# Numerical solution of (Fourier) fractional differential equation

I have a certain equation, which I would like to transform to the Laplace domain. The Laplace transform is defined as

$\hat{f}(s) = \int_0^\infty e^{-s x} f(x) {\rm d}x$.

Now, after transforming, I am dealing with the following term

$\int_0^\infty x^{3/2} e^{-sx} f(x){\rm d}x$.

If $3/2$ would be an integer $n$ instead, then we could make use of the fact that

$\frac{{\rm d}^n}{{\rm d}s^n}\hat{f}(s) = \int_0^\infty (-x)^n e^{-sx} f(x) {\rm d}x.$

Instead, I figured I would have to look into fractional derivatives. I noticed that Fourier's definition of a fractional derivative was exactly what I need (except that I am working with a Laplace transform rather than a Fourier transform, I'm not sure if this matters somehow):

$\frac{{\rm d}^\alpha}{{\rm d}s^\alpha}\hat{f}(s) = \int_0^\infty (-x)^\alpha e^{-sx} f(x) {\rm d}x,\,$ for $\alpha\in\mathbb{R}$.

This definition implies that

$i \frac{{\rm d}^{3/2}}{{\rm d}s^{3/2}}\hat{f}(s) = \int_0^\infty x^{3/2} e^{-sx} f(x){\rm d}x,$

for $i$ the imaginary unit.

So far so good, I think. Using this definition of the fractional derivative, I obtain a fractional differential equation of the form

$i s \frac{{\rm d}^{3/2}}{{\rm d}s^{3/2}}\hat{f}(s) = g(s) \hat{f}(s)$,

for some function $g$. Now I have two questions:

1. What kind of initial values are needed here, so that the solution is unique?
2. Is there a numerical way to solve this?

Relevant references to literature would also be appreciated.

• Can you specify what $f(x)$ is and how you go from the first equation to the second? The key here, I think, is that you must be able to calculate the fractional derivative of $f(x)$. And I also think you should interpret second equation in terms of the Laplace transform of the $n^{th}$ derivative of $f$, rather than as the derivative of the transform, $\hat{f}$. I have done something very similar with Fourier transforms. – Cye Waldman Apr 24 '17 at 18:53
• $f$ is a density function of a stationary process with a continuous state space that satisfies a certain balance equation. Since I do not know anything explicit about $f$, I can not calculate derivatives of $f$ (let alone fractional ones?). Are you suggesting that the fractional derivative of a Laplace transform is the same as the Laplace transform of a fractional derivative? – Stochast May 3 '17 at 9:32
• Yes it does. But it is beyond my ken. The solution you are using is called the Weyl differintegration and it refers the $s$-axis back to $-\infty$. As, I say, above my pay grade. But it might be okay for you because $s>0$. The formal definition is $$\frac{d^q e^{\nu s}}{[d(s+\infty)]^q}=\nu^q e^{\nu s}$$ – Cye Waldman May 3 '17 at 15:28