Well-posedness and simulation of weak solutions to the time-fractional Fokker–Planck equation with general forcing

Figure 1.  Nonuniform time meshes on the interval $ [0,T] $ with $ t_n = (n/N)^\gamma T $ for $ \gamma\in \{1,1.5,2,3\} $ (top to bottom) and $ N = 20 $; the red nodes are $ \{0,N/2,N\} $ in all cases

Figure 2.  Plot of the solution $ \psi $ for varying values of $ \alpha \in \big\{\frac14,\frac12,\frac34,1\big\} $; on the left (a) at $ t = 0.02 $ and on the right (b) at $ t = 0.5 $

Figure 3.  Plot of the solution $ \psi $ for varying time $ t \in \{0.02,0.045,0.08,0.125,0.245,0.5\} $; on the left (a) for $ \alpha = 1 $ and on the right (b) for $ \alpha = \frac12 $

Figure 4.  Plot of the solution $ \psi $ for varying values of $ \alpha \in \big\{\frac14,\frac12,\frac34,1\big\} $; on the left (a) at $ t = 0.02 $ and on the right (b) at $ t = 0.18 $

Figure 5.  Plot of the solution $ \psi $ for varying time $ t \in \{0.02,0.045,0.08,0.125,0.245,0.5\} $; on the left (a) for $ \alpha=1 $ and on the right (b) for $ \alpha=\frac12 $

Figure 6.  Plot of the solution $ \psi $ for different values of $ \alpha $ at different times $ t $; we consider the pairings $ (\alpha,t) \in \{(\frac14,t_5),(\frac12,t_{14}),(\frac34,t_{18}), (1,t_{37})\} $ for $ t_n $ as defined at the beginning of Section 5

Figure 7.  Plot of the solution to the time-fractional Fokker–Planck equation (20) (ⅰ) and the model with no right-hand side (ⅱ) for varying time $ t \in \{0.02,0.045,0.08\} $; on the left (a) for $ \alpha = \frac14 $ and on the right (b) for $ \alpha = \frac34 $