الفهرس 
A second order finite difference-spectral method for spaceOnly 14 pages are availabe for public view
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Abstract
The main goal of this thesis, which consists of six chapters, is to make a modification to the pseudo-spectral method and introduce a new procedure for solving some types of linear and nonlinear differential equations. This procedure is based on the Galerkin method and some kinds of numerical integrations. In chapter one, we present basic notations and mathematical concepts. We give a brief summary on the spectral methods, kinds of Gauss quadrature and the time integration methods. Also, we give a historical overview for the fractional calculus and the kinds of fractional differentiation. In chapter two, four numerical solutions for the Kortewegde Vries Burgers’ (KdVB) equation are presented. These methods are based on cardinal Chebyshev and Legendre basis functions with Galerkin method. Gauss-quadrature formula and El-gendi method are used to convert the problem into system of ordinary differential equations. Some numerical results are given to illustrate the efficiency and accuracy of the proposed methods. The numerical results are compared with those obtained by the pseudospectral method with preconditioning [26]. In chapter three, we apply the proposed methods which are derived in chapter two to the Generalized Burger’s—Huxley equation (GBH). This problem is more complicated than the Kortewegde Vries Burgers’ (KdVB) equation. So, we discuss this problem and present some numerical results. We compare our results with results obtained by the pseudospectral method with preconditioning and domain decomposition. In chapter four, we present the numerical solutions of coupled modified Korteweg-de Vries equations (mKdV) and the generalized Hirota–Satsuma coupled KdV equations by using El-gendi nodal Galerkin (EGG) approaches. The resulted system of ODES is solved by the fourth order Runge-Kutta solver. The convergence and the stability of these methods are analyzed numerically. The results are compared with those obtained by pseudo-spectral method. In chapter five, we introduce a new technique for solving problems which have non-smooth and discontinuous solutions. This technique depends on the division of the domain into subintervals to deal with the discontinuity. Cardinal Chebyshev basis functions are used to approximate the solution. Slope limiter is used to eliminate possible spurious oscillations in the approximate solution. Our numerical results are compared with the exact solution to show the efficiency of proposed method. In chapter six, numerical solutions of fractional time partial differential equations and the fractional space partial differential equations are presented by El-gendi Chebyshev Galerkin method. New formula of the Chebyshev fractional differentiation matrix is derived. Linear and nonlinear problems can be reduced to a set of linear algebraic equations by using the nodal Galerkin techniques. Solutions which are obtained by this method are in excellent agreement with those obtained by previous works.