Modified Crank-Nicolson Difference Schemes for Nonlocal Boundary Value Problem for the Schrödinger Equation

Allaberen Ashyralyev,Ali Sirma

doi:10.1155/2009/584718

Abstract

The nonlocal boundary value problem for Schrödinger equation in a Hilbert space is considered. The second-order of accuracy -modified Crank-Nicolson difference schemes for the approximate solutions of this nonlocal boundary value problem are presented. The stability of these difference schemes is established. A numerical method is proposed for solving a one-dimensional nonlocal boundary value problem for the Schrödinger equation with Dirichlet boundary condition. A procedure of modified Gauss elimination method is used for solving these difference schemes. The method is illustrated by numerical examples.

Highlights

In this article, the nonlocal boundary value problem for the Schrodinger equation iu t Au t f t, 0 < t < T, p u0 αmu λm φ, m10 < λ1 < λ2 < · · · < λp ≤ T in a Hilbert space H with the self-adjoint operator A is considered
The Schrodinger equation plays an important role in the modeling of many phenomena
The main aim of this paper is to study r modified Crank-Nicolson difference schemes for the approximate solution of problem 1.1

Summary

Introduction

The nonlocal boundary value problem for the Schrodinger equation iu t Au t f t , 0 < t < T, p u0 αmu λm φ, m10 < λ1 < λ2 < · · · < λp ≤ T in a Hilbert space H with the self-adjoint operator A is considered. The nonlocal boundary value problem for the Schrodinger equation iu t Au t f t , 0 < t < T, p u0 αmu λm φ, m1. The Schrodinger equation plays an important role in the modeling of many phenomena. Schrodinger equation have been studied extensively by many researchers see, e.g., 1–9 and the references given therein. In the articles 2, 3 the existence and the uniqueness of the solution of the nonlocal boundary value problem 1.1 and its general form under some conditions are studied. In the article 8 , to find an approximate solution of the problem 1.1 , first-order of accuracy Rothe difference scheme and second-order of accuracy Crank-Nicolson difference scheme are presented. The stability estimates for the solution of this problem and the stability of these difference schemes are established

Objectives

Methods

Results