On Exponentially Fitted Improved Runge-Kutta Method of Order Five for Solving Oscillatory Problems

Abstract

An exponentially fitted Improved Runge-Kutta (IRK) method with 5 stages is constructed. The new method integrates exactly differential equations where the solution is oscillatory, that is, whose solutions can be expressed as linear combinations of the set of functions {𝑒𝜆𝑡 , 𝑒−𝜆𝑡 }, 𝜆 ∈ ℂ, and in particular {sin(𝜔𝑡) , cos(ωt)}, where λ = 𝑖𝜔, 𝜔 ∈ ℝ. Analysis of algebraic order of the new method indicates that it is of order five like the classical five stages IRK method. Numerical experiments established the effectiveness and efficiency of the exponentially fitted method over the non-fitted method.