New methods for solving of nonlinear weakly singular integral equations.

Purpose ? Aims to present a boundary integral equation method for solving Laplace's equation ?u=0 with nonlinear boundary conditions. Design/methodology/approach ? The nonlinear boundary value problem is reformulated as a nonlinear boundary integral equation, with u on the boundary as the solution being sought. The integral equation is solved numerically by using the collocation method on smooth or nonsmooth boundary; the singularities of solution degrade the rates of convergence. Findings ? Variants of the methods for finding numerical solutions are suggested. So these methods have been compared with respect to number of iterations. Practical implications ? Numerical experiments show the efficiency of the proposed methods. Originality/value ? Provides new methods to solve nonlinear weakly singular integral equations and discusses difficulties that arise in particular cases. [ABSTRACT FROM AUTHOR]