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2015 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Linear Fractional Diffusion-Wave Equation for Scientists and Engineers

verfasst von: Yuriy Povstenko

Verlag: Springer International Publishing

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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SUCHEN

Über dieses Buch

This book systematically presents solutions to the linear time-fractional diffusion-wave equation. It introduces the integral transform technique and discusses the properties of the Mittag-Leffler, Wright, and Mainardi functions that appear in the solutions. The time-nonlocal dependence between the flux and the gradient of the transported quantity with the “long-tail” power kernel results in the time-fractional diffusion-wave equation with the Caputo fractional derivative. Time-nonlocal generalizations of classical Fourier’s, Fick’s and Darcy’s laws are considered and different kinds of boundary conditions for this equation are discussed (Dirichlet, Neumann, Robin, perfect contact). The book provides solutions to the fractional diffusion-wave equation with one, two and three space variables in Cartesian, cylindrical and spherical coordinates.

The respective sections of the book can be used for university courses on fractional calculus, heat and mass transfer, transport processes in porous media and fractals for graduate and postgraduate students. The volume will also serve as a valuable reference guide for specialists working in applied mathematics, physics, geophysics and the engineering sciences.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
Partial differential equations arise in various fields of science. Today, the literature on these equations is unbounded. Usually partial differential equations are divided into three basic types – elliptic, parabolic and hyperbolic.
Yuriy Povstenko
Chapter 2. Mathematical Preliminaries
Abstract
The integral transform technique allows us to remove partial derivatives from the considered equations and to obtain the algebraic equation in a transform domain. Here we briefly recall the integral transforms which are used in this book to reduce the differential operators to an algebraic form. The Laplace transform with respect to time is marked by an asterisk, the Fourier transforms are denoted by a tilde, the Hankel transforms are indicated by a hat and the Legendre transform is designated by a star.
Yuriy Povstenko
Chapter 3. Physical Backgrounds
Abstract
The conventional theory of heat conduction is based on the classical (local) Fourier law, which relates the heat flux vector q to the temperature gradient
$${\bf q}=-k\;\mathrm{grad}\;T$$
, where k is the thermal conductivity of a solid.
Yuriy Povstenko
Chapter 4. Equations with One Space Variable in Cartesian Coordinates
Yuriy Povstenko
Chapter 5. Equations with One Space Variable in Polar Coordinates
Yuriy Povstenko
Chapter 6. Equations with One Space Variable in Spherical Coordinates
Yuriy Povstenko
Chapter 7. Equations with Two Space Variables in Cartesian Coordinates
Yuriy Povstenko
Chapter 8. Equations in Polar Coordinates
Yuriy Povstenko
Chapter 9. Axisymmetric Equations in Cylindrical Coordinates
Yuriy Povstenko
Chapter 10. Equations with Three Space Variables in Cartesian Coordinates
Yuriy Povstenko
Chapter 11. Equations with Three Space Variables in Cylindrical Coordinates
Yuriy Povstenko
Chapter 12. Equations with Three Space Variables in Spherical Coordinates
Abstract
Consider the time-fractional diffusion-wave equation with a source term in spherical coordinates r,θand φ:
$$\frac{\partial^\alpha T}{\partial t^\alpha}\;=\;a\Bigg[\frac{\partial^2 T}{\partial r^2}\;+\frac{2}{r}\;\frac{\partial T}{\partial r}\;+\frac{1}{r^2 \mathrm{sin}\;\theta}\;\frac{\partial}{\partial\theta}\Bigg(\mathrm{sin}\;\theta\frac{\partial T}{\partial\theta}\Bigg)\;+\;\frac{1}{r^2\mathrm{sin}^2\theta}\frac{\partial^2 T}{\partial \varphi^2}\Bigg]+\;\Phi(r,\theta,\varphi,t),\qquad \qquad 0\leq r\leq\infty, 0\leq \theta\leq \pi, 0\leq\varphi\leq 2\pi.$$
Yuriy Povstenko
Backmatter
Metadaten
Titel
Linear Fractional Diffusion-Wave Equation for Scientists and Engineers
verfasst von
Yuriy Povstenko
Copyright-Jahr
2015
Electronic ISBN
978-3-319-17954-4
Print ISBN
978-3-319-17953-7
DOI
https://doi.org/10.1007/978-3-319-17954-4

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