This dissertation is focused on the varieties of numerical solutions of nonlinear Hammerstein integral equations. In the first part of this dissertation, several acceleration techniques for post-processed solutions of the Hammerstein equation are discussed. The post-processing techniques are implemented based on interpolation and extrapolation. In this connection, we generalize the results in [29] and [28] to nonlinear integral equations of the Hammerstein type. Post-processed collocation solutions are shown to exhibit better accuracy. Moreover, an extrapolation technique for the Galerkin solution of Hammerstein equation is also obtained. This result appears new even in the setting of the linear Fredholm equation.

In the second half of this dissertation, the wavelet-collocation technique of solving nonlinear Hammerstein integral equation is discussed. The main objective is to establish a fast wavelet-collocation method for Hammerstein equation by using a 'linearization' technique. The sparsity in the Jacobian matrix takes place in the fast wavelet-collocation method for Hammerstein equation with smooth as well as weakly singular kernels. A fast algorithm is based upon the block truncation strategy which was recently proposed in [10]. A multilevel augmentation method for the linearized Hammerstein equation is subsequently proposed which further accelerates the solution process while maintaining the order of convergence. Numerical examples are given throughout this dissertation.