Explicit schemes based on rational approximant for solving first order initial value problems

Abstract

A class of rational methods of the second, third and fourth-order are proposed in this study. The formulas are developed based on a rational function with the denominator of degree one. Besides that, the concept of the closest points of approximation is also emphasized in formulating these methods. The derived methods are not self-starting; thus, an existing rational method is applied to calculate the starting values. The stability regions of the methods are also illustrated in this paper and suggest that only the second-order method is A-stable, while the third and fourth-order methods are not. The proposed formulas are examined on different problems, in which the solution possesses singularity, stiff and singularly perturbed problems. The numerical results show the capability of the proposed methods in solving problems with singularity. It also suggests that the developed schemes are more accurate than the existing rational multistep methods for problems with integer singular point. It is also shown that the derived schemes are suitable for solving stiff and singularly perturbed problems, although some of the formulas are not A-stable.