Computational Mathmatics

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    Computational Methods for Solving Nonlinear VolterraIntegro- Differential Equation
    (جامعة النجاح الوطنية, 2019-12-01) ابو ثابت, فرح
    في هذه الأطروحة ركزنا على حل معادلة فولتيرا التكاملية التفاضلية الغير خطية لأنهاتحتوي على مجموعة واسعة من التطبيقات في الفيزياء الرياضية، والهندسة، والميكانيكا، والكيمياء، وعلم الفلك، وعلم الأحياء، والاقتصاد، ونظرية الإمكانات. بعد ان قدمنا بعض التعاريف والأساسيات التي نحتاجها، ركزنا اهتمامنا بشكل أساسي على الطرق العددية لحل معادلة فولتيرا التكاملية التفاضلية الغير خطية. هذه الطرق هي: طريقةالتحويل التفاضلي مع كثيرات الحدود الأدومية (DTM) طريقة تحليللابلاس أدوميان،(LADM) وطريقة التكرارالمتغير(VIM). حيث سيتم عرض الإطار الرياضي لهذه الطرق العددية مع خصائص التقارب الخاصة بها. حيث سيتم توضيح كفاءة هذه الطرق العددية من خلال بعض الأمثلة العددية. تظهر النتائج العددية بوضوح أن طريقة التكرار المتغير هي واحدة من أقوى التقنيات العددية لحل معادلة فولتيرا التكاملية التفاضلية الغير خطية بالمقارنة مع التقنيات العددية الأخرى بناءً على الأمثلة المستخدمة.
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    Numerical Methods for Solving Nonlinear Fredholm Integral Equations
    (An-Najah National University, 2019-11-14) Odeh, Hiba Jalal
    In this thesis we focus on the numerical treatment of nonlinear Fredholm integral equation of the second kind due to their enormous importance in many applications in various fields. After addressing the basic concepts of nonlinear Fredholm integral equation of the second kind, we focus on the numerical treatment of this equation. This will be accomplished by implementing two numerical methods, namely, Haar Wavelet method and Homotopy Analysis method (HAM). The mathematical framework of these numerical methods will be presented. These numerical methods will be illustrated by solving some numerical examples with known exact solutions. Numerical results show clearly that the Homotopy analysis method is more effective in solving nonlinear Fredholm integral equations in comparison with its counter parts.
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    A Dynamic Programming Approach to Control Heat Equation with Random Walk Process Using HJB Equation
    (An-Najah National University, 2019-04-29) Anabsa, Sally Mohammad Ali
    The heat equation is considered with Random Walk and Brownian motion under the assumption of Bernoulli's, Binomial, Geometric and Poisson distributions for Markov chain. Some numerical methods are also used to find a numerical solution of heat equation under certain conditions as finite difference method (explicit and implicit), Crank Nicolson method and method of lines. Separation of variables method also used to determine an analytic solution of heat equation. In addition, we have used the Hamilton Jacobi Bellman equation (HJB) and algebraic Riccati equation that arises in the linear quadratic regulator (LQR) to obtain the optimal control function for heat equation. Finally, a comparison between exact and approximate solution for state space equation using Euler's method.
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    Numerical Simulation for Computing the Number of Limit Cycles of Generalized Abel Equation
    (An-Najah National University, 2021-02-02) حواري, لجين مخلص
    Limit cycles (isolated periodic solutions) describe the phenomenon of oscillation that are considered and studied in different research fields such as physics, medicine, chemistry, populations…etc. In nature, some of biological and physical processes are represented by stable limit cycles. The interest point of this problem comes from the study of number of isolated closed orbits of a planar polynomial vector field, which is a part of Hilbert`s Sixteenth Problem; this problem has been one of the major problems in the qualitative theory of ordinary differential equations. Hilbert's problem is interested in the number of limit cycles for the planar polynomial differential system. In this work, both limit cycles in xy-plane and the stability types of limit cycles were exhibited, also direction field was considered, which describes graphically the behavior for the solution of the differential equation. Theorems related to the existence and non-existence of limit cycles were discussed. Moreover, a common nonlinear ordinary differential equation, named Abel differential equation, was discussed. Also this work presented limit cycles of first order polynomial differential equation where the coefficients are periodic, and presented some results concerned to the maximum number of limit cycles for polynomial differential equations, and work on proving these results numerically. Furthermore, limit cycles of planar differential system (planar vector field) were presented. Also the Poincaré map, multiplicity of limit cycles for planar differential system, and the multi-parameter of differential system were exhibited. Finally, the number of non-contractible limit cycles of a system in the cylinder was presented with numerical example. The most challenging problem in this work is to obtain numerical examples such that they contain more than one limit cycle by defining suitable interval, coefficients and initial conditions that satisfy relevant theorems and corollaries. While the second problem was to explore examples of limit cycles which have multiplicity greater than one.
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    Numerical Methods for Solving Hyperbolic Type Problems
    (An-Najah National University, 2017-03-29) Abd Al-Haq, Anwar Jamal Mohammad; Qatanani, Naji
    Hyperbolic Partial Differential Equations play a very important role in science, technology and arise very frequently in physical applications as models of waves. Hyperbolic linear partial differential equations of second order like wave equations are the ones to be considered. In fact, most of these physical problems are very difficult to solve analytically. Instead, they can be solved numerically using some computational methods . In this thesis, homogeneous and inhomogeneous wave equations with different types of boundary conditions will be solved numerically using the finite difference method (FDM) and the finite element method (FEM) to approximate the analytical (exact) solution of hyperbolic PDEs. The discretizing procedure transforms the boundary value problem into a linear system of n algebraic equations that can be solved by iterative methods. These iterative methods are: Jacobi, Gauss-Seidel, SOR, and Conjugate Gradient methods. A comparison between these iterative schemes is drawn. The numerical results show that the finite difference method is more efficient than the finite element method for regular domains, while the finite element method is more accurate for complex and irregular domains. Moreover, we observe that the Conjugate Gradient iterative technique gives the most efficient results among the other iterative methods.