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<p>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. The error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, we replace the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space by a system of algebraic equations and prove the existence and uniqueness of the solution to this system. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is demonstrated and the convergence rate of the method is determined.</p>

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Investigation of the Approximate Solution of One Class of Curvilinear Integral Equations by the Projection Method

  • 期刊论文

  • 发表日期：

    2025年3月1日

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  • Elnur H. Khalilov,
  • Araz R. Aliev,
  • Ali M. Musayev

摘要

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. The error estimates are obtained for the constructed quadrature formulas. By using these quadrature formulas, we replace the integral equation of the exterior Dirichlet boundary-value problem for the Helmholtz equation in two-dimensional space by a system of algebraic equations and prove the existence and uniqueness of the solution to this system. The convergence of the solution of the system of algebraic equations to the exact solution of the integral equation at the control points is demonstrated and the convergence rate of the method is determined.