Document Details

Document Type : Thesis 
Document Title :
Numerical Solution of Generalized Gardner Equation By High Order Compact Difference Methods
الحل العددي لمعادلة جاردنر بإستخدام طرق الفروق المدمجة عالية الرتبة
 
Subject : Faculty of Science 
Document Language : Arabic 
Abstract : The aim of this thesis is to solve numerically the Generalized Gardner Equation by high order compact difference methods. In the first chapter, we present the Generalized Gardner Equation and its exact solution, and we study the equation's conserved quantities. The solution of the penta-diagonal system and the solution of periodic Penta- diagonal system are derived. We describe the fixed point method, Newton's method and Runge Kutta of order 4 method for solving the nonlinear system. We also explain the Pade approximation. In these second chapter, we solve the Kortweg-deVries equation (KdV ) equations numerically by using implicit methods and the explicit Runge-Kutta method. The accuracy of the resulting implicit schemes are second order in time and fourth or- der in space and unconditionally stable. We use Newton's method for solving the nonlinear system obtained. In addition, we have used the explicit Runge Kutta of order 4 method where the accuracy of the resulting scheme is fourth order in both directions, space and time and it is conditionally stable. We give several numerical examples to show that these methods are conserving the conserved quantities. In the third chapter, we present several methods for solving the modified KdV equation (mKdV ) using Crank-Nicolson method and the linearization technique. We get schemes which are second order in time and fourth order in space and un- conditionally stable. We use Newton's method and fixed point method for solving the nonlinear system obtained. We use Crout's method for linear system. We give some numerical examples to show that this method is conserving quantities. In the fourth chapter, we solve the Generalized Gardner equation or the combined KdV- mKdV equation numerically by Crank-Nicolson method and linearizing the nonlinear system. The accuracy of the resulting schemes are second order in time and fourth order in space and unconditionally stable. We give some numerical examples to show that these methods are conserving the conserved quantities. In the fifth and final chapter, we offer a summary of what we have obtained in this research and some recommendations for future work. 
Supervisor : Prof. Dr. Daoud Suleiman Mashat 
Thesis Type : Doctorate Thesis 
Publishing Year : 1443 AH
2022 AD 
Co-Supervisor : Dr. Fouad Othman Mallawi 
Added Date : Monday, December 26, 2022 

Researchers

Researcher Name (Arabic)Researcher Name (English)Researcher TypeDr GradeEmail
فاخره مزعل العتيبيAlotaibi, Fakhirah MuzilResearcherDoctorate 

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