A numerical Approach for solving classes of Linear and Nonlinear Volterra Integral Equations by Chebyshev polynomial

In this paper we propose a numerical method for solving classes of of linear and nonlinear Volterraintegral equations having regular as well as weakly singular kernels. The method is based upon replacingthe unknown function by a truncated shifted Chebyshev series. This yeilds either a linear system ofalgebraic equations that can be solved using matrix algebra or a nonlinear system that can be solved byNewton’s iterative method. This method is effective and so easy to apply with low cost of computingoperations. The accuracy and efficiency of the method can be shown through the illustrated numericalexamples.keywords: Volterra integral equations, Newton’s method, Simpson’s method, Chebyshev polynomials,Shifted Chebyshev polynomials.Mathematics Subject Classification: 45B05 , 45Bxx , 65R1

Hermite collocation method for solving Hammerstein integral equations

In this paper, we are presenting Hermite collocation method to solve numer- ically the Fredholm-Volterra-Hammerstein integral equations. We have clearly presented a theory to …nd ordinary derivatives. This method is based on replace- ment of the unknown function by truncated series of well known Hermite expan-sion of functions. The proposed method converts the equation to matrix equation which corresponding to system of algebraic equations with Hermite coe¢ cients. Thus, by solving the matrix equation, Hermite coe¢ cients are obtained. Some numerical examples are included to demonstrate the validity and applicability of the proposed technique

A numerical method for functional Hammerstein integro-differential equations

In this paper, a numerical method is presented to solve functional Hammerstein integro-differential equations. The presented method combines the successive approximations method with trapezoidal quadrature rule and natural cubic spline interpolation to solve the mentioned equations. The existence and uniqueness of the problem is also investigated. The convergence and numerical stability of the problem are proved, and finally, the accuracy of the method is verified by presenting some numerical computations

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

An Efficient Numerical Method for Solving Volterra-Fredholm Integro-Differential Equations of Fractional Order by Using Shifted Jacobi-Spectral Collocation Method

The aim of this article is to solve the Volterra-Fredholm integro-differential equations of fractional order numerically by using the shifted Jacobi polynomial collocation method. The Jacobi polynomial and collocation method properties are presented. This technique is used to convert the problem into the solution of linear algebraic equations. The fractional derivatives are considered in the Caputo sense. Numerical examples are given to show the accuracy and reliability of the proposed technique