Solving frontier problems modelled by nonlinear partial differential equations

Abstract

We present a further development of the decomposition method [1,2], which leads to a single efficient and global method of solving linear or nonlinear, ordinary or partial differential equations for initial-value or boundary-value problems. No linearization, perturbation, or resort to discretized methods is involved. Potential savings in computation are very large (perhaps six orders of magnitude in some cases) and important implications exists for modeling and computational analysis. It is to be noted that once we realize that we can be less constrained by the mathematics by removing the necessity of techniques such as linearization, perturbation, or discretization to make analysis and computation of the models feasible and practical; we become able to develop more sophisticated and realistic models. Modeling effectively means retention of essential features while striving for simplicity so that the resulting equations can be solved. With fewer limitations imposed to achieve tractibility, the models can be more realistic and we have a convinient and global technique for solution.