Investigation of the approximate solution of one class of curvilinear integral equations by the projection method

Elnur H. Khalilov; Araz R. Aliev; Ali M. Musayev

doi:10.3842/umzh.v76i10.7762

Elnur H. Khalilov Azerbaijan State Oil and Industry University, Baku, Azerbaijan; Scientific and Innovation Center of the Western Caspian University, Baku, Azerbaijan
Araz R. Aliev Azerbaijan State Oil and Industry University, Baku, Azerbaijan; Institute of Mathematics and Mechanics, Baku, Azerbaijan; Center for Mathematics and Its Applications, Khazar University, Baku, Azerbaijan
Ali M. Musayev Azerbaijan State Oil and Industry University, Baku, Azerbaijan

DOI: https://doi.org/10.3842/umzh.v76i10.7762

Keywords: exterior Dirichlet boundary value problem, Helmholtz equation, boundary integral equations method, curvilinear singular integral, collocation method

Abstract

UDC 517.9

We prove the existence theorem for the normal derivative of the double-layer potential and establish the formula for its evaluation. A new method for the construction of quadrature formulas for the normal derivatives of simple- and double-layer potentials is developed, and the error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space is replaced by a system of algebraic equations, and the existence and uniqueness of the solution to this system is proved. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is proved and the convergence rate of the method is determined.

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