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.

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• Volterra-Prabhakar derivative of distributed order and some applications
• Generalised Diffusion and Wave Equations: Recent Advances
• Generalized diffusion equation with nonlocality of space-time. Memory function modelling
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• Generalized diffusion equation with nonlocality of space-time: analytical and numerical analysis
• Operational solutions for the generalized Fokker-Planck and generalized diffusion-wave equations
• A Fractional Calculus Framework for Open Quantum Dynamics: From Liouville to Lindblad to Memory Kernels