Numerical Solution For Mixed Volterra-Fredholm Integral Equations Of The Second Kind By Using Bernstein Polynomials Method

In this paper, we have used Bernstein polynomials method to solve mixed Volterra-Fredholm integral equations(VFIE’s) of the second kind, numerically. First we introduce the proposed method, then we used it to transform the integral equations to the system of algebraic equations. Finally, the numerical examples illustrate the efficiency and accuracy of this method. Keywords: Bernestein polynomials method, linear Volterra-Fredholm integral equations

Hybrid functions approach to solve a class of Fredholm and Volterra integro-differential equations

In this paper, we use a numerical method that involves hybrid and block-pulse functions to approximate solutions of systems of a class of Fredholm and Volterra integro-differential equations. The key point is to derive a new approximation for the derivatives of the solutions and then reduce the integro-differential equation to a system of algebraic equations that can be solved using classical methods. Some numerical examples are dedicated for showing efficiency and validity of the method that we introduce

Approximate Optimal Control of Volterra-Fredholm Integral Equations Based on Parametrization and Variational Iteration Method

This article presents appropriate hybrid methods to solve optimal control problems ruled by Volterra-Fredholm integral equations. The techniques are grounded on variational iteration together with a shooting method like procedure and parametrization methods to resolve optimal control problems ruled by Volterra - Fredholm integral equations. The resulting value shows that the proposed method is trustworthy and is able to provide analytic treatment that clarifies such equations and is usable for a large class of nonlinear optimal control problems governed by integral equations

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

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations

في هذه الورقة ، تم تقديم طريقة اضطراب homotopy لحل نظام خطي من  معادلات فولتيرا - فريدهولم التكاملية المختلطة (المدمجة) من النوع الثاني. تقوم الطريقة ببناء سلسلة يكون مجموعها هو حل المشكلة المدروسة. تمت مناقشة تقارب السلسلة التي تم بناؤها وتقديم برهانها ؛ تم الحصول على تقدير(تخمين) الخطأ أيضًا. للمزيد من التوضيح ، تم تطبيق الطريقة على العديد من الأمثلة وتم كتابة البرامج باستخدام R2015a)) MATLAB لحساب النتائج. و كذلك لبيان دقة النتائج وكفاءة الطريقة، تم حساب الحلول التقريبية لعدة امثلة ومن ثم مقارنتها بالحلول الحقيقية وذلك من خلال حساب الأخطاء المطلقة.           In this paper, the homotopy perturbation method (HPM) is presented for treating a linear system of second-kind mixed Volterra-Fredholm integral equations. The method is based on constructing the series whose summation is the solution of the considered system. Convergence of constructed series is discussed and its proof is given; also, the error estimation is obtained. Algorithm is suggested and applied on several examples and the results are computed by using MATLAB (R2015a). To show the accuracy of the results and the effectiveness of the method, the approximate solutions of some examples are compared with the exact solution by computing the absolute errors

Three-Dimensional Nonlinear Integral Operator with the Modelling of Majorant Function

 تقدم هذه الورقة البحثية طريقة  لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد غير الخطي في  R3. حيث يتم استخدام مفهوم (Majorant function) وباستخدام طريقة نيوتن المعدلة  لتحويل مؤثر فولتيرا التكاملي  الثلاثي الأبعاد غير الخطي  إلى متتالية  لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد الخطي ومن يتم استخدام طريقة (Gaussian-Legendre)  التربيعية لايجاد الحل التقريبي لمؤثر فولتيرا التكاملي  الثلاثي الأبعاد الخطي من خلال التعامل مع نظام جبري خطي.تم مناقشة وجود ووحدانية الحل للطريقة المستخدمة مع اعطاء أمثلة توضيحية لإظهار دقة وكفاءة الطريقة.In this paper, the process for finding an approximate solution of nonlinear three-dimensional (3D) Volterra type integral operator equation (N3D-VIOE) in R3 is introduced. The modelling of the majorant function (MF) with the modified Newton method (MNM) is employed to convert N3D-VIOE to the linear 3D Volterra type integral operator equation (L3D-VIOE). The method of trapezoidal rule (TR) and collocation points are utilized to determine the approximate solution of L3D-VIOE by dealing with the linear form of the algebraic system. The existence of the approximate solution and its uniqueness are proved, and illustrative examples are provided to show the accuracy and efficiency of the model. Mathematical Subject Classification (2010):  45P05, 45G10, 47H9

Bernstein Collocation Method for Solving Nonlinear Fredholm-Volterra Integrodifferential Equations in the Most General Form

A collocation method based on the Bernstein polynomials defined on the interval [a,b] is developed for approximate solutions of the Fredholm-Volterra integrodifferential equation (FVIDE) in the most general form. This method is reduced to linear FVIDE via the collocation points and quasilinearization technique. Some numerical examples are also given to demonstrate the applicability, accuracy, and efficiency of the proposed method

A hybrid functions method for solving linear and non-linear systems of ordinary differential equations

In the present paper, we use a hybrid method to solve linear or non-linear systems of ordinary differential equations (ODEs). By using this method, these systems are reduced to a linear or non-linear system of algebraic equations. In error discussion of the suggested method, an upper bound of the error is obtained. Also, to survey the accuracy and the efficiency of the present method, some examples are solved and comparisons between the obtained results with those of several other methods are carried out