Generalized diffusion-wave equation with memory kernel

Abstract

We study generalized diffusion-wave equation in which the second order time derivative is replaced by integro-differential operator. It yields time fractional and distributed order time fractional diffusion-wave equations as particular cases. We consider different memory kernels of the integro-differential operator, derive corresponding fundamental solutions, specify the conditions of their non-negativity and calculate mean squared displacement for all cases. In particular, we introduce and study generalized diffusion-wave equations with regularized Prabhakar derivative of single and distributed orders. The equations considered can be used for modeling broad spectrum of anomalous diffusion processes and various transitions between different diffusion regimes.

Crossover Dynamics from Superdiffusion to Subdiffusion: Models and Solutions

“…x 2 (t) ∼ t. Conversely for short times (3.22) behaves as η (s) ∼ s p 1 s α + p 2 s β . This form resembles the characteristic function of the double-order timefractional wave equation of natural type [2, 75, 23], which offers decelerating superdiffusive behaviour in the intermediate short-time domain, see Section 3.3 in[75]. Thus, the study of the characteristic function suggests the progress x 2 (t) ∼ t β+1 → t in the course of time.…”

“…It is noteworthy mentioning that GCE-I (3.2) can be written in terms of the generalised diffusion-wave equation with two powerlaw memory kernels [75]…”

“…In the case α = 1 we recover the characteristic property of the classical Cattaneo model, see equation (22) below, the crossover from ballistic to normal diffusion. The reader is also referred to generalisations of diffusion, Fokker-Planck, and diffusion-wave equations by introduction of a generalised memory kernel [73,74,75]. However, recent numerical simulations have shown different transitions from superdiffusion to subdiffusion which are not, in all cases, covered by equation (1.7), see e.g., quasiperiodic interacting systems [38] and dewettingspreading-wetting transitions of self-propelled (run-and-tumble) clustered disks in a substrate with randomly placed pinning sites [76].…”

Subordination Approach to Space-Time Fractional Diffusion

“…In [14,15], subordination principles for the multi-dimensional space-time fractional diffusion equation are deduced. In the case of time-fractional evolution equations with general time-fractional operators, subordination principles have been studied and employed in [23][24][25][26][27][28].…”

Generalized Cattaneo–Maxwell diffusion equation with fractional derivatives. Dispersion relations

“…[5,52,53]. An actual problem for description of nonequilibrium processes in complex systems is construction of generalized diffusion and wave equations [54,55] using fractional integrals and derivatives. The dispersion of heat waves in a dissipative environment using the Cattaneo-Maxwell heat diffusion equation with fractional derivatives has breen investigated in Ref.…”