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الحلول العددية للمعادلات التفاضلية باستخدام طريقة التفاضلات التربيعية المعتمدة على دوال السبلاي G == Numerical Solutions of Differential Equations Via G - Spline Based Differential Quadrature Method

Author name: مصطفى اكرم سعيد
Supervisor name: اسامة حميد محمد | فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Numerical Analysis
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: This thesis have two main objectives, namely : 1 - The first objective is to study the mathematical background of the differential quadrature method and its application to solve boundary value problems of the fourth order ordinary differential equations.2 - The second objective is first about function approximation by G - spline interpolation method. Secondly the numerical solution of two applications relating the vibration of a uniform beam problem which are represented by a boundary value problem of the fourth order ordinary differential equation and the vibration of a square thin plate given by a boundary value problem of the forth order partial differential equation, by using G - spline based differential quadrature method have been obtained

قابلية الاستقرارية لنظام سيطرة غير خطي متغير العشوائية مع الزمن بوساطة مسيطر استرجاعي لمخرجات النظام == Stabilization of Nonlinear Stochastic Control System via Output - Feedback Control

Author name: ايناس عاجل جاسم محمد الركابي
Supervisor name: راضي علي زبون الساعدي
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: Stochastic differential equations are one of the most useful areas of the theory of stochastic processes and its applications in mathematics.Some nonlinear (Itô) 1 dynamic stochastic control system driven by Brownian motion 2 based on dynamic observer have been considered.Output feedback (observer - based) robust and optimal control law which guarantees global (local) asymptotic stable in probability for the nonlinear stochastic dynamic system are discuss and developed. The necessary theorems regarding the globalty asymptotic stable in the probability of the equilibrium point at the origin of the closed loop stochastic system have been developed and proved. The Lyapunov function approach of stochastic dynamic system has been adapted to justify our proofs.The inverse optimal stabilization in probability with suitable performance index has also discussed and developed. The necessary mathematical requirements have also been provided. Concluding remarks, future work, computational algorithm based on the theoretical results and illustrations have been presented.

طرق متسلسلة تشيبتشيف لحل بعض المسائل الخطية == Chebyshev Series Methods for Solving Some Linear Problems

Author name: نور نبيل محمود القيسي
Supervisor name: احلام جميل خليل
General topic: Mathematics
Specific topic: Applied Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The main purpose of this work may be divided into the following aspects : 1.Study the Chebyshev polynomials of the first and second kinds defined on the intervals [0,1] and [ - 1,1] and modify some of their properties.2.Use two methods to solve the linear ordinary differential equations with nonconstant coefficients, namely, Chebyshev - matrix method and Chebyshev series method.3.Devote Chebyshev series method to solve system of linear Fredholm integral equations and integro - differential equations

قابلية الاستقرارية في نظام سيطرة غير خطي متغير العشوائية باستعمال الامثلية المعكوسة == Stochastic Nonlinear Control Stablizability Based on Invers Optimality

Author name: نورا علي عزيز
Supervisor name: راضي علي زبون الساعدي
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The main aim of this work is focused on studying the global asymptotic stability in the probability for some class of closed - loop control system of Itotype in the presence of system uncertainty.Some nonlinear continuous - time Ito - dynamic stochastic system deriven by unbounded stochastic noise input have been considered, where the equilibrium point of the stochastic system is preserved even in the presence of noise .The global asymptotic stability in probability has been developed by using stabilization controller and Lyapunov stochastic approach.The stochastic Lyapunov function is computed to guarantee the global asymptotic stability in probability. Some resulte of estimation of exponential stability is also discussed.The necessary theorem for finding the controller design and stability Lyapunov stochastic function have been stated and proved which are supported by some concluding remarks and illustrations.

حول امثلية انظمة السيطرة المتابعة التصادفية اللاخطية == On Optimality of Stochastic Non - Linear Tracking Control System

Author name: مریم یاقو یوسف رمو
Supervisor name: راضي علي زبون الساعدي
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The tracking problem for differential stochastic equations in the presentof stochastic uncertainty of white noise, and control input have beenconsidered.In this work, our consideration have been focused on the case whereboth original dynamic state stochastic system and the desired stochasticdynamic system, are driven by white noise stochastic process.The main aim of this work is to make the behavior of the originaldynamic system following the behavior of the desired one for arbitrarycontroller, using tracking control system approach.The tracking and stabilizing controller that guarantee the optimumtracking error system between the original system and the desired one havebeen derived and developed.The necessary theorems for optimum tracking have been stated andproved supported with some concluding remarks. The controller can also beendivided into robust one and optimal one.The optimum controller can be obtained as a solution of some lineardeterministic differential Riccati equation, while the robust one can be obtained so that some controllability properties are ensured.The Riccati equation associated with linear stochastic optimal controller and tracking one, have also been desired and discussed.Finally some illustration ranking for time varying system and for law order differential system to larger one, have been illustrated, with details and corresponding Riccati equation for justification of the present work.

دوال السبلاين G - لتقريب حلول المعادلات التفاضلية الاعتيادية باستخدام طرائق متعددة الخطوات == G - Spline Interpolation for Approximating the Solution of the Ordinary Differential Equations Using Linear Multistep Methods

Author name: زهراء جواد كاظم السوداني
Supervisor name: فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The main objectives of this thesis, is oriented toward function approximation using special type of spline functions, which is called the “G - spline “including the details of the subject.The second objective consider the 1st order ordinary differential equations of the form : . ]b,a[x),y,x(F)x(y∈=′y(a)=. 0yWhere the study concern the approximate solution of the above differential equation using linear multistep methods based on G - spline interpolation and then a generalization to this approach have been extended to solve Boundary value problems of the second order ordinary differential equations.

خوارزميات محورة لحل مسائل البرمجة الخطية

Author name: ياسمين معين محمد الاسدي
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this work, we studied the Path - FollowingAlgorithm, which is one of the family algorithms, calledInterior - Point Algorithms.We are discussed two modifications, the firstone concerned with the path solution, while the secondone is concerned with the feasibility solution. Thesetwo modifications are combined in a new manner, toconstruct a hybrid method. The same test problem hadbeen run for all the algorithms, as well as, number oftested problems had been implemented for comparison.From this comparision we have shown that ourmodifications give better results in the number of iterationsand the accuracy of the results.

نظريات وجود الحلول لمسائل القيم الحدودية للمعادلات التفاضلية الاعتيادية الدفعية == Existence Theorems of the Solutions for the Boundary Value Problems of the Impulsive Ordinary Differential Equations

Author name: نور شوقي كامل محمد
Supervisor name: احلام جميل خليل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The main theme of this work can be divided into three categories, which can be summarized as follows : First, we give some definitions of impulsive differential equations with or without delays with some illustrative examples and some real life applications.Second, we give the explicit forms of the solutions of the boundary value problems (periodic and nonperiodic) which consist of the first order linear ordinary differential equations with non - constant coefficients together with finite impulsive conditions and boundary condition (periodic and nonperiodic).Third, we transform the boundary value problems (periodic and nonperiodic) which consists of the first order nonlinear ordinary differential equations together with finite impulsive conditions and boundary condition (periodic and nonperiodic) into equivalent integral equations. Also the existence of the solutions for the above periodic boundary value problemsare discussed.

طريقة توسيع تيلر لحل المعادلات التكاملية والتكاملية التفاضلية اللاخطية == Taylor Expansion Method for Solving the Non - Linear Integral and Integro - Differential Equations

Author name: هدى عبـد الرزاق محمـد الجنابي
Supervisor name: احلام جميل خليل
General topic: Mathematics
Specific topic: Integral Equations
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: The main purpose of this work is to study Volterra - Fredholm integral and integro - differential equations.This study include the classification of Volterra - Fredholm integral and integro - differential equations.Also, some theorems for the existence and uniqueness of the solution for linear Volterra - Freadholm integral and integro - differential equations are presented.Moreover, Taylor expansion method for solving special types of nonlinear Volterra - Freadholm integral and integro - differential equations with some illustrate examples are discussed.

حول كمال الفضاءات المترية الضبابية == About the Completeness of Fuzzy Metric Spaces

Author name: اماني التفات كاظم
Supervisor name: فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: لهذه الاطروحة هدفين رئيسيين، وهما : الهدف الاول هو لدراسة المجموعات الضبابية (Fuzzy Sets) بالاضافة الى بعض الخواص الجبرية لهذه المجموعات وبعض النتائج النظرية المهمة.الهدف الثاني هو لدراسة الفضاءات المترية - D (D - Metric Spaces) والفضاءات المترية الضبابية - M (M - Fuzzy Metric Spaces) واعطاء بعضا من النتائج المهمة في هذين الفضائين. كما ويتضمن هدف الاطروحة دراسة كمال الفضاءات المترية الضبابية (Completeness of Fuzzy Metric Spaces) باستخدام الدوال المترية الضبابية - M. | The objective of this work may be oriented toward two objectives.The first objective is to study fuzzy set theory, as well as some of its basic algebraic properties and theoretical results. The second objective is to study D - metric spaces and M - fuzzy metric spaces, and some of their properties. Also, this objective includes the study of complete fuzzy metric spaces using M - fuzzy distance function. In addition, some additional results are presented and proved in this work.

حلول المعادلات التفاضلية الكسرية الحدودية == Solutions of Fractional Boundary Value Problems

Author name: سيماء عبد الستار محمد الفياض
Supervisor name: فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: في هذه الرسالة، قمنا بتقديم اسلوب مطور لحل المعادلات التفاضلية الحدودية ذات الرتب الكسرية (Fractional order boundary value problems). حيث اعتمدنا في هذا الاسلوب على تطبيق مؤثر رايسز - فيلر(Riesz - Feller operator) والحصول على الصيغة المطورة لمعادلة الفروقات المنتهية المناظرة للمعادلة التفاضلية الحدودية الكسرية.كما وان من اهداف هذا العمل هو دراسة مبرهنة وجود ووحدانية حلول المعادلات التفاضلية الحدودية الكسرية، وتقديم برهان لهتين المبرهنتين بالاعتماد على مبرهنة شاودر للنقطة الصامدة (Schauder fixed point theorem) للمؤثر التكاملي الكسري (Fractional integral operator). | In this thesis, we introduce a modified approach for solving fractional order boundary value problems. This approach is given by applying the Riesz - Feller operator to obtain a modified finite difference equation, which is symmetric to the equation of fractional boundary value problems.Also, the main objective of this work is to study the existence and uniqueness theorem of solutions of the fractional boundary value problems, and to present their proof depending on Schauder fixed point theorem for fractional order integral operator

حول تحويلات لابلاس المتعددة الابعاد == On the Multi - Dimensional Laplace Transforms

Author name: وسن عجيل حمود
Supervisor name: احلام جميل خليل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: الهدف من هذا العمل هو دراسة تحويلات لابلاس المتعددة الابعاد مع تطبيقاتها. هذه الدراسة شملت المحاور التالية : - 1 - تبني تحويلات لابلاس ذات البعد الواحد للدوال التي تعتمد على متغير مستقل واحد فقط مع بعض الخواص المهمة. اضافة الى ذلك بعض التطبيقات الرياضياتية لتحويلات لابلاس ذات البعد الواحد قدمت .2 - توسيع دراسة تحويلات لابلاس ذات البعد الواحد الى تحويلات لابلاس المتعددة الابعاد. اضافة الى ذلك قمنا باعطاء بعض الخواص المهمة الموسعة لتحويلات لابلاس المتعددة الابعاد.3 - استعمال تحويلات لابلاس المتعددة الابعاد لحل انواع خاصة من المعادلات التفاضلية الجزئية, المعادلات التكاملية المتعددة الابعاد والمعادلات التكاملية - التفاضلية المتعددة الابعاد. | The aim of this work is to study the multi - dimensional Laplace transforms and their applications. This study includes the following aspects : - 1 - Devote the one - dimensional Laplace transforms for functions of only one independent variable with some of their important properties. Also some mathematical applications for the one - dimensional Laplace transforms are presented.2 - Extend the study of the one - dimensional Laplace transforms to the multi - dimensional Laplace transforms. Also some generalized important properties of the multi - dimensional Laplace transforms are obtaind.3 - Use the multi - dimensional Laplace transforms to solve special types of the partial differential equations, the multi - dimensional integral equations and the multi - dimensional integro - differential equations

تكاملات مونت كارلو وتقنيات تخفيض التباين للتكاملات المتعدد الابعاد

Author name: اكرم عباس جاسم الصباغ
Supervisor name: اكرم محمد العبود
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this work, we consider two Monte Carlo methods for evaluating the ndimensional integrals for bounded integrand. Statistical properties of these methods are illustrated and unified. The supported number of trials to estimate the integrals, confidence interval and the efficiency for each method were derived theoretically and assessed practically. Variance Reduction for Monte Carlo methods is discussed theoretically and explained by algorithms where four techniques are considers, namely, the Importance Sampling, the Correlated Sampling, the Partition of the region, and the Biased Estimator.The computer programs are illustrated in appendices by the run is made by using MathCAD 2001i.

تخمين معلمات توزيع ويبل مع تطبيق باستخدام محاكاة مونت كارلو == Estimation of Parameters for Weibull Distribution with Application by Using Monte Carlo Simulation

Author name: سلام عادل احمد
Supervisor name: اكرم محمد العبود
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: تطرقنا في هذه الرسالة الى توزيع ويبل ذو المعلمتين لاهميته في مجالات الاحصاء وتطبيقاته من حيث استعراض لخواص التوزيع الرياضية والاحصائية والعزوم والعزوم العليا. ثم تطرقنا الى التخمين وخواصه ومناقشة اربعة طرق لتخمين معالم التوزيع وهي : طريقة الترجيح الاعظم, طريقة العزوم, طريقة العزوم المعدلة وطريقة المربعات الصغرى. نوقشت هذه الطرق نظريا وطبقت عمليا باستخدام ستة اساليب من محاكاة مونت كارلو لتوليد المتغيرات العشوائية من توزيع ويبل. اوجدت كفاءة بعض هذه الاساليب نظريا وقورنت عمليا. تمت المقارنة بين الطرائق الاربعة التخمينية باستخدام مقياس معدل مربعات الخطا. | In this work, we consider the Weibull distribution of two parameters for its importance in statistics and its applications. Mathematical and statistical properties of Weibull distribution are considered, moments and higher moments are illustrated and unified. Four methods of estimation to the distribution parameters namely (Maximum likelihood Method, Moments Method, Modified Moments Method, Least Square Method) are discussed theoretically and assessed practically by utilizing six procedures of Monte - Carlo simulation for generating random variates from the distribution. Efficiency of some procedures are found theoretically and compared practically. Comparisons are made among four methods of estimation by considering the mean square error measurement.

حلول المعادلات التفاضلية الاعتيادية المتجانسة من الرتب الكسرية ذات المعاملات المتغيرة == Solutions of Ordinary Homogenous Fractional Order Differential Equations with Variable Coefficients

Author name: ضمياء سالم
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: في هذا العمل تم دراسة حلول المعادلات التفاضلية الاعتيادية المتجانسة من الرتب الكسرية (والتي قيمها مابين الصفر والواحد) ذات المعاملات المتغيرة.لقد تم استنباط وجود هذه الحلول من خلال عرض نظرية استخدم فيها طريقة Power Series للحالات الاعتيادية(ordinary point) والمفردة(singular point) من المعادلات التفاضلية الاعتيادية من الرتب الكسرية ذات المعاملات المتغيرة وقد تم عرض مثال لكل نوع | In this work the solutions of ordinary homogenous fractional order (with values between zero and one) differential equations with variable coefficients are investigated. Also the existence of the solution is by presenting theorems, using the method of Power Series for ordinary and singular type of fractional order differential equations with variable coefficients. Example has been presented for each case

اساليب النمذجة الخطية في ادارة شبكة المشاريع == Linear Programming Techniques for Network Project Management

Author name: ايلاف محمد عبد
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this work, Linear Programming Problems have been implemented to build four linear models for projects management. An Interior - Point Method has been implemented to solve such linear models, instead of using the usual techniques "Simplex Method", by implementing the "what's Best 9.0 " software, and obtaining the critical path in minimum completion time, minimum crashing cost and optimal total ( direct & indirect ) costs for a simple real project. Then we are verified the results obtained by implementing " Project 2000 " software to construct the project network and obtain the same critical path.Finally, the Programming Evaluation Review Technique (PERT) has been used, to find the probabilities of completing the project.

مسالة هيرميت بيركهوف ذات الرتب الكسريه وتطبيقاتها لدوال السبلاين - G == HB - Problem with Fractional and It's Application to G - Spline Function

Author name: حسام عدي عبد الرسول
Supervisor name: فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
Key words:
  • التحليلات العددية
  • طرق النظرية التقريبة
  • الجبر الخطي
First pages:
Abstract: الهدف الرئيسي من هذه الرسالة ، هو اولا لدراسة التفاضل الكسري (Fractional Calculus) وطرق حساب المشتقات ذات الرتب الكسرية لبعض الــدوال وثانيا لدراســة دوال اندراج السبلاين - G وطريقة حساب هـذه الدوال باسـتخدام اسلوب جديـد وذلك لتكـوين مشكلـة هيرميت بيركهـوف (Heremite - Birkhoff problem) وذلك باستخدام مشتقات ذات رتب كســــــرية بدلا من مشتقات ذات رتب صحيحة | The objective of this thesis is to study first the theory of fractional calculus and some of well known methods for evaluating derivatives of fractional orders for certain functions.The second objective is to study the G - spline interpolation functions and its construction using a new approach in formulating the Heremite - Birkhoff problem using fractional derivatives instead of integer order derivatives

نظام تجفير معدل بالاعتماد على تطوير نظام الـ NTRU == Cryptosystem approach using modified NTRU

Author name: اياد حازم ابـراهيم
Supervisor name: عبد المنعم صالح رحمة ابو طبيخ | اكرم محمد العبود
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: يعتبر نظام التشفير NTRU نظام تشفيرحديث قدم عام 1998 ويمتاز هذا المفتاح بسرعة عالية مقارنة بالانظمة المعروفة RSA , ECC .من اهم مزايا هذا النظام سرعته العالية في توليد المفاتيح مقارنة بانظمة التشفيرالاخرى. وتاتي امنية نظام الـ NTRU من تداخل النظام المختلط المتعدد الحدود مع المستقبل لمعامل النقصان للعددين النسبيين p and q .تم في هذه الاطروحة مقترح نظام الـ NTRU كمفتاح جديد قصير بشكل معقول تتولد رموزه بسهولة وسرعة عالية كما لا يحتاج الى متطلبات ذاكرة كبيرة وهذه المزايا جعلته قابل للاستخدام في انظمة الاتصالات المتنقلة وقنوات البث الاذاعي لقلة احتياجاته من مكونات واجهزة .يستعمل نظام الـ NTRU نظام مختلط يعتمد على الجبر المتعدد الحدود ومبدا المجاميع المعتمد على نظرية الاحتمالات .اضافت الاطروحة نظام محور لنظام الـ NTRU اعتمدت مصفوفات الاعداد كاساس واظهرت كونها اعلى سرعة وكفاءة من نظام التشفير الاعتيادي .كما اضافت هذه الاطروحة وصف اسلوب وبرنامج البصمة الرقمي باستخدام الـ NTRU مع تحسين في تشفير الوثائق موثقة بخوارزميات وامثلة .تم فحص وتحليل الاداء باستخدام حاسبة شخصية ذات المواصفات التالية ( السرعة 1.7 dual cores ، الذاكرة 512 MB ،نظام الويندوزXP - SP2 ،وباستعمال لغة (Visual Basic ) لتطوير البرامج | The NTRU [Number theory research unit] cryptosystem is a relatively new public key cryptographic algorithm that was first introduced in 1998, and that key runs are much faster than conventional public key algorithms such as RSA, ECC. The main advantage of this cryptosystem is its high speed generation keys, which is often the most important part of public key cryptography.The security of NTRU cryptosystem comes from the interaction of the polynomial mixing system with the independence of reduction modulo two relatively prime integers' p and q. This thesis introduces the concepts behind NTRU as a new public key cryptosystem. NTRU features are reasonably short, easily created keys, high speed, and low memory requirement. These features make it favourable in mobile communication systems, broadcast and satellite channels for its low cost hardware needs. NTRU encryption and decryption use a mixing system suggested by polynomial algebra combined with a clustering principle based on elementary probability theory. Also an approach variant of the NTRU public key cryptosystem called Matrix NTRU cryptosystem is proposed and has been shown to be much faster and have higher efficiency than the classical NTRU cryptosystem.The thesis describes the NTRU Signature Scheme with enhanced document encoding, signature, verification, with provision of documented algorithms and examples.The test and performance analysis performed using a PC with the following specification (processor 1.7 dual cores,memory 512 MB with windows XP - SP2 operating system), and all programs are developed in Visual Basic

حلول المعادلات التفاضلية الكسرية باستخدام دوال السبلاين - G == Solutions of Fractional Differential Equations Using G - Spline Interpolation Functions

Author name: محمد صالح مهدي
Supervisor name: فاضل صبحي فاضل
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: الهدف الرئيس لهذه الاطروحة هو دراسة الحلول العددية للمعادلات التفاضلية الاعتديادية الكسرية (ordinary fractional differential equations) باستخدام دوال السبلاين - G (G - spline functions) للاستكمال. طبق اسلوبين في الطرائق العددية، الاسلوب الاول هو باستخدام طرائق متعددة الخطوات الصريحة(explicit linear multistep methods) والتي يمكن استخدامها وبسهولة لحل المعادلات التفاضلية الخطية وغير الخطية بينما الاسلوب الثاني هو اسلوب محسن وهو استخدام طرائق متعددة الخطوات الضمنية (implicit linear multistep methods) لحل معادلات تفاضلية كسرية اعتيادية وغير خطية والتي من الصعب حلها بالطرق الاعتيادية. وقد تم ذلك باقتراح اسلوب جديد الا وهو باستخدام قاعدة السلسلة(the chain rule) للمشتقات الكسرية. | The main objective of this work is to study the numerical solution of fractional ordinary differential equations using G - spline interpolation functions. Two numerical approaches are used, the first approach utilize the explicit linear multistep methods which can be applied easily for linear and nonlinear problems while the second approach is a modified approach by using the implicit linear multistep methods for solving nonlinear fractional ordinary differential equations which has so many difficulties in their solution. This is done by suggesting a new criterion by using the chain rule derivatives of fractional order.

حلول المعادلات التفاضلية التباطؤية ذات الرتب الكسرية == Solutions of Fractional Order Delay Differential Equations

Author name: فجر عدنان عبد الخالق
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
Key words:
  • Fractional differential equations
  • Delay differential equations
  • Numerical solution of differential equations
  • The existence and uniqueness theorem
First pages:
Abstract: في هذا العمل تمت دراسة المعادلات التفاضلية التباطؤية ذات الرتب الكسرية (FOBDDE’s) حيث تم اثبات نظرية الوجود والوحدانية لحل هذا النوع من المعادلات التفاضلية باستخدام طريقة التقريبات المتتابعة.وكذلك تم عرض اسلوب حل المعادلات التفاضلية التباطؤية ذات الرتب الكسرية (FOBDDE’s) تحليليا (Analytically) باستخدام تحويلات لابلاس (Laplace Transformation) وعدديا (Numerically)باستخدام طرائق متعددة الخطوات (linear multi - steps methods) وطرائق ذات الخطوة الواحدة (general - one step methods) وتمت مقارنة نتائج هذه الطرق مع قيم الحل المضبوطة (Exact solutions values). | The Fractional Order Bounded Delay Differential Equations (FOBDDE’s) has been studied in this work. The Existence and Uniqueness theorems of such type of differential equation have been proved, by using the successive approximation techniques. Also, the analytic solution of (FOBDDE’s) are presented, using Laplace Transformation, and the numerical solutions are discussed, using general one - step methods and linear multi - step methods. The comparison, among these methods and the exact solutions are presented.

الشروط الضرورية للامثلية لمسائل التغاير ذوات الرتب الكسرية المستمرة وغير المستمرة

Author name: علي خالد عطية
Supervisor name: علاء الدين نوري احمد | اكرم محمد العبود
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this thesis, some properties and basic definitions of fractional integral and derivatives of Riemann - Liouvill are presented , to construct the optimality conditions of mixed order unconstrained and constrained variational problems with continuous and discontinuous functional, on fixed and moving boundaries ,based on the classical product rule for Riemann - Liouvill , Several tested example are presented to demonstrate the implementation of the optimality necessary conditions.

الشروط الضرورية للامثلية في مسائل التغاير ذوات الرتب الكسرية == Optimality Necessary Conditions For Fractional order Variational Problems

Author name: نادية جاسم اسمير
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: تعد المشتقة ذات الرتب الكسرية فكرة جيدة من الناحيتين النظرية والتطبيقية. وتعد دراسة حسبان مسائل التغاير المتضمنة مشتقات ذات رتب كسرية (fractional variational problems) من المواضيع الحديثة.في هذا العمل، تم عرض بعض الخواص والتعاريف الاساسية للمشتقات والتكاملات ذات الرتب الكسرية لريمان لوفيل (Riemann Liouville). كما تم استنباط الشروط الضرورية لامثلية انواع مختلفة من المسائل التغاير ذات الرتب الكسرية التي تتضمن مشتقات ذات رتب كسرية ومتغيرات معتمدة (dependent variables) لمتغير مستقل (independent variable) واحد فقط على طول حدود ثابتة ومتحركة. كما تم عرض بعض الامثلة توضح تحقيق امثلية الشروط الضرورية. | Fractional (or non - integer) differentiation is an important concept both from theoretical and applicational points of view. The study of problems of the calculus of variations with fractional derivatives is a rather recent subject. In this work, some properties and basic definitions of fractional integral and derivatives of Riemann - Liouvill are presented. The optimality necessary conditions for fractional variational problems are constructed for different types of fractional problems of calculus of variations having one and different multi fractional order derivatives (FOD) on one and different multi - dependent variables with one independent variable, along fixed and moving boundaries. Several examples are presented to demonstrate the implementation of the optimality necessary conditions

الطرق العددية المطورة لحل المعادلات التكامليه الموسعة المتعددة الابعاد == MODIFIED NUMERICAL METHODS FOR SOLVING THE MULTI - DIMENSIONAL INTEGRAL EQUATIONS

Author name: يسر سهيل علي
Supervisor name: احلام جميل خليل
General topic: Mathematics
Specific topic: Numerical Analysis
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: الهدف الاول هو تصنيف ودراسة المعادلات التكاملية الموسعة ذات البعد الواحد والتي تحتوي على n من المؤثرات التكاملية ذات البعد الواحد. هذه الدراسة تتضمن وجود ووحدانية الحل لانواع خاصة من هذه المعادلات التكاملية وحل هذه المعادلات باستخدام بعض الطرق من الطرق التربيعية والتي هي قاعدة شبه المنحرف وقاعدة شبه المنحرف مع التصحيح وقاعدة سمسون. الهدف الثاني هو تصنيف ودراسة المعادلات التكاملية الموسعة المتعددة الابعاد والتي تحتوي على n من المؤثرات التكاملية المتعددة الابعاد. هذه الدراسة شملت حل هذه المعادلات باستخدام بعض الطرق من الطرق التربيعية والتي هي قاعدة شبه المنحرف وقاعدة شبه المنحرف مع التصحيح وقاعدة سمسون. | The first objective is to classify and study the generalized one - dimensional integral equations that contain n one - dimensional integral operators. This study includes the existence of a unique solution for special types of these integral equations and their solutions by using some quadrature methods, namely the trapezoidal rule, the modified trapezoidal rule and Simpson's rule.The second objective is to classify and study the generalized multi - dimensional integral equations that contain n multi - dimensional integral operators. This study includes their solutions by using some quadrature methods, namely the trapezoidal rule, the modified trapezoidal rule and Simpson's rule.

طريقة مهجنه لحل المعادلات التفاضلية الضبابية من الرتبة الثانية == A Hybrid Approach for Solving Fuzzy Differential Equation of Second Order

Author name: بسمه عبد الهادي نعمة
Supervisor name: علاء الدين نوري احمد
General topic: Mathematics
Specific topic: Mathematics
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this thesis, a hybrid approach is presented by combining fuzzy Laplace transformation method and fuzzy variational iteration methods which is employed to obtain approximate solutions of linear and nonlinear fuzzy differential equations with fuzzy initial and boundary values. Then our approach is implemented in which two approaches have been constructed according to the formula of the Lagrange multiplier obtaining the lower, upper and center solutions. The experimental results which are obtained shows every high accuracy in comparison with the exact results with less number of iterations other numerical or approximate method.

الحلول العددية للمعادلات التكاملية - الجبرية == Numerical Solutions of Integral - Algebraic Equations

Author name: صفاء حسن رسول
Supervisor name: اسامة حميد محمد
General topic: Mathematics
Specific topic: Numerical Analysis
Degree: Master
University: Al-Nahrain University - College Of Science
Language: English
University location: Baghdad
First pages:
Abstract: In this thesis two approximate methods are implemented in order to find the approximate solutions of the linear system of Volterra integral - algebraic equations which are so - called Hat basis functions and Sample - and - Hold functions.Both methods will transform the linear system of Volterra integral - algebraic equations to a linear lower triangular system of algebraic equations using the operational matrices of integration associated with the Hat basis functions and Sample - and - Hold functions.Convergence theorem and tested examples are given in order to check the validity and efficiency of the proposed methods
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