Hi,
I have to solve fem of fractional order parabolic partial differential equations (FPDE) for $y(x,t)$ in FEniCS:
$$ y^{\alpha}_{t}(x,t)+\mbox{div}(-A(x)\nabla y(x,t))=f(x,t), \ (x,t)\in \Omega\times [0,T] $$
$$ y(x,t)=0,\ (x,t) \in \partial\Omega\times [0,T], \quad y(x,0)=y_{0}(x),\ \mbox{in}\ \Omega, $$
Here $y^{\alpha}_{t}(x,t)$ is defined by the following Caputo fractional derivative of order $\alpha$ with respect to t and it is defined by
$$ y^{\alpha}_{t}(x,t)= \frac{\partial^{\alpha}y(x,t)}{\partial t^{\alpha}} = \frac{1}{\Gamma(1-
\alpha)}\int_{0}^{t}(t-\tau)^{-\alpha}\ \frac{\partial}{\partial \tau}y(x,\tau) \ d\tau, \quad 0
<\alpha<1 $$
where $\Gamma(\cdot)$ is the Gamma function.
I wouldn't able to get an idea about this.
Anybody clarify this to me.
Thank you
