Time fractional PDEs have been used in many applications for modeling and simulations. Many of these are multiscale contain high contrast variations the media properties. It requires very small time step size to perform detailed computations. On other hand, presence spatial grids, is required explicit methods. Explicit methods advantages as we discuss paper. In this paper, propose a partial met...

Fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. They are useful to model anomalous diffusion, where a plume of particles spreads in a different manner than the classical diffusion equation predicts. An initial value problem involving a space-fractional diffusion equation is an abstract Cauchy problem, whose analyt...

The fractional diffusion equation is derived from the master equation of continuous time random walks (CTRWs) via a straightforward application of the GnedenkoKolmogorov limit theorem. The Cauchy problem for the fractional diffusion equation is solved in various important and general cases. The meaning of the proper diffusion limit for CTRWs is discussed.

The inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements, with an exponential tempering that allows all moments to exist. This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its ‘‘folded’’ density solves a tempered time-fr...

The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). In this work, an explicit finite-difference scheme for TFDE is presented. Discrete models of a non-Markovian random walk are generate for simulating random variables whose spatial probability density evolves...

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N²M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, ...

We study distributed-order time fractional diffusion equations characterized by multifractal 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 fract...