Highly Accurate Numerical Method for First-Kind Singular Integral Equations

In this paper, we present a highly accurate numerical scheme for solving first-kind Cauchy-type singular integral equations (CSIEs). The method is based on approximating the unknown function using Chebyshev series and applying modified Gauss–Chebyshev quadrature formulas with Gauss–Lobatto nodes to evaluate weighted kernel integrals. The proposed approach is versatile and can effectively handle bounded, unbounded, and semi-bounded solution domains. The invertibility of the associated operator is established for both bounded and unbounded cases, ensuring the uniqueness of the solution. Furthermore, convergence rates are rigorously proven in Hilbert space. Several numerical examples are provided to demonstrate the performance of the method, along with detailed comparisons to existing numerical techniques. These results confirm the simplicity, efficiency, and reliability of the proposed scheme, making it a powerful tool for solving first-kind CSIEs in a wide range of applications.