Hammerstein integral equation with a generalized singular kernel

In this work, the Hammerstein integral equation (HIE), with a generalized singular kernel, is considered and solved numerically, using Product Trapezoidal rule, Toeplitz matrix method and Product Nystrom method. The existence and uniqueness solution, under certain conditions, are considered. Moreover, numerical results when the kernel takes a generalized logarithmic form, Carleman function and Cauchy kernel are investigated. Also the error, in each case, is estimated.