Computational Mathmatics
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- 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.
- ItemComputational Methods for Solving Nonlinear VolterraIntegro- Differential Equation(جامعة النجاح الوطنية, 2019-12-01) ابو ثابت, فرحفي 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.
- ItemNumerical Methods for Solving Nonlinear Fredholm Integral Equations(An-Najah National University, 2019-11-14) Odeh, Hiba JalalIn 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.