sigma\) is a given tolerance. Numerical results presented for three examples with known solutions indicate that the method leads to good approximations. Two methods are given for setting up the equation with a retarding argument.
singularly perturbed second order boundary value
scientific article; zbMATH DE number 4153800
scientific article