I come from an engineering background, so please forgive my inaccurate math grammar.
Currently I am looking at what we call a "spring-pot" system, where a material behaves somewhere between an elastic spring and a viscous dashpot. To perhaps oversimplify the physics, spring has force response proportional to deformation; for dashpot, we have force response proportional to the first time derivative of deformation. The "spring-pot" we are interested in has force response proportional to the fractional derivative of deformation, say:
$f(x) = D^{\alpha}y(x,t) = \frac{\partial^\alpha}{\partial t^\alpha} y(x,t)$
where $y(x,t)$ is some polynomial function of the position $x$, which varies over time $t$.
Now I want to include this expression of $f(x)$ inside an existing integral form that looks like:
$ q = \int_{\Omega} f(x)g(x) dx $
where, in this case, evaluation of $f(x)$ requires working with the fractional derivative of $y(x,t)$. $g(x)$ is another polynomial function.
In our existing framework, we can get closed-form expressions for $q$ when $\alpha$ is either 0 or 1, corresponding respectively to the purely elastic spring and purely viscous dashpot cases. It would be wonderful if an analytical approach is available. Even if not, I would love to reduce the amount of numerical iteration needed to evaluate the integral.
Based on a very rough literature review, I am aware that a convolution quadrature approach is effective for estimating the fractional derivative -- following: Lubich, C. (1986). Discretized fractional calculus. SIAM Journal on Mathematical Analysis, 17(3), 704-719. However, this leads to a pretty costly numerical evaluation process and still require another round of numerical evaluation for the integral outside.
Could someone point me to some other methods that may be useful here?
