The numerical solution of an Abel integral equation by the optimal quadrature formula

In this study, a novel and efficient approach utilizing optimal quadrature formulas is introduced to derive approximate solutions for generalizing Abel’s integral equations. The method, characterized by high accuracy and simplicity, involves constructing optimal quadrature formulas in the sense of Sard and providing error estimates within the Hilbert space of differentiable functions. The squared norm of the error functional for the quadrature formula in the space $W_2^{(2.1)}(0,t)$ is computed. To minimize this error, a system of linear equations regarding the formula’s coefficients is derived, leading to a unique solution. Then the explicit expressions for these optimal coefficients are obtained. The validity of the approach is demonstrated by solving several integral equations, with approximation errors presented in the corresponding tables.