Numerical treatments for solving nonlinear mixed integral equation

We consider a mixed type of nonlinear integral equation (MNLIE) of the second kind in the space C[0,T]×L2(Ω),T<1. The Volterra integral terms (VITs) are considered in time with continuous kernels, while the Fredholm integral term (FIT) is considered in position with singular general kernel. Using the quadratic method and separation of variables method, we obtain a nonlinear system of Fredholm integral equations (NLSFIEs) with singular kernel. A Toeplitz matrix method, in each case, is then used to obtain a nonlinear algebraic system. Numerical results are calculated when the kernels take a logarithmic form or Carleman function. Moreover, the error estimates, in each case, are then computed.