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# The Fractional Laplacian

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## Book Description

The fractional Laplacian, also called the Riesz fractional derivative, describes an unusual diffusion process associated with random excursions. **The Fractional Laplacian** explores applications of the fractional Laplacian in science, engineering, and other areas where long-range interactions and conceptual or physical particle jumps resulting in an irregular diffusive or conductive flux are encountered.

- Presents the material at a level suitable for a broad audience of scientists and engineers with rudimentary background in ordinary differential equations and integral calculus
- Clarifies the concept of the fractional Laplacian for functions in one, two, three, or an arbitrary number of dimensions defined over the entire space, satisfying periodicity conditions, or restricted to a finite domain
- Covers physical and mathematical concepts as well as detailed mathematical derivations
- Develops a numerical framework for solving differential equations involving the fractional Laplacian and presents specific algorithms accompanied by numerical results in one, two, and three dimensions
- Discusses viscous flow and physical examples from scientific and engineering disciplines

Written by a prolific author well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science, the book emphasizes fundamental ideas and practical numerical computation. It includes original material and novel numerical methods.

## Table of Contents

**The fractional Laplacian in one dimension **

Random walkers with constant steps

Ordinary diffusion

Random jumpers

Central limit theorem and stable distributions

Power-law probability jump lengths

A principal-value integral

Wires and springs

The fractional Laplacian

Fourier transform

Effect of fractional order

Numerical computation of the fractional Laplacian

Green’s function of the fractional Laplace equation

Fractional Poisson equation in a restricted domain

Green’s function of unsteady fractional diffusion

**Numerical discretization in one dimension **

Computation of a principal-value integral

Fractional Laplacian differentiation matrix

Fractional Poisson equation

Evolution under fractional diffusion

Differentiation by spectral expansion

**Further concepts in one dimension **

Fractional first derivative

Properties of the fractional first derivative

Laplacian potential

Fractional derivatives from finite-difference stencils

Fractional third derivative

Fractional fourth derivative

**Periodic functions **

Sine, cosines, and the complete Fourier series

Cosine Fourier series

Sine Fourier series

Green’s functions

Integral representation of the periodic Laplacian

Numerical discretization

Periodic differentiation matrix

Differentiation by spectral expansion

Embedding of the fractional Poisson equation

**The fractional Laplacian in three dimensions **

Stipulation on the Fourier transform

Integral representation

Fractional gradient

Laplacian potential

Green’s function of the fractional Laplace equation

The Riesz potential

Triply periodic Green’s function

Fractional Poisson equation

Evolution under fractional diffusion

Periodic functions and arbitrary domains

Fractional Stokes flow

**The fractional Laplacian in two dimensions **

Stipulation on the Fourier transform

Integral representation

Fractional gradient

Laplacian potential

Green’s function of the fractional Laplace equation

The Riesz potential

Doubly periodic Green’s function

Fractional Poisson equation

Evolution due to fractional diffusion

Periodic functions and arbitrary domains

**Appendix A: Selected definite integrals ****Appendix B: The Gamma function ****Appendix C: The Gaussian distribution ****Appendix D: The fractional Laplacian in arbitrary dimensions ****Appendix E: Fractional derivatives ****Appendix F: Aitken extrapolation of an infinite sum **

References

Index

## Author(s)

### Biography

**Constantine Pozrikidis** is a professor at the University of Massachusetts Amherst. He is well known for his contributions in fluid mechanics, biomechanics, applied mathematics, scientific computing, and computer science. He is the author of numerous research papers and books, including the highly recommended Chapman & Hall/CRC books *Introduction to Finite and Spectral Element Methods Using MATLAB ^{®}, Second Edition; XML in Scientific Computing; Computational Hydrodynamics of Capsules and Biological Cells; Modeling and Simulation of Capsules and Biological Cells*; and

*A Practical Guide to Boundary Element Methods with the Software Library BEMLIB*.

## Reviews

"The book under review includes an introductory discussion on the fractional Laplacian which should be accessible to scientists who may not be mathematicians. Practical numerical computations are particularly emphasized, and the book includes many exercises. The fundamental ideas are presented without the traditional organization into theorems and proofs. Here is the list of chapter headings: 1. The fractional Laplacian in one dimension. 2. Numerical discretization in one dimension. 3. Further concepts in one dimension. 4. Periodic functions. 5. The fractional Laplacian in three dimensions. 6. The fractional Laplacian in two dimensions. There are also several appendices: A. Selected de nite integrals. B. The Gamma function. C. The Gaussian distribution. D. The fractional Laplacian in arbitrary dimensions. E. Fractional derivatives. F. Aitken extrapolation of an in nite sum."

~Daniel Belita,Mathematical Reviews, 2017