Greetings MSE community,
I have recently conducted some rudimentary experiments in matlab coding of PDE's. I have explicit and implicit numerical solutions to both the heat and the wave equation, for some mundane boundary (and initial) conditions.
Basically, solved numerically $ \ \frac{\partial u}{\partial t}=k\frac{\partial^2 u}{\partial x^2}\ $ and $\ \ \frac{\partial^2 u}{\partial t^2}=k\frac{\partial^2 u}{\partial x^2},\ $ $u(x,t):[0,1]\times[0,1]\to\mathbb{R}$.
I am now curious about fractional (more specifically, non-integer) time derivatives.
$\ \ \ \frac{\partial^\alpha u}{\partial t^\alpha}=k\frac{\partial^2 u}{\partial x^2},\ $ for $\alpha\in[1,2].$
I am wondering if there is a continuous deformation between the two surface plots of the heat and wave equations as $\alpha$ moves from 1 to 2. Thus, my task is to code the following: $$\frac{1}{\Gamma(\beta)}\frac{\partial^2}{\partial t^2}\displaystyle\int^{t}_{0}(t-s)^{\beta-1}u(x,s)\ ds\ =\ k\frac{\partial^2 u}{\partial x^2}$$ where $0 < \beta \leq 1$. The above operator contains a "fractional integral", so that $\frac{\partial^\alpha}{\partial t^\alpha}=\frac{\partial^{2-\beta}}{\partial t^{2-\beta}}.$
This is somewhat beyond the scope of the initial experiment, and maybe too ambitious to someone with my programming skills. Where do I begin, since I cannot simply discretize the time derivative, which is what worked in the previous experiments? Any insight would be appreciated.
Maybe this is helpful.
The idea is the following. Let \begin{align} \mathcal{I}(x,t) = \frac{\partial^2}{\partial t^2} \int_0^t (t-s)^{\beta-1}u(x,s)\,ds \end{align} and \begin{align} I(x,t)= \int_0^t (t-s)^{\beta-1}u(x,s)\,ds \end{align} Then \begin{align} \mathcal{I}(x,t) \approx \frac{1}{\Delta t^2}( I(x,t-\Delta t) -2I(x,t)+I(x,t+\Delta t)) \end{align} The method is explained in detail in the paper. I hope this helps!