Simulating transient dynamics of the time-dependent time fractional Fokker–Planck systems

• An iteration method is proposed for the transient dynamics of time-dependent time fractional Fokker–Planck equations.
• The method is based on Fourier Series solution and the multi-step transition probability formula.
• With the time-modulated subdiffusion on finite interval as example, the polarized motion orientation is disclosed.
• With the time-modulated subdiffusion within a confined potential as example, the death of dynamic response is observed.

For a physically realistic type of time-dependent time fractional Fokker–Planck (FP) equation, derived as the continuous limit of the continuous time random walk with time-modulated Boltzmann jumping weight, a semi-analytic iteration scheme based on the truncated (generalized) Fourier series is presented to simulate the resultant transient dynamics when the external time modulation is a piece-wise constant signal. At first, the iteration scheme is demonstrated with a simple time-dependent time fractional FP equation on finite interval with two absorbing boundaries, and then it is generalized to the more general time-dependent Smoluchowski-type time fractional Fokker–Planck equation. The numerical examples verify the efficiency and accuracy of the iteration method, and some novel dynamical phenomena including polarized motion orientations and periodic response death are discussed.