We propose a method for solving linear and nonlinear hypersingular integral equations. For nonlinear equations the advantage of the method is in rather weak requirements for the nonlinear operator behavior in the vicinity of the solution. The singularity of the kernel not only guarantees strong diagonal dominance of the discretized equations, but also guarantees the convergence of a simple iterative scheme based on Lyapunov stability theory. The resulting computational method can be implemented with recurrent neural networks or analog computers.