quadrature rule extended spline method for nonlinear and linear volterra integral equations

In this research, we consider the linear and nonlinear volterra integral equations (vies). the main aims of research is to approximate the integral by gauss-tur´an quadrature rule and then using extended cubic b-spline as the bases function. the unknown coefficients in combination determine by collocation method. the arising system of linear and nonlinear can be solved via iterative method. error analysis is investigated theoretically. numerical text problems are considered to justify the applicability and efficient nature of our approach, comparison of the results justify the considerable accuracy and efficiency proposed methods. the extended parameter in valued in the spline can be chosen in such a way to improve the accuracy also.