A Fourier Analysis of Classical and Fractional Diffusion: Smoothing, Decay, and Comparison
DOI:
https://doi.org/10.32493/sm.v8i2.59338Keywords:
heat equation, fractional diffusion equation, Fourier transform, fractional LaplacianAbstract
This paper investigates the classical heat equation and the fractional diffusion equation using Fourier transform methods. By transforming both equations into the frequency domain, explicit representations of their solutions are obtained. These representations are then used to analyze several analytical properties, including smoothing effects, decay estimates, and the limiting behavior of the fractional model as the fractional order approaches one. The results show that the classical heat equation exhibits stronger smoothing and faster decay, while the fractional diffusion equation produces heavier spatial tails due to the intrinsic nonlocal behavior of the fractional Laplacian. This study highlights the main differences and connections between classical and fractional diffusion processes within a unified Fourier analysis framework.
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