# Solution to the fractional differential equation

What is the solution of the fractional differential equation $$f^{(\alpha-1)}(t) = tf(t)$$

where $(\alpha)$ denotes the fractional derivative of order $\alpha$

EDIT: Background behind this question.

I am interested in this equation in relation with the alpha-stable version of the Stein's lemma. Recall, that if $X \in N(0,1)$ then $$E(X g(X)) = E(g'(X))$$ for every function $g$ for which the expectations in the left and right parts exist.

The simplest way to prove this is to use the property of normal density $f'(x) = -x f(x)$ and integration by parts.

Let $\phi$ be a characteristic function of the standard symmetric stable density $S(\alpha, 1, 0)$. If I apply the usual derivative to $\phi$ and take a Fourier transform I will get the equation above (modulo signs and coefficients). A closed form solution would give an analytic presentation of the stable density (highly unlikely).

2nd question: Is it possible to obtain a differential equation for the stable density to be used in the proof of the Stein's lemma?

• Have you tried to take Laplace transforms? – András Bátkai Mar 26 '13 at 8:15
• since the fractional derivative of an elementary function is a higher transcendental function, I doubt that this differential equation has a solution that can be expressed in closed form. – Carlo Beenakker Mar 26 '13 at 12:31
• Where does your equation appear? If this is just a model example, consider equations whose coefficients depend not just on $t$, but on $t^\alpha$. For such equations there is a kind of analytic theory of differential equations''. – Anatoly Kochubei Mar 26 '13 at 15:55

The solution must be rather complicated. It is easy to verify that it cannot be expressed as a generalised Taylor series. On the other hand, using the Mellin transform, we obtain a difference equation with a variable coefficient depending on the ratio of two gamma functions. If $\alpha = 1$ the solution is the Gauss function $e^{t^2/2}$ for t in R. If t is in R+ this does not happen.