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Document Type |
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Thesis |
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Document Title |
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OPTIMIZED NUMERICAL STUDY OF COUPLED NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS BY APPLYING DIFFERENT NUMERICAL SCHEMES. الدراسة العددية الامثل للمعادلات التفاضلية الجزئية غير الخطية المزدوجة من خلال تطبيق أنظمة عددية ( رقمية ) مختلفة |
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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 main objective of this thesis is to find the optimize numerical solution of non-linear high order partial differential equations. Some of the more extensively used finite difference techniques for solving partial differential equations are explained. In particular the diffusion equation, and the convection diffusion equation, each in one, two, and three dimensions, have been considered. Both explicit and implicit techniques are considered and compared on the basis of computational economy. The stability, accuracy, consistency, and convergence of the solution produced by of the finite difference equations and of the solution produced by the partial differential equations, are considered in detail. Highly accurate finite difference schemes were used to test the given mathematical model. A numerical solution of one dimension coupled nonlinear system was discussed. The numerical method proposed by Crank and -Nicholson and proposed numerical (henceforth Nicholson) method is used to handle the problem. In current studies, algebraic difference equations have been solved at each time level. Data tables in this study illustrate the accuracy of such schemes. Numerical results revealed that the Crank- Nicholson scheme and the proposed numerical method are very efficient and reliable for solving one- dimensional coupled Burgers’ equation. To develop an efficient numerical scheme for a 2D convection diffusion equation using the Crank-Nicholson method and the alternating direction implicit (ADI) method, a time-dependent nonlinear system is discussed. The schemes used in this system have second- order accurate in space and time at each time level. The procedure is combined with iterative methods to solve nonlinear systems. Efficiency and accuracy were studied in terms of error norms confirmed by numerical results by choosing two test examples. The Crank Nicholson method and the ADI method were used to address the problems associated with the nonlinear two-dimensional coupled system. These schemes depict the second-order accuracy in space and time. Moreover, the system of these equations that is concerned with the implicit scheme is very reliable for solving coupled convection diffusion equations. These methodologies were unified with iterative methods to resolve nonlinear systems. Numerical results showed that the proposed alternating direction implicit scheme was very efficient and reliable for solving 2-D nonlinear coupled convection-diffusion equations. A three- dimensional advection diffusion equation, higher order ADI method is proposed. 2nd Second and fourth order ADI schemes are used to handle the problem. A Von-Neumann stability analysis shows that the ADI scheme is unconditionally stable. Numerical results for two test problems were carried out to establish the performance of the given method and to compare the method with other typical methods. A fourth order ADI method is found to be very efficient and stable for solving a three-dimensional advection diffusion equation. The proposed methods can be implemented for solving nonlinear problems arising in engineering and physics. |
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Supervisor |
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Prof. Dr. Daoud Suleiman Mashat |
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Thesis Type |
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Doctorate 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, December 20, 2017 |
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Researchers
| محمد ثاقب | Saqib, Muhammad | Researcher | Doctorate | |
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