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.

Approximate Methods for Solving Linear and Nonlinear Hypersingular Integral Equations

We propose an iterative projection method for solving linear and nonlinear hypersingular integral equations with non-Riemann integrable functions on the right-hand sides. We investigate hypersingular integral equations with second order singularities. Today, hypersingular integral equations of this type are widely used in physics and technology. The convergence of the proposed method is based on the Lyapunov stability theory of solutions of ordinary differential equation systems. The advantage of the method for linear equations is in simplicity of unique solvability verification for the approximate equations system in terms of the operator logarithmic norm. This makes it possible to estimate the norm of the inverse matrix for an approximating system. The advantage of the method for nonlinear equations is that neither the existence or reversibility of the nonlinear operator derivative is required. Examples are given illustrating the effectiveness of the proposed method.

A Quadrature Formula for a Hypersingular Integral Containing the Weight Function of Jacobi Polynomials with Complex Exponents

Although the concept of a hypersingular integral was introduced by Hadamard at the beginning of the 20th century, it began to be put into practical use only in the second half of the century. The theory of hypersingular integral equations has been widely developed in recent decades and this is due to the fact that they describe the governing equations of many applied problems in various fields: elasticity theory, fracture mechanics, wave diffraction theory, electrodynamics, nuclear physics, geophysics, theory vibrator antennas, aerodynamics, etc. It is analytically possible to calculate the hypersingular integral only for a very narrow class of functions; therefore, approximate methods for calculating such an integral are always in the field of view of researchers and are a rapidly developing area of computational mathematics. There are a very large number of papers devoted to this subject, in which various approaches are proposed both to approximate calculation of the hypersingular integral and to the solution of hypersingular integral equations, mainly taking into account the specifics of the behavior of the densi-ty of the hypersingular integral. In this paper, quadrature formulas are obtained for a hypersingular integral whose density is the product of the Hölder continuous function on the closed interval [–1, 1], and weight function of the Jacobi polynomials . It is assumed that the exponents α and β can be arbitrary complex numbers that satisfy the condition of non-negativity of the real part. The numerical examples show the convergence of the quadrature formula to the true value of the hypersingular integral. The possibility of applying the mechanical quadrature method to the solution of various, including hypersingular, integral equations is indicated.

On the Application of the Method of Hypersingular Integral Equations to Solving Problems for an Elastic Plane with a Collinear System of Cracks

In the present paper, using the method of hypersingular integral equations, based on the formulas of the inversion of the corresponding singular integral equations, the exact quadrature solution of the classical problems of the mechanics of an elastic plane with a collinear system of cracks is constructed. The elastic plane is in a state of antiplane deformation or plane deformation; in case of antiplane deformation, crack edges are symmetrically loaded by tangential forces, while in case of plane deformation, they are again loaded symmetrically but by normal forces. Mixed boundary-value problems for an elastic half-plane equivalent to these problems are formulated. Under plane deformation, the mixed boundary-value problem for an elastic half-plane is discussed as well when the plane boundary is reinforced by two similar and symmetrically located semi-infinite stringers between which a system of an arbitrarily final number of stringers is situated. It is considered that the stringers are absolutely rigid for expansion and compression and absolutely flexible for bending. A particular case of two similar symmetrically located cracks is considered more in detail. In this case, the exact solution to the problem is also constructed by the method of Chebyshev orthogonal polynomials.