Abstract
This paper presents a numerical approach for solving Fredholm integral equations of the first kind using the Bubnov–Galerkin method with Alpert wavelet bases. These equations are well-known for being ill-posed, meaning that small changes in input data can lead to large deviations in the solution. Therefore, robust and accurate numerical methods are essential. The proposed method utilizes orthonormal and compactly supported Alpert wavelets, which offer excellent localization properties and yield well-conditioned, sparse system matrices when projecting the integral operator. This enhances numerical stability and reduces computational complexity. A series of computational experiments was carried out using various refinement levels and polynomial degrees. The accuracy of the method was evaluated by comparing approximate solutions to the exact analytical solution. The results demonstrate exceptionally small absolute errors, often approaching machine precision. Additionally, a comparative analysis with power polynomial bases confirms the superiority of the Alpert wavelet approach in terms of convergence and approximation quality. Overall, the method proves to be efficient, stable, and suitable for further extension to more complex integral equations, including multidimensional and noisy-data problems. This confirms the potential of Alpert wavelet-based Galerkin schemes as a reliable tool for the numerical treatment of inverse and ill-posed problems in applied sciences.