Abstract
We consider a generalization of the diffusion equation on graphs. This generalized diffusion equation gives rise to both normal and superdiffusive processes on infinite one-dimensional graphs. The generalization is based on the k-path Laplacian operators Lk, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the k-path Laplacian operators are self-adjoint. Then, we study the transformed k-path Laplacian operators using Laplace, factorial and Mellin transforms. We prove that the generalized diffusion equation using the Laplace- and factorial-transformed operators always produce normal diffusive processes independently of the parameters of the transforms. More importantly, the generalized diffusion equation using the Mellin-transformed k-path Laplacians Σ∞k=1k-sLk produces superdiffusive processes when 1 < s < 3.
Original language | English |
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Pages (from-to) | 307-334 |
Number of pages | 28 |
Journal | Linear Algebra and its Applications |
Volume | 523 |
Early online date | 24 Feb 2017 |
DOIs | |
Publication status | Published - 15 Jun 2017 |
Keywords
- k-path Laplacian
- anomalous diffusion