Numerical Solution of Hirota Equation

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المملكة العربية السعودية وزارة التعليم العالي أم القرى جامعة العلوم التطبيقية فرع الطالبات كلية الرايضيات قسم العليا ادلراسات " " אא Numerical Solution of Hirota Equation { } א א. أستاذ التحليل العددي المشارك بجامعة المكل عبد العزيز بجده كلية العلوم قسم الرايضيات ٢٠٠٩/١٤٣٠
ABSTRACT Numerical Solution of Hirota Equation The aim of this thesis is to solve numerically the Hirota equation using finite difference method. This dissertation consists of three chapters and a list of references. In Chapter 1: we present in details, this equation and the exact solution, also we study its conserved quantities. The solution of the block penta-diagonal system and block septa-diagonal system are derived. We described the fixed point method and Newton’s method for solving the nonlinear system. In Chapter 2: we solve the Hirota equation numerically by using Crank-Nicolson method. The accuracy of the resulting scheme is second order in space and time and is unconditionally stable. Newton’s and fixed point methods are used for solving the nonlinear system. Also we use a linearized implicit technique. We give the some numerical examples to show that this method is conserving the conserved quantities. We give some experiments, like, single soliton and collision of two solitons. In Chapter 3: we use the same approach as we did in the previous chapter, but this time we are approximating the space derivatives in the Hirota equation by a fourth order replacement. The accuracy of the resulting scheme is fourth order in space and second order in time and is unconditionally stable. Newton’s and fixed point methods are used for solving the nonlinear system. Also we use a linearized implicit technique. We give the some numerical examples to show that this method is conserving quantities. We give some experiments like the one mentioned in chapter 2. - 1 -
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Numerical Solution of Hirota Equation

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