Optimal Hermite-Type Quadrature with Weakly Singular Weight Via Phi-Functions

Background: Many practical problems in physics and engineering involve calculating integrals with singularities. Traditional calculation methods lose accuracy near these singular points, so new methods are needed that can handle such difficult integrals. Purpose: This paper develops an improved calculation method for such integrals. The new method uses both function values and their slopes at selected points and is optimized to give the best possible accuracy. Method: We develop the mathematical framework by creating a special function that captures the behavior of the singular integral. By optimizing the parameters of this special function, we determine the best weights and coefficients for the calculation. Results: We find all the coefficients needed for the new calculation method and prove that the method works correctly. The method is accurate for polynomial functions, and numerical tests show it gives results that are three to five times more accurate than existing standard methods. Implication: The new method is practical and efficient for solving real problems that involve difficult integrals with singularities, and the approach can be extended to other similar problems.