# Fractional differential equation and inverse Laplace transform

I am writing my bachelor thesis to prove that a certain linear fractional differential equation of order $$(n,q)$$ has $$N$$ linearly independent solutions, where $$N$$ is the smallest possible integer greater than $$nv$$, where $$v=1/q$$. I understand the proof more or less, but now I desperately try to find an example. I have given an example which at first sight seemed relatively simple, but I have no idea how to come up with the solution.

So the example is

$$D^{4v}y(t)=0, ~~\text{with}~~ \ v=1/3$$

So, if I now take the Laplace Transform of it I get after a theorem, which I have proved in my bachelor thesis, that

$$L(D^{4v}y(t))=s^{4/3}Y(s)-sD^{-2/3}y(0)-D^{1/3}y(0)=0 .$$

So $$Y(s)=\frac{sD^{-4/3}y(0)+D^{1/3}y(0)}{s^{4/3}}$$ And now? Actually I would have to do the inverse Laplace transform now, wouldn't I? But how do I do that? Apart from the fact that I haven't really understood the inverse Laplace transformation yet, I don't even know how to read it in a table or something. When I google I only come across the inverse Laplace transformation for ordinary differential equations and not for fractional differential equations.

• So, what is it that you're really looking for. I can see 3 options (a) A counter example. (b) For someone to solve your specific counter example. (c) An inverse Laplace transform table suited for fractional calculus (or rather a methodology for that). Which is it of the three? Be specific. – Amir Sagiv Jan 12 at 16:13
• All three options would be good. I'd like to know how this example could be solved and whether I've even done it right so far. But I would also like to see another example. To the table: Is there such a thing at all for fractional differential equations? If not, how exactly do you calculate the inverse Laplace transform for fractional differential equations? – A.Soe Jan 12 at 16:56