Solution of nonlinear mixed integral equation via collocation method basing on orthogonal polynomials

In this paper, we study the existence of a unique solution of nonlinear mixed integral equation (NMIE) of the third kind in position and time in the space [Formula: see text] , [Formula: see text]. Moreover, the stability of the solution is discussed. Using a quadratic method, the NMIE is transformed into a system of NIEs in position. Using a collocation method with aid of two different polynomials, Hermite and Laguerre polynomials, we have two different nonlinear algebraic systems (NAS). The estimate of the error, in each numerical method is discussed. Many applications, for the NMIE of the first, second and third kind with continuous kernels in position and time, are considered. In addition, by considering different times of the proposed method and using Mable 18, many numerical results are computed. Moreover, the error estimate, in each case, is calculated.