Propagation speed of the maximum of the fundamental solution to the fractional diffusion–wave equation

Abstract

In this paper, the one-dimensional time-fractional diffusion-wave equation with the fractional derivative of order α, 1 < α < 2 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 timefractional 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.

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

“…Fujita proved that the fundamental solution takes its maximum at the point * = ± α α/2 for each > 0, where α > 0 is a constant determined by α. Recently, another proof of this formula for the location of the maximum as well as a formula for the maximum value of the fundamental solution and results of numerical calculations of the constant α > 0 for 1 < α < 2 were presented in [20]. In this paper we provide an extension of these results to the signalling problem for the time-fractional diffusion equation.…”

“…The rest of the paper is organized as follows: In the 2nd section, problem formulation and some analytical results for the fundamental solution to the signalling problem for the fractional diffusion-wave equation are given. In the 3rd section, we employ the method introduced in [20] to deduce some explicit formulas for the maximum location and the maximum value of the fundamental solution and for the propagation velocity of the maximum point. The 4th section is devoted to a short description of the numerical algorithms used to calculate the fundamental solution and its important characteristics including the location of its maximum and maximum value.…”

Travelling Waves in Space-Fractional Nonlinear Diffusion with Linear Convection

“…To determine these variables, we replace this equation in Equation (12). The calculus of the integral is given by: …”

“…Nowadays, many definitions have appeared in fractional calculus that shows us some difficulties and limitations of the theory with applications [4] [5] [6] [7]. The researchers have investigated in diffusion processes the anomalous related memory effects [8] [9] [10] [11], for example, some materials the moisture propagates according to the 2 x t α scaling, with 0 2 α < < [12] [13]. Costa et al [14] comment that water transport for large distance in a relatively short time (groundwater infiltration problem) can be described for a fractional space-time nonlinear diffusion equation.…”

Time-fractional extensions of the Liouville and Zwanzig equations

“…A wide variety of fractional equations involving the space and/or time derivatives of non-integer order have been suggested and investigated by many researchers over the past two decades for modelling various anomalous relaxation and transport processes. Examples of such equations are the subdiffusion and su-perdiffusion equations [1,2,[10][11][12], the diffusion-wave equations [13][14][15][16][17], the fractional Fokker-Plank equations [18][19][20], the fractional advection-dispersion equations [21][22][23], the fractional Bloch equation [24,25], the fractional Schrödinger equation [26,27] etc. Now there are a variety of techniques that can be used to deduce the fractional kinetic and transport equations.…”