978 research outputs found
Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation
In this paper, the one-dimensional time-fractional diffusion-wave equation
with the fractional derivative of order is revisited. This
equation interpolates between the diffusion and the wave equations that behave
quite differently regarding their response to a localized disturbance: whereas
the diffusion equation describes a process, where a disturbance spreads
infinitely fast, the propagation speed of the disturbance is a constant for the
wave equation. For the time fractional diffusion-wave equation, the propagation
speed of a disturbance is infinite, but its fundamental solution possesses a
maximum that disperses with a finite speed. In this paper, the fundamental
solution of the Cauchy problem for the time-fractional diffusion-wave equation,
its maximum location, maximum value, and other important characteristics are
investigated in detail. To illustrate analytical formulas, results of numerical
calculations and plots are presented. Numerical algorithms and programs used to
produce plots are discussed.Comment: 22 pages 6 figures. This paper has been presented by F. Mainardi at
the International Workshop: Fractional Differentiation and its Applications
(FDA12) Hohai University, Nanjing, China, 14-17 May 201
Flood Routing Based on Diffusion Wave Equation Using Lattice Boltzmann Method
AbstractOne-dimensional diffusion wave equation is a simplified form of the full Saint Venant equations by neglecting the inertia terms. In this study, the Lattice Boltzmann method for the linear diffusion wave equation was developed. In order to verify the calculation accuracy of it, the analytical solution and Muskingum method were also introduced. Excellent agreement was obtained between observed data and numerical prediction. The results show that the Lattice Boltzmann method is a very competitive method for solving diffusion wave equation in terms of computational efficiency and accuracy
Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation
We propose two stable and one conditionally stable finite difference schemes
of second-order in both time and space for the time-fractional diffusion-wave
equation. In the first scheme, we apply the fractional trapezoidal rule in time
and the central difference in space. We use the generalized Newton-Gregory
formula in time for the second scheme and its modification for the third
scheme. While the second scheme is conditionally stable, the first and the
third schemes are stable. We apply the methodology to the considered equation
with also linear advection-reaction terms and also obtain second-order schemes
both in time and space. Numerical examples with comparisons among the proposed
schemes and the existing ones verify the theoretical analysis and show that the
present schemes exhibit better performances than the known ones
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