There are two reasons for this research, the first which is the main was to clarify the use of a closed quadrature formulas included ( Trapizoidal , Simpson’s 1/3 rule and Simpson’s 3/8 rule )which are the most familiar formula of numerical integration ,to evaluating the integral part to find the approximate solution of the 2nd kind of FIDE’s of the 1st order and reducing it to linear system of (n) equation with n unknowns of the solution sample value y(ti) ,i=0,1,2,3,…,n .The other reason was to explain the differences between three Quadrature formulas in solving equation according to the specified period , has been clarified through examples. Finally, Acomparison was made between the three methods ,programs for methods were written in MATLAB language and examples with satisfactory results are given .

There has recently been considerable focus on finding reliable and more effective numerical methods for solving different mathematical problems with integral equations. The Runge-Kutta methods in numerical analysis are a family of iterative methods, both implicit and explicit, with different orders of accuracy, used in temporal and modification for the numerical solutions of integral equations. Fuzzy Integral equations (known as FIEs) make extensive use of many scientific analysis and engineering applications. They appear because of the incomplete information from their mathematical models and their parameters under fuzzy domain. In this paper, the sixth order Runge-Kutta is used to solve second-kind fuzzy Volterra integral equations numerically. The proposed method is reformulated and updated for solving fuzzy second-kind Volterra integral equations in general form by using properties and descriptions of fuzzy set theory. Furthermore a Volterra fuzzy integral equation, based on the parametric form of a fuzzy numbers, transforms into two integral equations of the second kind in the crisp case under fuzzy properties. We apply our modified method using the specific example with a linear fuzzy integral Volterra equation to illustrate the strengths and accurateness of this process. A comparison of evaluated numerical results with the exact solution for each fuzzy level set is displayed in the form of table and figures. Such results indicate that the proposed approach is remarkably feasible and easy to use.

The aim of this paper is to study and to obtain an approximate solution of non-linear Volterra integral equation of the second kind ,the researcher implemented the modified method by using specific examples involving volterra integral equation to show the capability and efficiency of our approximate method according to the exact solution in addition to the ease in programming the approximate method Keywords: General 2 nd order of non-linear Volterra integral equation, 6 th order Improved Range-Kutta methods. ________________________________________________________________________________________________________ 2-DEFINITION OF VOLTERRA INTEGRAL EQUATIONSThe general form of volterra integral equation is given by the form