On the use of nonstandard finite difference methods†

Abstract

Many real life problems are modelled by differential equations, for which analytical solutions are not always easy to find. One of the most difficult problems is how to solve these differential equations efficiently. Several researchers have tried to do this in various different ways (e.g. via Finite Element Methods, Standard Finite Difference Methods, Spline Approximation Methods, etc.). In recent years, to get reliable results with less effort, researchers have applied nonstandard finite difference methods (NSFDMs) and obtained competitive results to those obtained with other methods. In this survey article, the author tries to provide as much stimulating information as available regarding these NSFDMs to the researchers, which will be helpful for them as the research proceeds in this direction. While the author made the utmost efforts to include whatever he could, he would like to apologize if there are any omissions which are totally unintentional.