Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 9 2 2025 09 13 Quadrature Rule Extended Spline Method for Nonlinear and Linear Volterra Integral Equations 150 160 1899 10.22128/ansne.2025.3006.1141 EN Zahra Mahmoodi Department of Mathematics‎, ‎WT.C.‎, ‎Islamic Azad University‎, ‎Tehran‎, ‎Iran 0000-0002-0855-6017 Journal Article 2025 08 02 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$\acute{a}$n 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. https://ansne.du.ac.ir/article_1899_12bb1e342d3e8eddd4dc5ad277a6c954.pdf