Beyond Classical Diffusion: A Sparse Legendre Wavelet Method for Fractional Heat Equations With Heat Sources

ABSTRACT Fractional heat equations (FHEs) are crucial for modeling systems exhibiting memory effects and non‐local thermal behavior, commonly found in advanced materials, microelectronics, and biological systems. However, their numerical solution remains challenging due to the complexity of fractional‐order derivatives. This paper presents an efficient numerical approach based on the Legendre wavelet method (LWM) to solve time and space fractional heat equations with and without heat sources. By employing sparse operational matrices, the proposed scheme transforms fractional boundary and initial value problems into algebraic systems, significantly reducing its computational cost. Numerical simulations demonstrate the high accuracy of the method, which is validated through comparisons with some well‐established techniques such as the homotopy analysis method (HAM), Laplace residual power series method (LRPSM), Chebyshev collocation method (CCM), and fractional Laplace‐adomian decomposition method (FLADM). The considered method is tested on several benchmark problems, including anomalous diffusion and fractional models with heat sources. Graphical and tabulated results confirm that LWM delivers a better approximation when compared to the aforementioned techniques. The results suggest that LWM is a reliable, accurate, and computationally efficient tool for solving complex fractional heat equations in engineering and environmental modeling contexts. Importantly, our contribution is algorithmic not a new physical model via a sparse, block‐structured Legendre‐wavelet collocation for Caputo operators that unifies time‐ and space‐fractional Heat equation with or without sources.