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Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

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Povstenko Y.

Open Physics

2013

vol. 11

issue 10

1284-1294

Heat conduction in two joint half-lines is considered under the condition of perfect contact, i.e. when the temperatures at the contact point and the heat fluxes through the contact point are the same for both regions. The heat conduction in one half-line is described by the equation with the Caputo time-fractional derivative of order α, whereas heat conduction in another half-line is described by the equation with the time derivative of order β. The fundamental solutions to the first and second Cauchy problems as well as to the source problem are obtained using the Laplace transform with respect to time and the cos-Fourier transform with respect to the spatial coordinate. The fundamental solutions are expressed in terms of the Mittag-Leffler function and the Mainardi function.

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

100%

Luchko Y., Mainardi F.

Open Physics

2013

vol. 11

issue 6

666-675

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations 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. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.

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2 Open Physics

2 2013

1 Luchko Y.

1 Mainardi F.

1 Povstenko Y.

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Fundamental solutions to time-fractional heat conduction equations in two joint half-lines

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation