@ Andreas,

First, good job on your attempts of simulating fractional derivatives. The problem you stating that error increases for higher times is also a typical characteristic of numerical approximation of regular derivatives. Though people have found ways to circumvent the error in the latter case.

In fractional derivatives it is somewhat an issue for (x, or t > 1) and that's why you see most of the research work only display results in 0

I do not use ever any of the integral formulations unless for theoretical aspects. I suggest you to use the Grunwald-Letnikov derivatives. Please, read the papers of P. Ostalczyk  and Brezinsky. There is also an approach based on polynomial approximations.