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The Fractional Laplacian




ISBN 9781498746151
Published February 24, 2016 by Chapman and Hall/CRC
294 Pages 74 B/W Illustrations

 
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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

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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