Exponentially fitted explicit fifth order improved Runge-Kutta method for solution of initial value problems with oscillatory behaviour

Abstract

In this article, we constructed exponentially fitted Improved Runge-Kutta (EFIRK) method for the numerical integration of initial value problems whose solutions are linear combinations of functions of the form {x^j e^ωx,x^j e^(-ωx) },(ω∈R or iR,j=0,1,⋯jmax), where 0≤jmax≤⌊s/2 -1⌋,s being the number of stages of the method. The number is of order five with five stages, wherein the coefficients depend on the frequency and step size. A heuristically chosen algorithm is adopted to determine the frequency (ω) in each integration interval [x_n,x_(n+1)]. The results of numerical experiments with sample initial value problems with oscillatory solutions established the superiority of the exponentially fitted method over the non-fitted method.