The Fractional Laplacian
Taylor and FrancisAmazon Contents (in pdf) Several new concepts are introduced in this book. Those who study fluid mechanics will be interested in the concept of fractional Stokes flow (Section 5.11). Those who study graphs and networks will be interested in the concept of fractional graph Laplacian (Section 4.8.2).
The ordinary Laplacian is defined as the ordinary second derivative of a function of one variable or the sum of the ordinary second partial derivatives of a function of a higher number of variables in a physical or abstract Caresian space. Physically, the ordinary Laplacian describes an ordinary diffusion process in an isotropic medium mediated by non-idle random walkers who step into neighboring or nearby sites of an idealized grid, but are unable to perform long jumps.
In the physical sciences, the ordinary Laplacian appears as a contribution to a conservation law or evolution equation due to a diffusive species flux according to Fick's law, a conductive thermal flux according to Fourier's law, or a viscous stress according to the Newtonian constitutive equation. An implied assumption is that the rate of transport of a field of interest at a certain location is determined by an appropriate field variable at that location, independent of the global structure of the transported field.
The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighboring or nearby sites, and also perform excursions to remote sites by way of L\'evy flights. Literal or conceptual flights have been observed or alleged to occur in a variety of applications, including turbulent fluid motion and material transport in fractured media. In the context of mechanics, the fractional Laplacian describes the motion of a chain or array of particles that are connected by elastic springs not only to their nearest neighbors, but also to all other particles. The spring constant diminishes with the particle separation, while the particle array may describe an ordinary or fractal configuration.
A key physical concept underlying the notion of the fractional Laplacian is the fractional diffusive flux, arising as a generalization of the ordinary diffusive flux expressed by Fick's law, the ordinary conductive flux expressed by Fourier's law, or the expression for the viscous stress according to the Newtonian constitutive equation. The generalized flux associated with the fractional Laplacian provides us with expressions for the rate of transport at a certain location as an integral of an appropriate field variable over an appropriate domain of influence. The fractional diffusive flux at a certain location is affected by the state of the field in the entire space.
The extraordinary effect of the fractional flux can be demonstrated by considering species diffusion or heat conduction in two isolated patches that are separated by an insulating material. Assume that the first patch is devoid of a diffusing species, or else isothermal, whereas the second patch hosts a diffusive species, or else supports a temperature field. Under the influence of a fractional flux, the first patch develops a concentration or temperature field due to the second patch in a process that may appear as an optical illusion or an instance of the paranormal. The physical reason is that material and energy can be transported over long distances by physical or conceptual splattering.
In the most general abstract context, the fractional Laplacian describes the contribution to a conservation law of a non-local process that is affected not only by the local conditions, but also by the global state of a field of interest at a given time. Non-local dependencies are familiar to those who study non-Newtonian mechanics. Applications can be envisioned in a broad range of disciplines in mainstream science and engineering, image processing, but also in sociology, entomology, health care management, and finance.
The notion of the fractional Laplacian provides us with an interesting tool for mathematical modeling when traditional approaches appear to fail. The subtlety of the underlying mathematical concepts has motivated a substantial body of literature in applied mathematics and physics. Despite a long history and considerable progress made in recent years, the general subject is still emerging and a number of conceptual and computational issues require further elaboration or development.
My goal in this book is to offer a concise introduction to the fractional Laplacian at a level that is accessible to mainstream scientists and engineers with a rudimentary background in ordinary differential and integral calculus. Emphasis is placed on fundamental ideas and practical numerical computation. Original material is included throughout the book and novel numerical methods are developed.
There are two intentional peculiarities in the presentation. First, the fractional Laplacian in three dimensions is discussed in Chapter 5, followed by the fractional Laplacian in two dimensions in Chapter 6, and then followed by the fractional Laplacian in arbitrary dimensions in Appendix D. This ordering is due to certain unusual properties of Laplace's equation in two dimensions coupled with the author's belief that the most general case should not necessarily be treated first. The second peculiarity relates to the occasional near repetition of discussion and equations in one, two, or three dimensions. Although consolidation would have abbreviated the discourse, it would have compromised the reader's ability to study the material in a non-sequential fashion.