Numerical treatments to nonlocal Fredholm –Volterra integral equation with continuous kernel

In this paper, we consider the nonlocal FredholmVolterra integral equation of the second kind, with continuous kernels. We consider three different numerical methods,the Trapezoidal rule, Simpson rule and Collocation method to reduce the nonlocal F-VIE to a nonlocal algebraic system of equations. The algebraic system is computed numerically, when the historical memory of the problem (nonlocal function) takes three cases: when there is no memory, when the memory is linear and when the memory is nonlinear. Moreover, the estimate error, in each method and each case, is computed. Here, we deduce that, the error in the absence of memory is larger than in the linear memory. Moreover, the error of the linear memory is larger than the nonlinear memory. Keyword: nonlocal Fredholm-Volterra integral equation (nonlocal F-VIE), numerical methods, algebraic system (AS), the error estimate. MSC (2010): 45B05, 45G10, 60R.