Exponentially-fitted pseudo Runge-Kutta method

Abstract

This article is devoted to the development of an embedded pseudo-Runge-Kutta method of order three (EPRK3) and its exponential-fitting. The motivation behind the development is to minimise the cost of computation for existing Runge-Kutta (RK) type method and also to make EPRK method compatible to solve the initial value problem (IVP) having periodic solutions. Here we assume that ef-EPRKM exactly integrates two exponential functions e±ωx, with unknown frequency ω. The proposed methods are applied to two IVPs of order two. The computation cost in terms of total function evaluations is compared in Table 1. In Tables 2 and 3, a comparison of norms of endpoint errors is made between EPRK3 method, ef-EPRK3 method, Berghe's ef-RK method and Simos's ef-RK method from which it is quite evident that errors by ef-EPRKM are smallest. To compute unknown frequency ω in e±ωx, the local truncation error (LTE) for ef-EPRKM is computed.