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

Improved polynomial approximations for the solution of nonlinear integral equations

AbstractIn this paper, the solutions of nonlinear integral equations, including Volterra, Fredholm, Volterra–Fredholm of first and second kinds, are approximated as a linear combination of some basic functions. The unknown parameters of an approximate solution are obtained based on minimization of the residual function. In addition, the existence and convergence of these approximate solutions are investigated. In order to use Newton’s method for minimization of the residual function, a suitable initial point will be introduced. Moreover, to confirm the efficiency and accuracy of the proposed method, some numerical examples are presented. It is shown that there are considerable improvements in our results compared with the results of the existing methods. All numerical computations have been performed on a personal computer using Maple 12

Numerical Solution of Some Nonlinear Volterra Integral Equations of the First Kind

In this paper, the solving of a class of the nonlinear Volterra integral equations (NVIE) of the first kind is investigated. Here, we convert NVIE of the first kind to a linear equation of the second kind. Then we apply the operational Tau method to the problem and prove convergence of the presented method. Finally, some numerical examples are given to show the accuracy of the method

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

Solution of the Volterra-Fredholm integral equations via the Bernstein polynomials and least squares approach

We develop a numerical scheme to solve a general category of VolterraFredholm integral equations. For this purpose, the Bernstein polynomials and their features have been used. We convert the main equation into a set of algebraic equations in which the coefficient matrix is obtained by the least squares approximation approach. The error analysis is given to corroborate the precision of the proposed method. Numerical results are presented to demonstrate the success of the scheme for solving integral equations.Publisher's Versio

The Convergence Study of the Homotopy Analysis Method for Solving Nonlinear Volterra-Fredholm Integrodifferential Equations

We aim to study the convergence of the homotopy analysis method (HAM in short) for solving special nonlinear Volterra-Fredholm integrodifferential equations. The sufficient condition for the convergence of the method is briefly addressed. Some illustrative examples are also presented to demonstrate the validity and applicability of the technique. Comparison of the obtained results HAM with exact solution shows that the method is reliable and capable of providing analytic treatment for solving such equations

Numerical solution of the higher-order linear Fredholm integro-differential-difference equation with variable coefficients

AbstractThe main aim of this paper is to apply the Legendre polynomials for the solution of the linear Fredholm integro-differential-difference equation of high order. This equation is usually difficult to solve analytically. Our approach consists of reducing the problem to a set of linear equations by expanding the approximate solution in terms of shifted Legendre polynomials with unknown coefficients. The operational matrices of delay and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre polynomials. Illustrative examples are included to demonstrate the validity and applicability of the presented technique and a comparison is made with existing results

Sparse spectral methods for integral equations and equilibrium measures

In this thesis, we introduce new numerical approaches to two important types of integral equation problems using sparse spectral methods. First, linear as well as nonlinear Volterra integral and integro-differential equations and second, power-law integral equations on d-dimensional balls involved in the solution of equilibrium measure problems. These methods are based on ultraspherical spectral methods and share key properties and advantages as a result of their joint starting point: By working in appropriately weighted orthogonal Jacobi polynomial bases, we obtain recursively generated banded operators allowing us to obtain high precision solutions at low computational cost. This thesis consists of three chapters in which the background of the above-mentioned problems and methods are respectively introduced in the context of their mathematical theory and applications, the necessary results to construct the operators and obtain solutions are proved and the method's applicability and efficiency are showcased by comparing them with current state-of-the-art approaches and analytic results where available. The first chapter gives a general scope introduction to sparse spectral methods using Jacobi polynomials in one and higher dimensions. The second chapter concerns the numerical solution of Volterra integral equations. The introduced method achieves exponential convergence and works for general kernels, a major advantage over comparable methods which are limited to convolution kernels. The third chapter introduces an approximately banded method to solve power law kernel equilibrium measures in arbitrary dimensional balls. This choice of domain is suggested by the radial symmetry of the problem and analytic results on the supports of the resulting measures. For our method, we obtain the crucial property of computational cost independent of the dimension of the domain, a major contrast to particle simulations which are the current standard approach to these problems and scale extremely poorly with both the dimension and the number of particles.Open Acces