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We present a short overview of the recent results in the theory of diffusion and wave equations with generalised derivative operators. We give generic examples of such generalised diffusion and wave equations, which include time-fractional, distributed order, and tempered time-fractional diffusion and wave equations. Such equations exhibit multi-scaling time behaviour, which makes them suitable for the description of different diffusive regimes and characteristic crossover dynamics in complex systems.
The change from the diffusion-limited to the reaction-limited cooperative behaviour in reaction-diffusion systems is analysed by comparing the universal long-time behaviour of the coagulation-diffusion process on a chain and on the Bethe lattice. On
We consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate $r$. We consider several generalisations of the model of M. R. Evans
We consider an evolution equation with the Caputo-Dzhrbashyan fractional derivative of order $alpha in (1,2)$ with respect to the time variable, and the second order uniformly elliptic operator with variable coefficients acting in spatial variables.
It is pointed out that, for the fractional Fokker-Planck equation for subdiffusion proposed by Metzler, Barkai, and Klafter [Phys. Rev. Lett. 82 (1999) 3563], there are four types of infinitely many exact solutions associated with the newly discovere
We present a classical, mesoscopic derivation of the Fokker-Planck equation for diffusion in an expanding medium. To this end, we take a conveniently generalized Chapman-Kolmogorov equation as the starting point. We obtain an analytical expression fo