We study distributed order time fractional diffusion equations characterized by multi-fractal memory kernels, in contrast to the simple power-law kernel of common time fractional diffusion equations. Based on the physical approach to anomalous diffusion provided by the seminal Scher-Montroll-Weiss continuous time random walk, we analyze both natural and modified-form distributed order time fractional diffusion equations and compare the two approaches. The mean squared displacement is obtained and its limiting behavior analyzed. We derive the connection between the Wiener process, described by the conventional Langevin equation and the dynamics encoded by the distributed order time fractional diffusion equation in terms of a generalized subordination of time. A detailed analysis of the multi-fractal properties of distributed order diffusion equations is provided.
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