scholarly journals On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equation

Mathematics ◽  
2017 ◽  
Vol 5 (4) ◽  
pp. 76 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Diffusion Wave ◽  
Space And Time

scholarly journals On Some New Properties of the Fundamental Solution to the Multi-Dimensional Space- and Time-Fractional Diffusion-Wave Equatio

2017 ◽  
Author(s):  
Yuri Luchko
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Open Problem ◽  
Special Functions ◽  
Dimensional Space ◽  
Fractional Diffusion ◽  
Two Dimensional ◽  
Diffusion Wave ◽  
Space And Time ◽  
Mellin Convolution

In this paper, some new properties of the fundamental solution to the multi-dimensional space- and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to get two new representations of the fundamental solution in form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed form formulas for particular cases of the fundamental solution are derived. In particular, we solve an open problem of representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

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

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Yuri Luchko ◽  
Francesco Mainardi
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Wave Equations ◽  
Propagation Speed ◽  
Fractional Diffusion ◽  
Maximum Point ◽  
Diffusion Wave ◽  
Constant Coefficients ◽  
The One ◽  
Signalling Problem

AbstractIn 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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