This study presents a comprehensive numerical approach for solving Fredholm integral equations of the first kind in both one and two dimensions through the application of Haar wavelet collocation methods to Tikhonov regularized formulations. The study establishes rigorous convergence analysis in the \(L^2\) norm for both one and two-dimensional cases, while addressing the critical challenge of regularization parameter selection through the implementation of the Morozov principle and adaptive parameter selection rules within an a posteriori framework that operates independently of prior knowledge regarding solution smoothness. The methodology demonstrated its practical efficacy through extensive numerical experiments across one and two-dimensional problems, with convergence rate evaluations conducted using optimally selected regularization parameters. A comparative analysis with existing methodologies from recent literature validates the superior performance and computational efficiency of the proposed Haar wavelet collocation approach, establishing it as a robust and reliable technique for addressing ill-posed integral equation problems in multiple dimensions.
