Document Details
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Document Type |
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Thesis |
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Document Title |
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Numerical Solution of Generalized Nonlinear Schrödinger Equation الحل العددي لمعادلة شرودنجر العامة |
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Subject |
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Faculty of Sciences |
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Document Language |
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Arabic |
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Abstract |
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The aim of this thesis is to solve numerically the generalized nonlinear Schrödinger equation using finite difference method. The flow chart of the thesis as follows:
Chapter 1: we derive the exact solution of the nonlinear Schrödinger equation, and the conserved quantities. We describe Crout's method for solving the block tridiagonal system. We describe Newton's method for solving nonlinear systems.
Chapter 2: we solve nonlinear Schrödinger equation numerically using Crank-Nicolson method. The accuracy of the resulting scheme is second order in space and time, and it is unconditionally stable. Also we used Newton's method for solving the nonlinear system. We give some numerical examples to show that this method is conserving the mass, momentum and energy conserved quantities. We give some numerical experiments, like, single soliton, collision of two and three solitons. We present another method for solving the nonlinear Schrödinger equation using Douglas idea with implicit midpoint rule, we get a scheme which is fourth order in space and second order in time, and it is unconditionally stable using von-Neumann stability analysis. We use Newton's method for solving the nonlinear system. We give some numerical examples to show that this method is conserving the mass, momentum and energy conserved quantities.
Chapter 3: we study in details the generalized nonlinear Schrödinger equation, we study the conserved quantities, so we solve the generalized nonlinear Schrödinger equation numerically by using explicit method. The accuracy of the resulting scheme is second order in space first order in time and unconditionally unstable. We give one numerical example to show that this method is conserved for the mass, momentum and energy conserved quantities and it works well for short time. We present another method for solving the generalized nonlinear Schrödinger equation numerically, using Crank-Nicolson method. The accuracy of the resulting scheme is second order in space and time and it is unconditionally stable. To overcome the difficulty of solving the nonlinear system obtained from Crank-Nicolson method, we present a fixed point method to solve this system, the scheme is of second order in time and space, it is unconditionally stable.
Chapter 4: we solve the generalized nonlinear Schrödinger equation numerically using a linearized implicit scheme. To avoid the difficulty of solving the nonlinear scheme obtained in chapter 3. A linearized implicit scheme is obtained and can be solved directly using Crout's method. The scheme is of second order accuracy in both directions time and space, and it is unconditionally stable. We give some numerical examples like chapter 3, to show that this method is conserving the mass, momentum and energy quantities. We give some experiments like, single solution, and collision of two solitons.
Chapter 5: we will give a summary of our numerical results, in this thesis and give some comments. |
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Supervisor |
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Prof. Dr. Mohammad Said Hammoudeh |
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Thesis Type |
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Master Thesis |
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Publishing Year |
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1439 AH
2017 AD |
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Added Date |
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Wednesday, October 18, 2017 |
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Researchers
| أنيس صالح حربي | Harbi, Anees Saleh | Researcher | Master | |
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