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About 10,000 papers for “Diffusion equation”

Diffusion Equation

Christopher K. W. Tam · 2012

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Diffusion Equation

Fidel Santamarı́a, Toma M. Marinov · Encyclopedia of Computational Neuroscience · 2022

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Diffusion equation

Christian Huber · 2025

Abstract Diffusion is ubiquitous in Earth and planetary sciences and occurs over a wide range of spatial and temporal scales. In this chapter, the diffusion equation is discretized according to three methods: explicitly, implicitly, and with the Crank–Nicolson method. While the explicit discretization is easier to implement because a simple marching equation exists for each spatial node, it is shown to be conditionally stable. The implicit discretization is more challenging to implement and requires a matrix inversion, but is unconditionally stable. The matrix inversion can be made very efficient because the matrix is sparse and tridiagonal. Finally, the Crank–Nicolson method requires an approach similar to the implicity scheme (matrix inversion) but offers a second-order accuracy over time (instead of first order for the other schemes). A potential complication with implicit schemes is how their matrix inversion solution procedure generalizes to higher dimensions (two or three). This chapter shows that the alternative direction implicit (ADI) method provides an elegant solution to this issue and preserves the unconditional stability of the implicit scheme. Applications are shown for diffusion kinetics within crystals and crater diffusion on planetary surfaces caused by small meteoric impacts.

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Diffusion Equation

Fidel Santamarı́a, Toma M. Marinov · Encyclopedia of Computational Neuroscience · 2015

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A diffusion equation

G. F. Newell · Lecture notes in economics and mathematical systems · 1984

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The Diffusion Equation

Karoline Horgmo Jæger, Aslak Tveito · Simula SpringerBriefs on computing · 2023

Abstract The diffusion equation appears in many applications in science and engineering, and computational physiology is no exception. In its most basic form, the diffusion equation is also useful as an example of how to deal with a PDE using numerical methods. We will start by considering it as a stand-alone model, but in the next chapterswe will study it in combination with non-linear ODEs. This chapter therefore serves as a warm-up for the more complex models. We will also follow the path we started above. In the very simplest case of an ODE, we found a formula for the solution of both the differential equation and the numerical scheme approximating the equation. One nice consequence of this is that, as we saw above, we can explicitly study the error introduced by the numerical approximation. In this chapter, we will use the same approach to study the error of the numerical approximation of the diffusion equation.

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The Diffusion Equation

John S. Gulliver · Cambridge University Press eBooks · 2007

In this chapter, we will review various solution techniques for the diffusion equation, which is generally defined as the mass transport equation with diffusive terms. These techniques will be applied to chemical transport solutions in sediments. There are also a number of applications to chemical transport in biofilms. There are many other applications of the diffusion equation, including most of the topics of this text, but they require more background with regard to the physics of mixing processes, which will be addressed in later chapters.

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Hyperbolic diffusion equation

Leopold Herrmann · AIP conference proceedings · 2012

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Leopold Herrmann; Hyperbolic diffusion equation. AIP Conf. Proc. 10 December 2012; 1504 (1): 1337–1340. https://doi.org/10.1063/1.4772177 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

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Paring Diffusion Equation

BENTHAM SCIENCE PUBLISHERS eBooks · 2013

In this section, we consider reaction between impurity, point defects, and charges in diffusion phenomenon and diffusion equations for both of impurity and point defects. There are many levels of sophistication of treatment of the interaction of impurity and point defects. We perform approximations step by step, and derive diffusion models in various supplication levels.

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