Recent experimental findings on anomalous diffusion have demanded novel models that combine annealed (temporal) and quenched (spatial or static) disorder mechanisms. The comb-model is a simplified description of diffusion on percolation clusters, where the comb-like structure mimics quenched disorder mechanisms and yields a subdiffusive regime. Here we extend the comb-model to simultaneously account for quenched and annealed disorder mechanisms. To do so, we replace usual derivatives in the comb diffusion equation by different fractional time-derivative operators and the conventional comb-like structure by a generalized fractal structure. Our hybrid comb-models thus represent a diffusion where different comb-like structures describe different quenched disorder mechanisms, and the fractional operators account for various annealed disorders mechanisms. We find exact solutions for the diffusion propagator and mean square displacement in terms of different memory kernels used for defining the fractional operators. Among other findings, we show that these models describe crossovers from subdiffusion to Brownian or confined diffusions, situations emerging in empirical results. These results reveal the critical role of interactions between geometrical restrictions and memory effects on modeling anomalous diffusion. * eklenzi@uepg.br 1 arXiv:2002.05433v1 [cond-mat.stat-mech]

We investigate a time-dependent Schrödinger-like equation in presence of a nonlocal term by using the method of variable separation and the Green function approach. We analyze the Green function for different forms of the memory kernel and the nonlocal term. Results for delta potential energy function are presented. Distributed order memory kernels are also considered, and the asymptotic behaviors of the Green function are derived by using Tauberian theorem. The obtained results for the Green function for the considered Schrödinger-like equation may be transformed to those for the probability distribution function of a diffusion-like equation with memory kernel and can be used to explain various anomalous diffusive behaviors.

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