http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
1 online resource (294 pages) : 84 illustrations, text file, PDF
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.
The fractional Laplacian in one dimension Random walkers with constant steps Ordinary diffusionRandom 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 Greens function of the fractional Laplace equation Fractional Poisson equation in a restricted domain Greens 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 derivativeProperties 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 Greens functionsIntegral 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 Greens function of the fractional Laplace equation The Riesz potential Triply periodic Greens 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 Greens function of the fractional Laplace equation The Riesz potential Doubly periodic Greens 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.