This paper gives an ecient numerical method for solving the nonlinear systemof Volterra-Fredholm integral equations. A Legendre-spectral method based onthe Legendre integration Gauss points and Lagrange interpolation is proposedto convert the nonlinear integral equations to a nonlinear system of equationswhere the solution leads to the values of unknown functions at collocationpoints.

In this paper we investigate the existence and uniqueness for Volterra-Fredholm type integral equations and extension of this type of integral equations. The result is obtained by using the  coupled fixed point theorems in the framework of Banach space $ X=C([a,b],mathbb{R})$. Finally, we  give an example to illustrate the applications of our results.

One of the numerical methods for solving linear Volterra integral equations is block-by-block method, which is explained in [L.M. Delves, J.L. Mohamed, Computational Methods for Integral Equations, Cambridge University Press, 1985; L.M. Delves, J. Walsh, Numerical Solution of Integral Equations, Oxford University Press, 1974] and [P.K. Kyte, P. Puri, Computational Methods for Linear Integral Eq...

in this paper, we use the parametric form of fuzzy numbers, and aniterative approach for obtaining approximate solution for a classof fuzzy nonlinear fredholm integral equations of the second kindis proposed. this paper presents a method based on newton-cotesmethods with positive coefficient. then we obtain approximatesolution of the fuzzy nonlinear integral equations by an iterativeapproach.

Keywords: Integro-differential equations Volterra equations Abel's integral equations Bernstein polynomials Singular volterra integral equations a b s t r a c t In this study, a new collocation method based on the Bernstein polynomials is introduced for the approximate solution of a class of linear Volterra integro-differential equations with weakly singular kernel. If the exact solution is pol...