Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media

Version 1 : Received: 25 July 2021 / Approved: 26 July 2021 / Online: 26 July 2021 (11:16:34 CEST)

How to cite: Bueno-Orovio, A.; Burrage, K. The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media. Preprints 2021, 2021070569 (doi: 10.20944/preprints202107.0569.v1). Bueno-Orovio, A.; Burrage, K. The Complex-Order Fractional Laplacian: Stable Operators, Spectral Methods, and Applications to Heterogeneous Media. Preprints 2021, 2021070569 (doi: 10.20944/preprints202107.0569.v1).

Abstract

Fractional differential equations have become a fundamental modelling approach for understanding and simulating the many aspects of nonlocality and spatial heterogeneity of complex materials and systems. Yet, while real-order fractional operators are nowadays widely adopted, little progress has been made in extending such operators to complex-order counterparts. In this work, we introduce new definitions for the complex-order fractional Laplacian, fully consistent with the theory of averaging of smooth functions over fractal sets, and present tailored spectral methods for their numerical treatment. The proposed complex-order operators exhibit a dual particle-wave behaviour, with solutions manifesting wave-like features depending on the magnitude of the imaginary part of the fractional order. Reaction-diffusion systems driven by the complex-order fractional Laplacian exhibit unique spatio-temporal dynamics, such as equilibrium of diffusion in random materials by interference of scattered waves, conduction block and highly fractionated propagation, or the generation of completely novel Turing patterns. Taken together, our results support that the proposed complex-order operators hold unparalleled capabilities to advance the description of multiscale transport phenomena in physical and biological processes highly influenced by the heterogeneity of complex media.

Keywords

Fractional calculus; fractional Laplacian; anomalous diffusion; spectral methods; reaction-diffusion equations; nonlocality; spatial heterogeneity

Subject

MATHEMATICS & COMPUTER SCIENCE, Algebra & Number Theory

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