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

Abstract

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.