The Fractional Laplacian
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Opis: The Fractional Laplacian - Constantine Pozrikidis

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 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


Szczegóły: The Fractional Laplacian - Constantine Pozrikidis

Tytuł: The Fractional Laplacian
Autor: Constantine Pozrikidis
Wydawnictwo: Productivity Press Inc
ISBN: 9781498746151
Rok wydania: 2016
Ilość stron: 294
Oprawa: Twarda
Waga: 0.54 kg


Recenzje: The Fractional Laplacian - Constantine Pozrikidis
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Książka

The Fractional Laplacian

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 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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