414 research outputs found
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
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
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
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
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
Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method
A numerical method to solve nonlinear Fredholm integral equations of second kind is presented in this work. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Taylor series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of algebraic equations. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method
Numerical Solution of Nonlinear Fredholm Integrodifferential Equations of Fractional Order by Using Hybrid Functions and the Collocation Method
A numerical method to solve nonlinear Fredholm integral equations of second kind is presented in this work. The method is based upon hybrid function approximate. The properties of hybrid of block-pulse functions and Taylor series are presented and are utilized to reduce the computation of nonlinear Fredholm integral equations to a system of algebraic equations. Some numerical examples are selected to illustrate the effectiveness and simplicity of the method
COLLOCATION METHOD FOR VOLTERRA-FREDHOLM INTEGRO-DIFFERENTIAL EQUATIONS NEHZAT EBRAHIMI a1 AND JALIL RASHIDINIA
ABSTRACT This paper introduces an approach for obtaining the numerical solution of the linear and nonlinear Volterra-Fredholm integro-differential equations based on quintic B-spline functions.The solution is collocated by quintic B-spline and then the integrand is approximated by 5-points Gauss-Tur´an quadrature formula with respect to the Legendre weight function.The main characteristic of this approach is that it reduces linear and nonlinear Volterra -Fredholm integro-differential equations to a system of algebraic equations, which greatly simplifying the problem. The error analysis of proposed numerical method is studied theoretically. Numerical examples illustrate the validity and applicability of the proposed method
- …