Computational Mathmatics

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https://hdl.handle.net/20.500.11888/3205

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Analysis of The Logistic Distribution Use in The Suppression Technique for Scalability in Multicast Routing
(2007) Hadi Ali Khalil Hamad; Dr. Mohammad Najib Ass'ad
The immense growth of the computer-supported communication systems, especially the internet, made it imperative to design protocols that have to be efficient and scalable to support the work of the networks infrastructure. By scalable is meant the ability of the protocol to cope with the requirements of groups of the communicating processes when they grow very large in size. The ever increasing demand on communication and the high capability of modern networks call continuously for efficient solutions to problems of communication. Among these solutions was the introduction of multicast routing and also the use of periodic unacknowledged messaging. Related to these two solutions of the problem of scalability, certain techniques were used to overcome this problem, including the suppression technique. This study deals with utilizing probabilistic distribution functions (pdfs) in the suppression technique with the aim of improvement of scalability of multicast routing in communication networks. The two most employed distributions in the suppression techniques are the uniform and the exponential distributions, the first outperforms the second in the performance time metric, while the exponential excels in the performance metric of extra messages. This study introduces a modified form of the logistic distribution as a candidate for use in the suppression technique and compares it with the two other above mentioned distributions. The MATLAB software was used in calculating the values of the performance metrics and in drawing the corresponding figures for comparing the results. The logistic distribution was proved to excel or compete with the other two pdfs in time performance metrics and to have a comparable performance in the overhead metrics.
Analytical and Numerical Solutions of Magnetohydrodynamic Flow Problems
(2013) Abdel Latif Khaleel Sa’ad Aldin; rof. Naji Qatanani
The MagnetoHydreoDynamic flow ( MHD ) is one of the most important topics in mathematical physics due to its wide range of applications . In this work we present some analytical and numerical solutions for some MHD problems . The MHD flow past an impulsively started infinite horizontal plate in a rotating system and unsteady MHD flow through two parallel porous flat plates are considered. In this work an exact analytical solution for these problems based on Laplace transform method has been constructed and analyzed. This involves transforming the coupled partial differential equations into a single equation. For the numerical treatment of these problems we use the finite difference scheme and then implementing a computer software “ MAPLE 15“ to obtain some numerical results.
Analytical and Numerical Solutions of Volterra Integral Equation of the Second Kind
(2014) Feda’ Abdel Aziz Mustafa Salameh; Prof. Naji Qatanani
In this thesis we focus on the analytical and numerical aspects of the Volterra integral equation of the second kind. This equation has wide range of applications in physics and engineering such as potential theory, Dirichlet problems, electrostatics, the particle transport problems of astrophysics, reactor theory, contact problems, diffusion problems and heat transfer problems. After introducing the types of integral equations, we will investigate some analytical and numerical methods for solving the Volterra integral equation of the second kind. These analytical methods include: the Adomian decomposition method, the modified decomposition method, the method of successive approximations, the series solution method and the conversion to initial value problem. For the numerical treatment of the Volterra integral equation we will implement the following numerical methods: Quadrature methods (Trapezoidal rule, Runge-Kutta method of order two, the fourth order Runge-Kutta method), Projection methods including collocation method and Galerkin method and the Block method. The mathematical framework of these numerical methods together with their convergence properties will be presented. These numerical methods will be illustrated by some numerical examples. Comparisons between these methods will be drawn. Numerical results show that the Trapezoidal rule has proved to be the most efficient method in comparison to the other numerical methods.
Analytical and Numerical Treatment of Maxwell's Equations
(2012) Mai "Muhammad Ribhe" Asad Musmar; Prof. Naji Qatanani
Maxwell's equations are one of the most important models in different fields. It describes electromagnetic phenomena such as micro, radios and radar waves. The modeling of systems involving electromagnetic waves is widely spread and has attracted the attention of many authors and researchers. In this work, we will present some important analytical and numerical aspects of Maxwell's equations. We will review some basic properties of electromagnetic theory, namely: electromagnetic fields, magneto-static fields, and time varying fields. Moreover, we will use these physical properties to derive Maxwell's equations in various forms. Though, it is well known that Maxwell's equations are hard to solve analytically, however, we will attempt to use some well known analytical methods to solve these equations in some particular domains such as a sphere and a circular cylinder. Such analytical methods include: separation of variables, series expansion method, conformal mapping and integral methods such as Laplace transforms and cosine and sine Fourier transforms. Numerical methods for solving Maxwell's equations are extensively used nowadays and are usually referred to as Computational Electro-magnetic (CEM). Here the Finite Difference and Finite Difference Time Domain Method (FDTDM) known for its simplicity and efficiency will be proposed to solve Maxwell's equations. And the Yee Algorithm will also be illustrated. Moreover, the convergence, stability and error analysis for these numerical methods will also be investigated.
AUTOMATED OPTIC DISC SEGMENTATION FOR FUNDUS IMAGES BASED ON ARTIFICIAL NEURAL NETWORKS: U-NET
(An-Najah National University, 2024-08-27) Alhendi, Nour
Optic 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.
Bridging Centrality in Scale- Free Network Using Bridging Nodes as the Boundary of Clustering
(2009) Hind Ali Ahmed Eid; Dr. Sobhi Rozieh
Graph theory is one of the most popular fields in mathematics because if it's important applications in solving many problems in the real world and understanding many natural phenomena. This work focuses mainly on studying the scale-free networks and their properties. Moreover, it deals with the study of clustering methods and developing a new a new clustering algorithm by using the properties of scale-free networks. Bridging centrality of the graph together with Between ness centrality and bridging coefficients will also be investigated. Finally we will illustrate how bridging centrality is used in clustering. This will result in a new algorithm of clustering that is called Highest Bridging Centrality Cut algorithm (HCRC algorithm). We concluded that the HCRC algorithm depends on bridging centrality of the nodes.
Closeness Centrality and Epidemic Spreading in Networks
(2008) Fares Masuod Abdelgani Rabaya'; Dr . Sobhi Rosyea'; Dr . Adwan Yasin
This thesis is about the relation between the closeness centrality of the first infected node in the network and each of the total infection time that needs to infect all nodes in that network ,the infection rate for spreading epidemics in that network ,which measures the fraction of nodes those infected per unit time and the infection spreading power of that node ,that measures the power for each node to spread the epidemic to other uninfected nodes in that network. In this thesis, I deal with four types of networks ,unweight small and large networks and weighted small and large networks and study that relation in these four types. The importance of this work is when we find the closeness centrality and the infection spreading power of any node that help us understand which weakness or advantages this node has for maintenance or blocking dangers at the right time. In this work, I made some development in the SI model for the epidemic network in which most of authors consider the infection rate in that model assumed and constant. In this work I found that this infection rate is not constant but it depends on the closeness centrality of the first infected node in the network ,hence I suggest to replace the infection rate in the SI model by the closeness centrality of the first infected node in the network. The results obtained from this work show that each of the total infection time, the infection rate and the infection spreading power when any node infected first in the network depend on the closeness centrality for that node.
A Comparable Study of Hiding Information in Images Using Least Significant Bit (LSB) Substitution and Pixel Value Differencing (PVD) Methods
(2016) Rana Tayseer Sabbah; Dr. Mohammad Assad; Dr. Loa’i Malhis
Steganography is one of the most powerful techniques that conceal the existence of hidden secret data inside a cover object. In this thesis, we take an image for a carrier of secret data which is known as a host or cover image. After embedding the secret data into the host image, the output image of this hiding process is called a stego-image, data hiding schemes are used, by applying some pixel adjustment process to the stego-image obtained by the simple LSB substitution method, and using PVD method, the image quality of the stego-image can be greatly improved with low extra computational complexity. The mean-square-error between the stego-image and the cover-image and Peak Signal to noise ratio are computed. Experimental results show that the stego-image is visually indistinguishable from the original cover-image. As we make a comparable study, the results show that PVD method gives a better image quality combined with the capacity of hidden data, as well as the complexity and robustness of embedded data are increased where the secret data is stored in a difficult way to understand by any intruder.
COMPARISON OF NUMERICAL METHODS FOR SOLVING SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS
(An-Najah National University, 2024-10-31) Boshnaq, Abdul Fattah
In 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.
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.
Confidence – based Optimization for the Single Period Inventory Control Model
(2016) Thana’a Hussam eddin Amin Abu Sa’a; Dr.Mohammad Ass’ad
In this thesis we introduce the issue of demand estimation. We study a problem of controlling the inventory of a single item over a single period with stochastic demand in which the distribution of the demand has an unknown parameter. We assume that the decision maker has a past demand sample and the demand distribution is known but some of its parameters are not known. We introduce some approaches to estimate the unknown parameter and depending on results from estimating the unknown parameter we identify a range of order quantities that-with confidence coefficient – contains the optimal order quantity, and then we construct an interval for the estimated expected cost that the manager will pay if he orders any quantity from the range of candidate quantities. We consider three cases, the demand has a Binomial distribution with unknown parameter , and the demand has a Poisson distribution with unknown parameter , also we consider the case in which the demand has an Exponential distribution with unknown parameter . We present numerical examples in order to clarify our strategy and to show how the confidence interval approach complements with the point estimation approach in order to give the best outlook to the manager to take a decision that achieve an optimal profit.
Cost Value Function Of Water Distribution Networks A Reliability-Based Approach Using Matlab
(2008) Khalid Ahmad “Mohammad” Hasan As-Sadiq; Dr. Mohammad Najeeb Ass'ad; Dr. Mohammad Nihad Almasri
Every community on Earth ought to find the appropriate means to distribute water from different sources to a consumption centers. In general, water distribution networks (WDN) attain this. Each network is composed of arches (to deliver water) and nodes (to consume the water delivered). The ability of any WDN to satisfy the requirements at each node under normal and abnormal conditions is one of the dimensions of its reliability. A method was introduced in this work to evaluate the reliability of a WDN for several combinations of diameters. A network solver was used to find the diameter combination successively. The results obtained from the network solver were saved in a text file. This file is then read by MATLAB, in order to do the necessary calculations. The system reliability for each diameter combination was computed. The values of the system reliability of each diameter combination are recorded as a vector in the MATLAB environment. The most important objective is the maximum of the system reliability of all these combinations, which has been achieved by MATLAB. For this maximum reliability value, we have determined the corresponding values of the cost, where each one represents a diameter combination. The minimum of these values is then determined using a computer code that was developed within the method. MATLAB was used to develop the computer program that converts the information into matrices, which make the required outcome easy to obtain and process. A hypothetical case study was developed to demonstrate methodology implementation. The results were composed of two important things: The computed reliability of any WDN and the way to find the least cost design with a value of reliability that is over a minimum boundary value. Another important thing is all the alternatives of reliability values that can be achieved with a specific budget that we already have.
Data Compression with Wavelets
(2009) Rana Bassam Da'od Ismirate; Dr. Anwar Saleh
There are two types of data compression; the first is lossless(exact) and the second is lossy (approximate). In lossless compression, all details are reserved but high compression ratios can not be achieved and this type is not considered in this thesis. The other type is the loss compression where some details are lost in the process of compression. The size of the lost details is proportional with the the desired compression ratio which is controlled by the user. Using this type, high compression ratios can be achieved with acceptable resolution in the reconstructed data. In this thesis, a computational study of the classical Fourier transform and the relatively new wavelet transform is done. In addition, a computational comparison between the two major transforms shows that the wavelet transform is more efficient than the classical Fourier transform. The high compression ratios that can be achieved by wavelet transform lead to the introduction of several wavelet-based lossy data compression software. Examples of these are the image compressor JPEG2000 and the text compressor DJVU.
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.
Error Analysis and Stability of Numerical Schemes for Initial Value problems “IVP’s”
(2013) Imad Omar Faris Kayid; أ.د. ناجي قطناني
Most of initial value problems are natural phenomena written in the language of mathematics. Solving these initial value problems is one of the most challenging fields in mathematics, because of the mathematicians’ continuous desire of exactness. This work focuses mainly on developing algorithms and programs to construct higher order Taylor’s methods for approximating the solution of first order initial value problems, systems of first order initial value problems and higher order initial value problems. Moreover, it concentrates on studying error and stability of numerical methods for solving initial value problems. For this purpose, we developed programs to find the error amplification functions of Taylor’s and Runge-Kutta methods and to plot boundaries of stability regions for these methods and other methods. We concluded that with the programs we developed, higher order Taylor’s methods could be a good choice for approximating solutions of a wide range of initial value problems.
Error-Detecting and Error-Correcting Using Hamming and Cyclic Codes
(2009) Ne'am Hashem Ibraheem Ibraheem; Dr. "Mohammad Othman" Omran
In this thesis we provide an overview of two types of linear block codes: Hamming and cyclic codes. We study the generation, encoding and decoding of these codes as well as studying schemes and/or algorithms of error-detecting and error-correcting of these codes.
FINITE VOLUME AND WEIGHTED RESIDUALS METHODS FOR SOLVING MAXWELL’ S EQUATIONS
(An-Najah National University, 2025-03-06) Dwayyat, Nada
The 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.
FINITE VOLUME METHOD FOR SOLVING NAVIER STOKES EQUATIONS IN FLUID DYNAMICS
(An-Najah National University, 2025-06-24) Abu Arrah, Abdulraheem
Several 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.
An Inventory Control Model with (M/M/1) Queueing System and Lost Sales
(2011) Imad Ramzi Mohammed Jomah; Dr. Mohammed N. Asad
In this thesis We investigate M/M/1/ - queuing systems with inventory management, continuous review, and lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective system. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/ -system. We compute performance measures and derive optimality conditions under different order distributions. Although we can completely determine analytically the steady state probabilities for the system. We are able to derive functional relations for replenishment order size distributions that is in single server system with inventory. A computer programs were developed in this thesis to obtain the optimal policy.
Numerical Methods for Solving Differential Algebraic Equations
(2010) Samer Amin Kamel Abu Saa'; Dr.Sameer Matar
This study involves the implementation of two numerical methods for solving linear Deferential-algebraic equations (DAEs): Power series and Backward Difference Formula (BDF). It aims to facilitate dealing with DAEs by analyzing these two numerical methods, designing simple algorithms, and using MATLAB programs to code the two algorithms. Some numerical examples were implemented by the two methods, tables and figures were also presented in order to make comparisons and conclusions. This study concluded that Power Series is easier to use, time and effort saving. Furthermore, it is direct in tackling the problems. Finally, some difficult problems were solved by using this method. Other methods were unable to deal with.

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