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Journal ArticleDOI

A fully discrete difference scheme for a diffusion-wave system

Zhi-zhong Sun, +1 more
- 01 Feb 2006 - 
- Vol. 56, Iss: 2, pp 193-209
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TLDR
In this paper, a fully discrete difference scheme is derived for a diffusion-wave system by introducing two new variables to transform the original equation into a low order system of equations. And the solvability, stability and L∞ convergence are proved by the energy method.
About: 
This article is published in Applied Numerical Mathematics.The article was published on 2006-02-01. It has received 949 citations till now. The article focuses on the topics: Diffusion (business).

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Citations
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Journal ArticleDOI

A Space-Time Spectral Method for the Time Fractional Diffusion Equation

Xianjuan Li, +1 more
- 01 Apr 2009 - 
TL;DR: Thanks to the spectral accuracy in both space and time of the proposed method, the storage requirement due to the “global time dependence” can be considerably relaxed, and therefore calculation of the long-time solution becomes possible.
Journal ArticleDOI

A new difference scheme for the time fractional diffusion equation

Anatoly A. Alikhanov
- 01 Jan 2015 - 
TL;DR: A new difference analog of the Caputo fractional derivative (called the L 2 - 1 σ formula) is constructed and some difference schemes generating approximations of the second and fourth order in space and the second order in time for the time fractional diffusion equation with variable coefficients are considered.
Journal ArticleDOI

A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications

Guang-hua Gao, +2 more
- 15 Feb 2014 - 
TL;DR: The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L2 formula when solving time-fractional differential equations numerically.
Journal ArticleDOI

A compact finite difference scheme for the fractional sub-diffusion equations

Guang-hua Gao, +1 more
- 01 Feb 2011 - 
TL;DR: The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis.
Journal ArticleDOI

Compact difference schemes for the modified anomalous fractional sub-diffusion equation and the fractional diffusion-wave equation

Zhibo Wang, +1 more
- 15 Nov 2014 - 
TL;DR: In this paper, compact finite difference schemes for the modified anomalous fractional sub-diffusion equation and fractional diffusion-wave equation are studied, and based on the idea of weighted and shifted Grunwald difference operator, they establish schemes with temporal and spatial accuracy order equal to two and four respectively.
References
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BookDOI

Fractals and fractional calculus in continuum mechanics

Alberto Carpinteri, +1 more
TL;DR: Panagiotopoulos, O.K.Carpinteri, B. Chiaia, R. Gorenflo, F. Mainardi, and R. Lenormand as mentioned in this paper.
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Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics

Francesco Mainardi
- 01 Jan 2001 - 
TL;DR: In this article, the authors review some applications of fractional calculus developed by the author (partly in collaboration with others) to treat some basic problems in continuum and statistical mechanics.
Journal ArticleDOI

Fractional diffusion and wave equations

W. R. Schneider, +1 more
- 01 Jan 1989 - 
TL;DR: In this article, the Green's function of fractional diffusion is shown to be a probability density and the corresponding Green's functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited.
Journal ArticleDOI

Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena

Francesco Mainardi
- 01 Sep 1996 - 
TL;DR: In this article, the authors introduced fractional derivatives of order α in time, with 0 for relaxation, diffusion, oscillations, and wave propagation, and showed that they are governed by simple differential equations of order 1 and 2 in time.
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The fundamental solution of the space-time fractional diffusion equation

Francesco Mainardi, +2 more
- 18 Feb 2007 - 
TL;DR: In this paper, the Cauchy problem for the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation.