Wavelet collocation method and multilevel augmentation method for Hammerstein equations (Q2882791)

The Hammerstein nonlinear integral equation is presented in the form of the Fredholm nonlinear integral equation of the second kind NEWLINE\[NEWLINE u(t)- \int_I K(t,s)\Psi(s,u(s))ds=f(t),\qquad t\in I\subset R, NEWLINE\]NEWLINE where the kernel \( K(t,s)\) is considered weakly singular and there exist the positive constants \( \sigma < 1 , \theta \) such that NEWLINE\[NEWLINE| D^{\alpha}_{t}D^{\beta}_{s}K(t,s)|\leq\frac{\theta}{|t-s|^{\sigma+ 2m}},\qquad \alpha,\beta\leq m .NEWLINE\]NEWLINE The approximate solution of this problem is obtained using the collocation method with a special wavelet basis and the functionals. The nonlinear approximate system is solved by the Newton method. The authors also develop the multilevel iterative method to solve this problem and give the numerical results for many examples.