Numerical Integration Formulas for Hypersingular Integrals

It is known that the solution of the Cauchy problem for partial differential equations of hyperbolic type can be reduced to singular integrals of a unique form. Laterly, singular integrals were called integrals in the sense of Hadamard or Hadamard integrals. In addition to equations of the hyperbolic type, Hadamard integrals are widely used in elasticity, electrodynamics, aerodynamics, and other vital areas of mechanics and mathematical physics. The exact evaluation of Hadamard integrals is possible only in exceptional cases, so that there is a need to develop approximate methods for their evaluation. In the present paper, we develop an optimal algorithm for the approximate calculation of Hadamard integrals. Here, we deal with finding the analytical form of the coefficients of an optimal quadrature formula. Numerical results show the validity and accuracy of the method.

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