Global Conference on Mathematics and Applications (GCMA)

First-kind Fredholm integral equations with algebraically weakly singular kernels are severely ill-posed, making numerical solutions highly sensitive to data noise. To overcome this instability, we introduce a robust two-stage numerical framework. First, the unknown solution is projected onto a basis of orthogonal Legendre polynomials, transforming the integral equation into a discrete linear system whose weakly singular integrals are computed using a Jacobi–Gauss quadrature rule. Second, Tikhonov regularization is incorporated to stabilize the ill-conditioned system. We provide an a priori strategy for parameter selection and establish a theoretical error bound of order $\mathcal{O}(\delta^{2/3})$. Numerical experiments validate the theoretical estimates, demonstrating high accuracy and stability even under substantial noise levels.