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 L 2 ( 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.