Computational Mathmatics
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- ItemFINITE VOLUME METHOD FOR SOLVING NAVIER STOKES EQUATIONS IN FLUID DYNAMICS(An-Najah National University, 2025-06-24) Abu Arrah, AbdulraheemSeveral equations, particularly the Navier-Stokes equations, govern fluid dynamics. These equations are essential for describing fluid motion, which helps us understand many natural phenomena. The Navier-Stokes equations present significant challenges for researchers in mathematics and engineering due to their complexity and the difficulties in obtaining analytical solutions. As a result, it has become necessary to explore alternative methods for solving these equations, particularly through numerical approaches. Since numerical methods yield approximate solutions, it is vital to evaluate the effectiveness of this approach in addressing the Navier-Stokes equations. One such numerical method is the finite volume method (FVM), which provides approximate solutions to the Navier-Stokes equations. In this thesis, we conducted a thorough examination of the finite volume method using various examples of the Navier Stokes equations that have analytical solutions. We began with simpler cases and gradually increased the complexity while also comparing our numerical results with the analytical solutions to assess how closely they aligned with the exact solutions. This comparison enabled us to evaluate the effectiveness of the method. We encountered issues related to the stability and accuracy of the numerical solutions based on the specific conditions we examined while employing this method. As a result, we discussed the numerical schemes related to the Finite Volume Method (FVM) and the criteria for selecting a specific scheme, especially concerning the Peclet number. We then evaluated the effectiveness of each scheme by applying them to the same case. The results obtained from the finite volume method for solving one-dimensional steady state Navier-Stokes equations, with a suitable choice of discretization scheme, provided accurate solutions with excellent stability. However, we observed that when high Peclet numbers were used, solution instability emerged, necessitating the implementation of higher-order discretization schemes. Future research could build on this method by looking at flow situations in two or three dimensions and improving computing efficiency with adaptive mesh refinement (AMR) and better discretization schemes.
- ItemFINITE VOLUME AND WEIGHTED RESIDUALS METHODS FOR SOLVING MAXWELL’ S EQUATIONS(An-Najah National University, 2025-03-06) Dwayyat, NadaThe Finite Volume Method (FVM) and the Method of Weighted Residuals (MWR), two numerical techniques for solving Maxwell's equations—which characterize electromagnetic fields—are compared in this work. Numerous technological fields, including electrical engineering, wireless communication, and electromagnetic wave patterns, depend on Maxwell's equations. The Finite Volume Method ensures that the flow of quantities (fluxes) is preserved at the boundaries by dividing the problem region into small volumes and applying conservation principles to each one of them separately. This method is frequently implemented in fluid dynamics and electromagnetic simulations and is particularly of use for problems demanding exact flux conservation. The Method of Weighted Residuals uses specific weighting functions to reduce the errors (residuals) in the governing equations after making a precise assumption of an approximate solution. If the appropriate pilot functions are used, this approach can be highly precise and is adaptable, making it appropriate for complex forms and boundary conditions. This work scrutinizes the two approaches MATLAB implementations and evaluates how well they solve Maxwell's equations. Precision, computational efficacy, and convenience are the main comparing points. The findings demonstrate that MWR provides greater resilience and can be more precise in complex scenarios, even though FVM is more accurate at saving flux in particular. All things taken into account, this work contributes to the progress of effective numerical techniques in problem solving strategies, which may be of exceptional advantage in future cutting-edge technologies.
- ItemNUMERICAL SOLUTION FOR FREDHOLM INTEGRAL EQUATIONS(An-Najah National University, 2024-09-08) Khalaf, FatenThe integral Fredholm-Hammerstein integral equation has many applications in various areas of mathematical physics, including heat transfer problems fluid dynamics and quantum mechanics . In this thesis, we focus on the numerical treatment of the Fredholm-Hammerstein equations using the Galerkin method and the shifted Chebyshev Polynomial method. To test the efficiency and the accuracy of this method, one numerical example with known exact solution is presented. Numerical results show that the conveyance of this method is in good agreement with the exact solution. Moreover, we conclude that the in Galerkin method and shifted Chebyshev method provides very accurate results.
- ItemCOMPARISON OF NUMERICAL METHODS FOR SOLVING SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS(An-Najah National University, 2024-10-31) Boshnaq, Abdul FattahIn this work, we shed the light on the numerical handling of systems of ordinary differential equations. These systems have wide range of applications in mathematical physics, chemistry, biology, stereology, heat conducing and engineering models. After introducing some important aspects of systems of differential equations including the solvability of homogeneous and non-homogeneous systems, we focus on the numerical techniques for solving systems of differential equations. Namely; one step and multistep methods. The one step methods include Euler and Runge-Kutta methods. The multistep methods involve Adams-Bashforth method, Adams-Moulton method and the Predictor-Corrector method. The mathematical framework of these numerical methods together with their convergence properties and their error bound associated with these methods will be presented. The proposed numerical methods will be illustrated by solving some numerical examples. Numerical results show clearly that the multistep methods are more efficient and give faster convergence than other methods.
- ItemAUTOMATED OPTIC DISC SEGMENTATION FOR FUNDUS IMAGES BASED ON ARTIFICIAL NEURAL NETWORKS: U-NET(An-Najah National University, 2024-08-27) Alhendi, NourOptic disc (OD), located at the back of the eye, is a significant part of the retina. It represents the entry point for the optic nerve and blood vessels. Accurate OD segmentation provides critical information about the anatomy and health state of the retina, aiding in diagnosing and managing various eye conditions such as glaucoma, diabetic retinopathy (DR), and optic nerve abnormalities. With automatic OD segmentation, computer-based systems can efficiently analyze large numbers of retinal images, enabling early detection and monitoring of eye diseases. This automation not only enhances the speed and accuracy of diagnosis but also facilitates cost-effective and accessible healthcare, especially in areas with limited ophthalmic expertise. In this study, an automatic method for OD segmentation in retinal images using a convolutional neural network (CNN) architecture, known as U-Net, was introduced. First, a region of interest (ROI) was extracted from the fundus images using the bounding box technique. For faster calculations, the cropped images were resized to 128 × 128 pixels. Then, these images were enhanced using the contrast limited adaptive histogram equalization (CLAHE) to eliminate the noise and improve their qualities. After that, a U- Net model was constructed and trained to obtain segmented images. The proposed model was trained and evaluated using the public dataset ORIGA, and the predicted results were compared with the ground truth (GT) images. This method competed with other studies and achieved average accuracy of 98.42%, average precision of 97.46%, and average sensitivity of 95.33%. As the execution time is short, this enables the proposed method to be an online implemented method.