An approximate solution of nonlinear hypersingular integral equations

This paper describes numerical schemes based on spline-collocation method and their justifications for approximate solutions of linear and nonlinear hypersingular integral equations with singularities of the second kind. Collocations with continuous splines and piecewise constant functions are examined for solving linear hypersingular integral equations. Uniqueness of the solution has been proved. An error of approximation has been obtained for collocation with continuous spline in case a solution of equation has derivatives up to the second order. Collocation with piecewise constant functions are examined for nonlinear hypersingular equations. The convergence of the method has been justified. An estimate of error has been obtained. Illustrative examples demonstrate the accuracy and efficiency of the developed algorithms.