Numerical Solution of a Singular Integral Equation of the First Kind with Hilbert Kernel

In solving practical problems in the fields of physics and engineering, singular integral equations are frequently encountered. Among these, singular integral equations with the Hilbert kernel constitute the periodic cases. In this article, we discuss the construction of an optimal quadrature formula for the numerical solution of Fredholm-type singular integral equations of the first kind with Hilbert kernels using the functional approach in the space L2(1)(0,2π). Using the constructed optimal quadrature formula, the error between the exact solution and the approximate solution of the integral equation is demonstrated through examples. Graphs illustrate how the approximate value converges to the exact value as the number of nodes in the optimal quadrature formula increases.