13,036 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
Distributed order fractional constitutive stress-strain relation in wave propagation modeling
Distributed order fractional model of viscoelastic body is used in order to
describe wave propagation in infinite media. Existence and uniqueness of
fundamental solution to the generalized Cauchy problem, corresponding to
fractional wave equation, is studied. The explicit form of fundamental solution
is calculated, and wave propagation speed, arising from solution's support, is
found to be connected with the material properties at initial time instant.
Existence and uniqueness of the fundamental solutions to the fractional wave
equations corresponding to four thermodynamically acceptable classes of linear
fractional constitutive models, as well as to power type distributed order
model, are established and explicit forms of the corresponding fundamental
solutions are obtained
On the Speed of Spread for Fractional Reaction-Diffusion Equations
The fractional reaction diffusion equation u_t + Au = g(u) is discussed,
where A is a fractional differential operator on the real line with order
\alpha between 0 and 2, the C^1 function g vanishes at 0 and 1, and either g is
non-negative on (0,1) or g < 0 near 0. In the case of non-negative g, it is
shown that solutions with initial support on the positive half axis spread into
the left half axis with unbounded speed if g satisfies some weak growth
condition near 0 in the case \alpha > 1, or if g is merely positive on a
sufficiently large interval near 1 in the case \alpha < 1. On the other hand,
it shown that solutions spread with finite speed if g'(0) < 0. The proofs use
comparison arguments and a new family of traveling wave solutions for this
class of problems
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