A new algorithm appropriate for solving singular and singularly perturbed autonomous initial-value problems

Abstract

A nonlinear explicit scheme is proposed for numerically solving first-order singular or singularly perturbed autonomous initial-value problems (IVP) of the form y ^′=f(y). The algorithm is based on the local approximation of the function f(y) by a second-order Taylor expansion. The resulting approximated differential equation is then solved without local truncation error. For the true solution the method has a local truncation error that behaves like either 𝒪(h ³) or 𝒪(h ⁴) according to whether or not some parameter vanishes. Some numerical examples are provided to illustrate the performance of the method. Finally, an application of the method for detecting and locating singularities is outlined.

Keywords:

• Initial value-problems
• Non-standard algorithm
• Stability
• Singular problems
• Singularly perturbed problems