Abstract
Exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type (MRKT) methods for solving y′′′x=fx,y,y′ are derived in this paper. These methods are constructed which exactly integrate initial value problems whose solutions are linear combinations of the set functions eωx and e-ωx for exponentially fitted and sinωx and cosωx for trigonometrically fitted with ω∈R being the principal frequency of the problem and the frequency will be used to raise the accuracy of the methods. The new four-stage fifth-order exponentially fitted and trigonometrically fitted explicit MRKT methods are called EFMRKT5 and TFMRKT5, respectively, for solving initial value problems whose solutions involve exponential or trigonometric functions. The numerical results indicate that the new exponentially fitted and trigonometrically fitted explicit modified Runge-Kutta type methods are more efficient than existing methods in the literature.
Highlights
IntroductionThis work deals with exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods for solving third-order ordinary differential equations (ODEs) y (x) = f (x, y (x) , y (x)) , y (x0) = y0, y (x0) = y0 , y (x0) = y0,
This work deals with exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods for solving third-order ordinary differential equations (ODEs) y (x) = f (x, y (x), y (x)), y (x0) = y0, y (x0) = y0, y (x0) = y0, (1) x ≥ x0.This sort of problems is often found in numerous physical problems like thin film flow, gravity-driven flows, electromagnetic waves, and so on
Paternoster [1] developed Runge-Kutta-Nystrom methods for ODEs with periodic solutions based on trigonometric polynomials
Summary
This work deals with exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods for solving third-order ordinary differential equations (ODEs) y (x) = f (x, y (x) , y (x)) , y (x0) = y0, y (x0) = y0 , y (x0) = y0,. Kalogiratou et al [[4, 5]] constructed trigonometrically and exponentially fitted Runge-Kutta-Nystrom methods for the numerical solution of the Schrodinger equation and related problems which is eighth algebraic order. In this paper we construct explicit exponentially fitted and trigonometrically fitted modified Runge-Kutta type methods with four-stage fifth-order, called EFMRKT5 and TFMRKT5, respectively. This section discusses the oscillatory and nonoscillatory properties of the third-order linear differential equation y (x) + p (x) y + q (x) y = 0.
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