Applications of extended simple equation method on unstable nonlinear Schrödinger equations

In the present paper we compare the two methodologies for the development of exponentially and trigonometrically fitted methods. One is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),xexp (±wx),…,x
 m
 exp (±w
 x)} and the second is based on the exact integration of the functions of the form: {1,x,x
 2,…,x
 p
 ,exp (±wx),exp (±2wx),…,exp (±mwx)}. The above functions are used in order to improve the efficiency of the classical methods of any kind (i.e. the method (5) with constant coefficients) for the numerical solution of ordinary differential equations of the form of the Schrodinger equation. We mention here that the above sets of exponential functions are the two most common sets of exponential functions for the development of the special methods for the efficient solution of the Schrodinger equation. It is first time in the literature in which the efficiency of the above sets of functions are studied and compared together for the approximate solution of the Schrodinger equation. We present the error analysis of the above two approaches for the numerical solution of the one-dimensional Schrodinger equation. Finally, numerical results for the resonance problem of the radial Schrodinger equation are presented.

Exponentially and Trigonometrically Fitted Methods for the Solution of the Schrödinger Equation

High order closed Newton–Cotes trigonometrically-fitted formulae for the numerical solution of the Schrödinger equation

In this paper, we investigate the connection between • closed Newton-Cotes formulae, • trigonometrically-fitted methods, • symplectic integrators and • efficient integration of the Schrodinger equation. The study of multistep symplectic integrators is very poor although in the last decades several one step symplectic integrators have been produced based on symplectic geometry (see the relevant literature and the references here). In this paper we study the closed Newton-Cotes formulae and we write them as symplectic multilayer structures. Based on the closed Newton-Cotes formulae, we also develop trigonometrically-fitted symplectic methods. An error analysis for the one-dimensional Schrodinger equation of the new developed methods and a comparison with previous developed methods is also given. We apply the new symplectic schemes to the well-known radial Schrodinger equation in order to investigate the efficiency of the proposed method to these type of problems.

High order closed Newton-Cotes trigonometrically-fitted formulae for the numerical solution of the Schrodinger equation

In this work we construct new Runge-Kutta-Nystrom methods especially designed to integrate exactly the test equation y^''=-w^2y. We modify two existing methods: the Runge-Kutta-Nystrom methods of fifth and sixth order. We apply the new methods to the computation of the eigenvalues of the Schrodinger equation with different potentials such as the harmonic oscillator, the doubly anharmonic oscillator and the exponential potential.

New modified Runge-Kutta-Nyström methods for the numerical integration of the Schrödinger equation

In this work we consider the numerical integration of second order ODEs where the first derivative is missing. We construct two step hybrid methods with six and seven stages and seventh and eighth algebraic order. We apply the new methods on the numerical integration of several test problems.

Two step hybrid methods of 7th and 8th order for the numerical integration of second order IVPs

Fourth order Trigonometrically-fitted and Exponentially-fitted Symplectic Methods for the Numerical Integration of the Schrödinger Equation 1

The general framework of constructing trigonometrically fitted Runge‐Kutta methods based on the approach introduced by Simos [8] is presented in this paper. Methods of high exponetial order are constructed and tested on the numerical solution of the Schrodinger equation.

Conditions for Trigonometrically Fitted Runge‐Kutta Methods

Two derivative Runge Kutta methods with minimum phase-lag and amplification error

In this work we consider Trigonometrically Fitted Symplectic Runge Kutta Nystrom methods. We present a new Trigonometrically fitted symplectic three stage method with algebraic order two.

Th. Monovasilis

Papers

Two step hybrid methods of 7th and 8th order for the numerical integration of second order IVPs

Fourth order Trigonometrically-fitted and Exponentially-fitted Symplectic Methods for the Numerical Integration of the Schrödinger Equation 1

Conditions for Trigonometrically Fitted Runge‐Kutta Methods

Two derivative Runge Kutta methods with minimum phase-lag and amplification error

A Trigonometrically Fitted Symplectic Runge‐Kutta‐Nyström Method