Algorithms based on a 3-derivative method for singular differential equations including equations with blow-up solutions

Abstract

We propose block algorithms based on a 3-derivative Nyström-type method (TDNM) for integrating singular second-order ordinary differential equations (ODEs), including parabolic and hyperbolic partial differential equations (PDEs) with blow-up solutions. The block algorithms are implemented in a block-by-block mode for initial value problems (IVPs) and block unification mode for boundary value problems (BVPs). In particular, the algorithms are applied to PDEs by first reducing them into ODEs via the method of lines, whereby the space variable is discretized to yield a system of initial value problems, or the time variable is discretized to yield a system of boundary value problems. It is shown that the numerical solution given by the block-by-block and block unification algorithms are superior to those available in the literature. It is also demonstrated that the implementation of the variable step version of the TDNM, which is generally recommended for IVPs, has no advantage over the block-by-block algorithm.