الفهرس 
Chapter 1 IntroductionOnly 14 pages are availabe for public view
 |  | from | 167 |  |  |
| | | from | 167 | | |
Abstract
Nonlinear equations describe the most phenomena in real world. Large class of these equations have no an analytic solution, so numerical methods are used to handle these equations.Many physical phenomena are described by the interaction between reaction mechanisms, convection effects and diffusion transports. from a physical point of view, the convection-diffusion process and the diffusion-reaction process are quite fundamental in describing a wide variety of problems in physical, chemical, biological and engineering sciences. Some nonlinear partial differential equations (NPDEs) that model these processes provide many new insights into the question of interaction of nonlinearity, convection and diffusion. The generalized Burger’s–Huxley equation (GBHE), the generalized Burger’s–Fisher equation (GBFE), the two-dimensional unsteady Burger’s equation (2D – unsteady Burger’s equation) and the generalized regularized long wave (GRLW) equation are nonlinear model equations which arise in a wide variety of problems involving diffusion and reaction processes. The GRLW equation is reduced to the regularized long wave (RLW) equation and the modified regularized long wave (MRLW) equation.The research problem: How to introduce and construct new numerical techniques to solve any NPDEs in one dimension, two dimensions and higher dimensions.And then the research aims:1. Carry out a comprehensive survey on the new numerical techniques for the solution of NPDEs.2. Construct and introduce some of the new numerical techniques to solve the NPDEs.3. Implement these new methods by using Mathematica package.Steps of study: In this thesis, I introduced and constructed seven new numerical techniques in detail which are not used before to solve the GBHE, GBFE, 2D – unsteady Burger’s equation, RLW and MRLW equations. These seven techniques are studied in the following steps:We introduce and construct a general formula to the compact finite difference (CFD) method for any 2N order, we apply a sixth-order compact finite difference scheme (CFD6) to solve GBHE, GBFE, RLW and MRLW equation..We introduce and construct a general formula to another CFD method also for any 2N order. After using the CFD method, we apply a collocation method to improve the solution of NPDEs. We apply a fourth-order compact finite difference scheme (CFD4) to solve GBHE, GBFE and 2D – unsteady Burger’s equation.We introduce and construct Chebyshev spectral collocation method (C SCM), it is applied to solve the equations RLW, MRLW, GBHE and GBFE.. We introduce and construct the Legendre - Chebyshev spectral collocation method (L-C SCM), it is applied to solve the equations RLW and MRLW.location method (C-L SCM), it is applied to solve the equations RLW and MRLW We introduce and construct the Legendre spectral collocation method (L SCM), it is applied to solve the equations RLW and MRLW.We introduce and construct the Chebyshev collocation method (C CM)), it is applied to solve GBHE, GBFE and 2D – unsteady Burger’s equation.The study concludes: All the previous rustles and comparisons in this thesis insure that the suggested numerical techniques are efficient and more accurate, hence the suggested schemes are capable for solving any NPDEs.