|
Document Type |
: |
Thesis |
|
Document Title |
: |
Numerical Solution of Coupled Nonlinear Schrödinger Equations In (2+1) Dimensions الحل العددي لمعادلة شرودنجر المزدوجة غير الخطية في الابعاد (2+1) |
|
Subject |
: |
Faculty of Sciences |
|
Document Language |
: |
Arabic |
|
Abstract |
: |
The aim of this thesis is to present a highly accurate schemes to solve coupled nonlinear Schrödinger equations in (2+1) dimensions numerically and to study the properties of this scheme.
In chapter 1: We present in detail, these equations and the exact solution of the system under consideration, and also we will give how to solve the block tridiagonal system by Crout's method. We describe the fixed point method for solving the nonlinear system, also we study conserved quantities of equations.
In chapter 2: We introduce some numerical methods to solve parabolic equation in (2+1) dimensions. We start with studying the properties of some explicit schemes, which we fined that they are conditionally stable, first order in time, and second order in space. Next, we describe the Crank-Nicolson implicit method, then we will give the details of the Alternating Direction Implicit (ADI) method and present four different schemes depending on the splitting region applied to the ADI method. The schemes are Peaceman-Rachford scheme, D'yakonov scheme, Douglas Rachford scheme and Mitchell and Fairweather scheme. Analyzing the properties of these schemes, such as stability and accuracy, was under consideration in this chapter.
In chapter 3: We present the ADI method for solving numerically the coupled nonlinear Schrödinger equation. We derive Douglas Rachford scheme, Peaceman-Rachford scheme and Mitchell and Fairweather scheme for solving this system. In addition, we discuss the properties of these methods to justify their stability and accuracy.
In chapter 4: We illustrate the numerical solution obtained by using the derived methods for solving the CNLS equations and then compare the results with the exact solution through applying the infinity norm. We give some numerical examples to show that this method is conserving the conserved quantity. The exact solution are varied in some numerical examples to also show the ability of the schemes in preserving the conserved quantity during the time integration. We conclude this chapter with our final remarks and observation. |
|
Supervisor |
: |
Dr. Mohamed Said Ismail |
|
Thesis Type |
: |
Master Thesis |
|
Publishing Year |
: |
1438 AH
2017 AD |
|
Co-Supervisor |
: |
Dr. Hala Abdullah Ashi |
|
Added Date |
: |
Monday, May 29, 2017 |
|
Researchers
| فايزه أحمد الرخمي | Al-Rakhmi, Fayza Ahmed | Researcher | Master | |
|