We consider the Cauchy problem of fractional pseudo-parabolic equation on the whole space R n , n ≥ 1. Here, the fractional order α is related to the diffusion-type source term behaving as the usual diffusion term on the high frequency part. It has a feature of regularity-gain and regularity-loss for α > 1 and 0 < α < 1, respectively. We establish the global existence and time-decay rates for small-amplitude solutions to the Cauchy problem for α > 0. In the case that 0 < α < 1 , we introduce the time-weighted energy method to overcome the weakly dissipative property of the equation.
In this paper, the finite difference scheme is developed for the time‐space fractional diffusion equation with Dirichlet and fractional boundary conditions. The time and space fractional derivatives are considered in the senses of Caputo and Riemann‐Liouville, respectively. The stability and convergence of the proposed numerical scheme are strictly proved, and the convergence order is O(τ2−α+h2). Numerical experiments are performed to confirm the accuracy and efficiency of our scheme.
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