An Efficient HaarWavelet–Finite Difference Scheme for Solving Heat-Type Parabolic Equations

This paper presents a detailed numerical analysis of a particular class of heat equations. To approximate the solutions, we develop a hybrid computational framework that combines the Haar wavelet collocation method with finite difference techniques. In the proposed approach, finite difference schemes are first applied to discretize the time variable, thereby reducing the original problem to a semi-discrete system. Subsequently, Haar wavelets defined on a uniform spatial grid are employed to approximate the spatial derivatives, allowing for efficient representation of both smooth and non-smooth solution profiles. Within the framework of Sobolev spaces, rigorous proofs of stability and convergence are provided, ensuring the reliability of the method from a theoretical standpoint. To validate the practical performance of the proposed scheme, extensive numerical experiments are conducted across a variety of test problems. The accuracy of the method is quantitatively assessed using both the <img src=image/13443061_01.gif> and <img src=image/13443061_02.gif> error norms. The results consistently demonstrate that the method delivers high accuracy while maintaining strong computational efficiency. Furthermore, the computed solutions exhibit excellent agreement with exact analytical solutions, thereby confirming the robustness and effectiveness of the hybrid scheme for solving heat equations. In addition, the findings of this research make several contributions to the field of numerical analysis of partial differential equations.