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

• Date: 2025
• Main Authors: Khalilov, Elnur H., Aliev, Araz R., Musayev, Ali M.
• Format: Article
• Language: English
• Published: Institute of Mathematics, NAS of Ukraine 2025
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/7762
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

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.