Hypersingular Integral Equations Encountered in Problems of Mechanics

In the paper, for hypersingular integral equations with new kernels, a solution is constructed using an approach based on Chebyshev orthogonal polynomials and the principle of contraction mappings. Integrals in hypersingular integral equations are understood in the sense of Hadamard finite-part integrals. The hypersingular integral equations under consideration in some cases of kernels are solved exactly in closed form using the Chebyshev orthogonal polynomial method, and with other kernels by the same method, they are reduced to infinite systems of linear algebraic equations. In addition, hypersingular integral equations with the kernels considered in the article are reduced to finite systems of linear algebraic equations using Gauss–Chebyshev type quadrature formulas. To assess the effectiveness of the two methods, a comparative analysis of the results for hypersingular integral equations with the corresponding kernels is carried out.