Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations

Due to the ability of function representation, hybrid functions and wavelets have a special position in research. In this thesis, we state elementary definitions, then we introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby- shev, sine-cosine and linear Legendre multi wavelets. The construction of most wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them. The properties of the Floor function are used to and a function which is one on the interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters lead us to get similar function corresponding to the interval [a; b). These functions and their combinations enable us to represent the stepwise functions as a function of floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet, Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new representations of these functions. The main advantage of the wavelet technique for solving a problem is its ability to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential and Fredholm integral equations of the second kind. We also use collocation points and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic fields. Illustrative examples are included to reveal the sufficiency of the technique. In linear integro-differential equations and Fredholm integral equations of the second kind cases, comparisons are done with CAS wavelets and differential transformation methods and it shows that the accuracy of these results are higher than them. Homotopy Analysis Method (HAM) is an analytic technique to solve the linear and nonlinear equations which can be used to obtain the numerical solution too. We extend the application of homotopy analysis method for solving Linear integro- differential equations and Fredholm and Volterra integral equations. We provide some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW, LLMW and CAS wavelets methods. For future studies, some problems are proposed at the end of this thesis

Numerical Solution of Mixed Volterra – Fredholm Integral Equation Using the Collocation Method

         معادلات فولتيرا- فريدهولم التكاملية المختلط ((MVFIEs لديها اهتمام كبير من قبل الباحثين مؤخرا . الطريقة العددية الي اقترحت لحل هذا النوع من المعادلات تستعمل نقاط التجميع وتقريب الحل بواسطة الدالة  اساس الشعاعي (radial basis function)  و متعددة حدود من الدرجة الثانية واندراج النقطة من دون استخدام الشبكة, ولسهولة  الحل تم استخدام اصفار متعددة حدود ليجندر كنقاط تجمع. الغرض الرئيسي من استخدام دالة أساس الشعاعي ومتعدد الحدود هو التغلب على التفرد الذي قد يرتبط بأساليب التجميع. علاوة على ذلك، فإن وظيفة الاستيفاء التي تم الحصول عليها تمر عبر كل النقاط المنتشرة في مجال ما ، وبالتالي فإن وظائف الشكل هي من خصائص خاصية دلتا. تمت مقارنة الحل الدقيق للحلول الانتقائية بالنتائج التي تم الحصول عليها من التجارب العددية من أجل التحقق من دقة وكفاءة طريقتنا.Volterra – Fredholm integral equations (VFIEs) have a massive interest from researchers recently. The current study suggests a collocation method for the mixed Volterra - Fredholm integral equations (MVFIEs)."A point interpolation collocation method is considered by combining the radial and polynomial basis functions using collocation points". The main purpose of the radial and polynomial basis functions is to overcome the singularity that could associate with the collocation methods. The obtained interpolation function passes through all Scattered Point in a domain and therefore, the Delta function property is the shape of the functions. The exact solution of selective solutions was compared with the results obtained from the numerical experiments in order to investigate the accuracy and the efficiency of scheme

Collocation Orthonormal Berntein Polynomials method for Solving Integral Equations

In this paper, we use a combination of Orthonormal Bernstein functions on the interval  for degree ,and 6 to produce anew approach implementing Bernstein Operational matrix of derivative as a method for the numerical solution of linear Fredholm integral equations of the second kind and Volterra integral equations. The method converges rapidly to the exact solution and gives very accurate results even by low value of m. Illustrative examples are included to demonstrate the validity and efficiency of the technique and convergence of method to the exact solution. Keywords: Bernstein polynomials, Operational Matrix of Derivative, Linear Fredholm Integral Equations of the Second  Kind and Volterra Integral Equations

Solving Mixed Volterra - Fredholm Integral Equation (MVFIE) by Designing Neural Network

الهدف الاساسي في هذا البحث هو تقديم طريقه عدديه جديده لحل هذا النوع من المعادلات باستخدام الشبكات العصبية ANN)). حيث تم تصميم شبة عصبيه ذات تغذية اماميه(FFNN ) سريعة, هذا التصميم ذو الطبقات المتعددة والذي يحوي على طبقه واحده خفيه تحتوي على خمسة وحدات خفيه وتستخدم الدالة التحويل (log_sigmoid  ) وطبقة واحدة للإخراج, وتم تدريب الشبة باستخدام خوارزمية ليفن برك (Levenberg – Marquardt) . ولبيان دقة و كفاءة الطريقة المقدمة تم مقارنة نتائج الامثلة التوضيحية مع الحلول المضبوطة لهذه الأمثلة, و من خلال المقارنة تبين بان الطريقة ذات كفاءة و دقة عالية وذات خطاء قليل جدا.       In this paper, we focus on designing feed forward neural network (FFNN) for solving Mixed Volterra – Fredholm Integral Equations (MVFIEs) of second kind in 2–dimensions. in our method, we present a multi – layers model consisting of a hidden layer which has five hidden units (neurons) and one linear output unit. Transfer function (Log – sigmoid) and training algorithm (Levenberg – Marquardt) are used as a sigmoid activation of each unit. A comparison between the results of numerical experiment and the analytic solution of some examples has been carried out in order to justify the efficiency and the accuracy of our method.                                 &nbsp

Computational Block-Pulse Functions Method for Solving Volterra Integral Equations with Delay

يتمثل الهدف من هذا العمل في اتباع نهج طريقة الضغط النبضي في الحل العددي لمعادلات فولتيرا التكاملية مع التأخير. تستخدم هذه الطريقة للحصول على حل رقمي. علاوة على ذلك، تتم كتابة البرنامج بلغة MATLAB. تم عمل تحليل للخطأ وتم توضيح التطبيقات من خلال المثال التوضيحيThe aim of this work is to present method of the Block-pulse function approach to numerical solution of Volterra integral equations with delay. This method is used to obtain numerical solution. Moreover, programs for his method is written in MATLAB language. An error analysis is worked out and applications demonstrated through illustrative example

Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique

Legendre Wavelets Method for Solving Fractional Population Growth Model in a Closed System

A new operational matrix of fractional order integration for Legendre wavelets is derived. Block pulse functions and collocation method are employed to derive a general procedure for forming this matrix. Moreover, a computational method based on wavelet expansion together with this operational matrix is proposed to obtain approximate solution of the fractional population growth model of a species within a closed system. The main characteristic of the new approach is to convert the problem under study to a nonlinear algebraic equation