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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?

Thanks, Vladimir

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    $\begingroup$ Have you tried to take Laplace transforms? $\endgroup$ Mar 26, 2013 at 8:15
  • $\begingroup$ 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. $\endgroup$ Mar 26, 2013 at 12:31
  • $\begingroup$ 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''. $\endgroup$ Mar 26, 2013 at 15:55

3 Answers 3

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Some classes of stable densities can be obtained as solutions of parabolic pseudo-differential equations, for example, with fractional Laplacian in spatial variables. See, for the simplest case, the semi-physical paper

http://link.springer.com/article/10.1134/1.558856

whose results can be made rigorous. In fact, there are several papers about classes of equations of the above type producing probability densities. See

V. Kolokoltsov, Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc., III. Ser. 80, No.3, 725-768 (2000).

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Let's substitute $\alpha - 1$ with $\alpha - 1 = \beta$ to simplify the Fractional Differential Equation ($f^{\left( \alpha - 1 \right)} = f^{\left( \beta \right)}$): $$ \begin{align*} f^{\left( \alpha - 1 \right)}\left( t \right) &= t \cdot f\left( t \right)\\ f^{\left( \beta \right)}\left( t \right) &= t \cdot f\left( t \right)\\ \end{align*} $$

Leonhard Euler came up with a nice method for some Fractional Derivatives (see Euler's Approach). He has sorted the derivatives of the function by order and is trying to find a pattern in it. He then tried to transfer this pattern from integer orders to fractional ones, which I always affectionately call the "Euler method". We can also use this procedure for some Fractional Differential Equation like this.

But for this we need a group of special functions, namely the Generalized Hypergeometric Functions $\operatorname{_{p}F_{q}\left( \cdots;\, \cdots;\, \cdot \right)}$ (I will abbreviate them to GHF in the following). The Generalized Hypergeometric Function has a definition given by: $$ \begin{align*} \operatorname{_{p}F_{q}\left( A_{1},\, A_{2},\, \cdots,\, A_{p};\, B_{1},\, B_{2},\, \cdots,\, B_{q};\, z \right)} &= \sum\limits_{k = 0}^{\infty}\left[ \frac{\left( A_{1} \right)_{k} \cdot \left( A_{2} \right)_{k} \cdots \left( A_{p} \right)_{k}}{\left( B_{1} \right)_{k} \cdot \left( B_{2} \right)_{k} \cdots \left( B_{q} \right)_{k} \cdot k!} \cdot z \right]\\ \end{align*} $$ wehre $\left( x \right)_{y}$ is the Rising Factorial aka Pochhammer Symbol.

$\beta$ Solution: $f(t)$ In the form of GHFs
$1$ $c \cdot e^{\frac{1}{2} \cdot t^{2}}$ $c \cdot \operatorname{_{0}F_{0}}\left[ \cdot;\, \cdot;\, \frac{1}{2} \cdot t^{2} \right]$
$2$ $c_{1} \cdot \operatorname{Ai}\left( t \right) + c_{2} \cdot \operatorname{Bi}\left( t \right)$ $\cdots$
$3$ $f\left( t \right)$ $c_{1} \cdot \operatorname{_{0}F_{2}}\left[ \cdot;\, \frac{1}{2},\, \frac{3}{4};\, \frac{1}{64} \cdot t^{\beta + 1} \right] + c_{2} \cdot t \cdot \operatorname{_{0}F_{2}}\left[ \cdot;\, \frac{3}{4},\, \frac{5}{4},\, \cdots;\, \frac{1}{64} \cdot t^{4} \right] + c_{3} \cdot t^{2} \cdot \operatorname{_{0}F_{2}}\left[ \cdot;\, \frac{5}{4},\, \frac{3}{2},\, \cdots;\, \frac{1}{64} \cdot t^{4} \right]$
$4$ $f\left( t \right)$ $c_{1} \cdot \operatorname{_{0}F_{3}}\left[ \cdot;\, \frac{2}{5},\, \frac{3}{5},\, \frac{4}{5};\, \frac{1}{615} \cdot t^{5} \right] + c_{2} \cdot t \cdot \operatorname{_{0}F_{3}}\left[ \cdot;\, \frac{3}{5},\, \frac{4}{5},\, \frac{6}{5};\, \frac{1}{615} \cdot t^{5} \right] + c_{3} \cdot t^{2} \cdot \operatorname{_{0}F_{3}}\left[ \cdot;\, \frac{4}{5},\, \frac{6}{5},\, \frac{7}{5};\, \frac{1}{615} \cdot t^{5} \right] + c_{4} \cdot t^{3} \cdot \operatorname{_{0}F_{3}}\left[ \cdot;\, \frac{6}{5},\, \frac{7}{5},\, \frac{8}{5};\, \frac{1}{615} \cdot t^{5} \right]$
$\vdots$ $\vdots$ $\vdots$
$n$ $f\left( t \right)$ $\sum\limits_{k = 1}^{n \in \mathbb{N}}\left[ c_{k} \cdot t^{k - 1} \cdot \operatorname{_{0}F_{n - 1}}\left[ \cdot;\, \frac{k + 1}{n + 1},\, \frac{k + 2}{n + 1},\, \cdots;\, \frac{1}{\left( n + 1 \right)^{n}} \cdot t^{n + 1} \right] \right]$

where $\operatorname{Ai}\left( \cdot \right)$ and $\operatorname{Bi}\left( \cdot \right)$ are the $\operatorname{Ai}$- and $\operatorname{Bi}$- Airy Functions.

Note: The formulas marked in blue lead to Wolfram|Alpha, which confirms the results. Also Note: The series skips integer fractions.

We can generalize this simply by putting $\beta$ in a rounding function, such as the Ceil Function $\left\lceil \cdot \right\rceil$, Floor Function $\left\lfloor \cdot \right\rfloor$, or round $\operatorname{round}\left( \cdot \right)$.

$$ \begin{align*} f\left( t \right) &= \sum\limits_{k = 1}^{\left\lfloor \beta \right\rfloor}\left[ c_{k} \cdot t^{k - 1} \cdot \operatorname{_{0}F_{\beta - 1}}\left[ \cdot;\, \frac{k + 1}{\beta + 1},\, \frac{k + 2}{\beta + 1},\, \cdots;\, \frac{1}{\left( \beta + 1 \right)^{\beta}} \cdot t^{\beta + 1} \right] \right]\\ \end{align*} $$

So there is a solution given for $\alpha - 1 \in \mathbb{R}^{+}$ by: $$ \begin{align*} f\left( t \right) &= \sum\limits_{k = 1}^{\left\lfloor \alpha - 1 \right\rfloor}\left[ c_{k} \cdot t^{k - 1} \cdot \operatorname{_{0}F_{\alpha - 1 - 1}}\left[ \cdot;\, \frac{k + 1}{\alpha - 1 + 1},\, \frac{k + 2}{\alpha - 1 + 1},\, \cdots;\, \frac{1}{\left( \alpha - 1 + 1 \right)^{\alpha - 1}} \cdot t^{\alpha - 1 + 1} \right] \right]\\ f\left( t \right) &= \sum\limits_{k = 1}^{\left\lfloor \alpha - 1 \right\rfloor}\left[ c_{k} \cdot t^{k - 1} \cdot \operatorname{_{0}F_{\alpha - 2}}\left[ \cdot;\, \frac{k + 1}{\alpha},\, \frac{k + 2}{\alpha},\, \cdots;\, \frac{1}{\alpha^{\alpha - 1}} \cdot t^{\alpha} \right] \right]\\ \end{align*} $$

So $\alpha - 1 \in \mathbb{R}^{+}$ gives: $$\fbox{$ \begin{align*} f\left( t \right) &= \sum\limits_{k = 1}^{\left\lfloor \alpha - 1 \right\rfloor}\left[ c_{k} \cdot t^{k - 1} \cdot \operatorname{_{0}F_{\alpha - 2}}\left[ \cdot;\, \frac{k + 1}{\alpha},\, \frac{k + 2}{\alpha},\, \cdots;\, \frac{1}{\alpha^{\alpha - 1}} \cdot t^{\alpha} \right] \right]\\ \end{align*} $}$$

However, determining the whole thing for $\alpha - 1 \in \mathbb{R}^{-}$ is much more difficult but that's another topic. You can see with Wolfram|Alpha that the formula also works for higher derivatives like $\alpha = 21$.

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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.

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