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## HANS ENGLER’S TRAVELING WAVE SOLUTIONS FOR TIME-FRACTIONAL DIFFUSIONS

The idea is to replace Engler’s stable density with the Wright function and systematically replace $1/\alpha$ (where $\alpha$ is a parameter for the stable distribution with $\beta/\alpha$).

The fundamental solution of $D^\beta_t u + A u = 0$ is

$W(x,t) = t^{-\beta/\alpha} f( |x| t^{-\beta/\alpha})$

Following Engler () consider $U_{\tau}(\xi) = F(\xi \tau^{-\beta/\alpha})$ where F is the cumulative distribution function, and set $u_{c\tau}(x,t) = U_{\tau}(x+ct)$.  Then verify that

$D^{\beta}_t u_{c\tau}(x,t) + A u_{c\tau}(x,t) = c\tau^{-\beta/\alpha} g_0(u_{c\tau}(x,t) + \beta/\alpha \tau g_1(u_{c\tau}(x,t))$

with $g_0(\zeta) = f(F^{-1}(\zeta)$ and $g_1(\zeta) = F^{-1}(\zeta) f(F^{-1}(\zeta))$.

These traveling wave solutions provide unbounded speed for spread, i.e., $lim_{t\rightarrow \infty} u(t,x) = 1$