Approximate solution of hypersingular integral equations on the number axis

In the paper we investigate approximate methods for solving linear and nonlinear hypersingular integral equations defined on the number axis. We study equations with the second-order singularities because such equations are widely used in problems of natural science and technology. Three computational schemes are proposed for solving linear hypersingular integral equations. The first one is based on the mechanical quadrature method. We used rational functions as the basic ones. The second computational scheme is based on the spline-collocation method with the first-order splines. The third computational scheme uses the zero-order splines. Continuous method for solving operator equations has been used for justification and implementation of the proposed schemes. The application of the method allows to weaken the requirements imposed on the original equation. It is sufficient to require solvability for a given right-hand side. The continuous operator method is based on Lyapunov's stability for solutions of systems of ordinary differential equations. Thus it is stable for perturbations of coefficients and of right-hand sides. Approximate methods for solving nonlinear hypersingular integral equations are presented by the example of the Peierls - Naborro equation of dislocation theory. By analogy with linear hypersingular integral equations, three computational schemes have been constructed to solve this equation. The justification and implementation are based on continuous method for solving operator equations. The effectiveness of the proposed schemes is shown on solving the Peierls - Naborro equation.