The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion
Reads0
Chats0
TLDR
In this article, the authors describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research, including the linear heat equation and the theory of parabolic equations of different types.Abstract:
We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic partial differential equations gives a foundation to the probabilistic study of diffusion.read more
Citations
More filters
Journal ArticleDOI
The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)
Journal ArticleDOI
What is the fractional Laplacian? A comparative review with new results
Anna Lischke,Guofei Pang,Mamikon Gulian,Fangying Song,Christian A. Glusa,Xiaoning Zheng,Zhiping Mao,Wei Cai,Mark M. Meerschaert,Mark Ainsworth,George Em Karniadakis +10 more
TL;DR: A comparison of several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties, and a collection of benchmark problems to compare different definitions on bounded domains using a sample of state-of-the-art methods.
Posted Content
What Is the Fractional Laplacian
Anna Lischke,Guofei Pang,Mamikon Gulian,Fangying Song,Christian A. Glusa,Xiaoning Zheng,Zhiping Mao,Wei Cai,Mark M. Meerschaert,Mark Ainsworth,George Em Karniadakis +10 more
TL;DR: This work provides a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and presents new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions.
References
More filters
Book
Partial Differential Equations
TL;DR: In this paper, the authors present a theory for linear PDEs: Sobolev spaces Second-order elliptic equations Linear evolution equations, Hamilton-Jacobi equations and systems of conservation laws.
Book
Singular Integrals and Differentiability Properties of Functions.
TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.
Journal ArticleDOI
The Chemical Basis of Morphogenesis
TL;DR: In this article, it is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis.
Book ChapterDOI
Elliptic Partial Differential Equations of Second Order
Piero Bassanini,Alan R. Elcrat +1 more
TL;DR: In this paper, a class of partial differential equations that generalize and are represented by Laplace's equation was studied. And the authors used the notation D i u, D ij u for partial derivatives with respect to x i and x i, x j and the summation convention on repeated indices.
Journal ArticleDOI
The random walk's guide to anomalous diffusion: a fractional dynamics approach
Ralf Metzler,Joseph Klafter +1 more
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.