Nonlinear Fredholm-Volterra Integral Equation and its Numerical Solutions with Quadrature Methods

In this work, the existence and uniqueness of solution of the nonlinear Fredholm-Volterra integral equation (NF-VIE), with continuous kernels, are discussed and proved in the space L2(Ω)—C(0,T). The Fredholm integral term (FIT) is considered in position while the Volterra integral term (VIT) is considered in time. Using a numerical technique we have a nonlinear system of Fredholm integral equations (SFIEs). This system of integral equations can be reduced, using quadrature methods, to a nonlinear algebraic system (NAS). Then, the NAS can be solved, using two numerical methods. These methods are: Trapezoidal rule method and Simpson's rule method. Finally, some numerical examples are considered and the error estimate, in each case, is computed.