# Motion described by $m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12 }x}{\mathrm{d}t^{\frac12}}$

What kind of motion would a (preferably dimensionless for simplicity) body do if the force acted on it was proportional to the semi-derivative of displacement, i.e.

$$m \frac{\mathrm{d}^2 x}{\mathrm{d}t^2}=-k\frac{\mathrm{d}^{\frac12}x}{\mathrm{d}t^{\frac12}} \, \, ?$$

It would be helpful if someone with a copy of Mathematica plotted this for various values of the constants.

• Might be relevant: physics.stackexchange.com/q/4005 Oct 8, 2014 at 18:56
• More on fractional derivatives. Oct 8, 2014 at 19:05
• Hi user148432. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. Oct 8, 2014 at 19:13
• @user148432 you might find math.stackexchange.com a good place for detailed help. Oct 8, 2014 at 20:26
• It is good practice to state what you have done so far instead of letting answerers start from scratch. At its current state, your (interesting) question does unfortunately sound quite like a "hey, anyone please work this out for me"... Oct 15, 2014 at 11:37

If $$D^n$$ denotes the $$n$$th derivative and $$D^{-n}$$ the $$n$$th integral, then we have that,

$$D^n f(t) = D^m[D^{-(m-n)}f(t)]$$

providing $$m \geq \lceil{n}\rceil$$. For our half derivative, we choose $$n=1/2$$, and $$m=2$$, in which case we have,

$$D^{1/2}f(t) = D^2[D^{-(3/2)}f(t)]$$

There is a general formula for the $$n$$th integral of a function, one of my favorite results of Cauchy:

$$f^{-(n)}(t) = \frac{1}{\Gamma(n)}\int_{0}^t (t-u)^{n-1}f(u) \, du$$

which is essentially a convolution $$f(t) \ast t^{n-1}$$. Applying it, we find,

$$D^{1/2}f(t) = \frac{d^2}{dt^2} \left[ \frac{2}{\sqrt{\pi}}\int_0^t (t-u)^{1/2}f(u) \, du\right]$$

Given the differential equation,

$$\frac{d^2 x(t)}{dt^2} = -\frac{k}{m} \frac{d^{1/2} x(t)}{dt^{1/2}}$$

we can substitute in our definition of $$D^{1/2}x(t)$$, and conclude,

$$x(t) = -\frac{2k}{m\sqrt{\pi}}\int_{0}^t (t-u)^{1/2}x(u)\, du + c_1t +c_2$$

for $$c_1,c_2 \in \mathbb{R}$$ which is an integral equation. If we can assume $$x(t)$$ is supported on $$[0,\infty)$$ only, then the integral is a convolution $$x(t) \ast \sqrt{t}$$ and taking the Laplace transform, we find,

$$X(s) = \left( 1+ \frac{k}{ms^{3/2}}\right)^{-1} \left( \frac{c_1}{s^2} + \frac{c_2}{s} \right) = \frac{m(c_1 + c_2 s)}{k\sqrt{s}+ms^2}$$

The solution $$x(t)$$ is then the inverse Laplace transform of $$X(s)$$. Formaly, this is given by,

$$x(t) = \frac{1}{2\pi i} \int_{\Gamma} e^{st} \frac{m(c_1 + c_2 s)}{k\sqrt{s}+ms^2} \, ds$$

where the contour $$\Gamma$$ is in the complex plane; it is a vertical line of infinite length with all poles of $$F(s)$$ to its left. In practice, we close the contour with an additional contour, ensure the second integral tends to zero (e.g. by the estimation lemma), and use the residue theorem.

The integrand, which we denote $$F(s)$$, has three poles located at $$s^3 = k^2/m^2$$, or equivalently,

$$s_1 = \omega^{4/3}_0, \quad s_2 = \frac{1}{2}(1+i\sqrt{3})\omega^{4/3}_0, \quad s_3 = \frac{1}{2}(i\sqrt{3}-1)\omega^{4/3}_0$$

as well as at $$s_0= 0$$, where we define $$\omega^2_0 := k/m$$. The vertical contour should begin after $$s_1$$ so all poles are to the left. However, doing so analytically is somewhat tedious. I chose to use a numerical method for the evaluation of inverse Laplace transforms due to H.E. Salzer which uses aquadrature formula. With Mathematica, I managed to reconstruct $$x(t)$$ partially: in the simplified case when $$c_1 = c_2 = k/m = 1$$. It seems, by visual inspection, the solution resembles that of damped harmonic motion, such as when one introduces a damping term $$\gamma \dot{x}$$ in the equations of motion of a standard harmonic oscillator.

• Not trying to be rude or anything, I'm grateful you answered so I upvoted, by I am not trying to find a way to solve the differential equation analytically, so that's why I don't accept your answer. I posted this on PSE so I can get some intuition on the kind of motion the body will do under the influence of a force proportional to that fractional derivative. I'm looking forward to a solution other than hypergeometric but I fear it will just be an alternative representation of hypergeometric functions, but we never know what intuition a different solution might provide us. Oct 8, 2014 at 20:35
• @user148432: Yes, I understood your question, but I started fiddling with the equation, and found a cute result, so I thought I'd share it as it is partially relevant, and may help you with the problem in general. Oct 8, 2014 at 20:37
• @JamalS. Didn't you forget the initial condition while applying the Laplace transform ? The Laplace transform of $\ddot x$ is $s^2X(s)-s x(0)-\dot x(0)$. Oct 9, 2014 at 8:32
• @Tom-Tom: No, I did not. I applied the Laplace transform to the equation $x(t) = x(t) \ast t^{1/2}$ which has no derivatives; it is a convolution equation. Oct 9, 2014 at 8:37
• @fibonatic: No, it's not the square root of a derivative, it is literally the application of a differentiation half times, it is like half an integral and half a derivative. Oct 9, 2014 at 14:19

I am no mathematician and am a little afraid that my answer is too simple to be true, but here goes:

I use Fourier transforms to define the fractional derivative. $x(\omega)$ is defined such that

$$x(t) = \int_{-\infty}^\infty \, \frac{\text{d}\omega}{2\pi} \text{e}^{i \omega t} \, x(\omega) \, .$$

Then any integer derivatives is

$$\frac{\text{d}^n}{\text{d} t^n} x(t) = \int_{-\infty}^\infty \, \frac{\text{d}\omega}{2\pi} \text{e}^{i \omega t} \, (i \omega )^n \, x(\omega) \, ,$$

so that

$$\left(\frac{\text{d}^n x}{\text{d} t^n}\right)(\omega) = (i \omega )^n \, x(\omega) \, .$$

Then this can be generalised to any real number $n$. In particular,

$$\frac{\text{d}^{1/2}}{\text{d} t^{1/2}} x(t) = \int_{-\infty}^\infty \, \frac{\text{d}\omega}{2\pi} \text{e}^{i \omega t} \, \sqrt{i \omega } \, x(\omega) \, .$$

Since your equation is linear it can be solved separately for each Fourier modes. In Fourier space it becomes

$$\left(- m \omega^2 + k \sqrt{i \omega}\right) x(\omega) = 0 \, .$$

This tells us that either $x(\omega) = 0$ or $\omega^2 = \frac{k}{m} \sqrt{i \omega}$. The first solution is trivial. It is equivalent to $x(t)= 0$. The second however has four solutions:

\begin{align} & \omega_1 = \left(\frac{k}{m}\right)^{2/3} \text{e}^{i\,\pi/6} \, ,\\ & \omega_2 = \left(\frac{k}{m}\right)^{2/3} \text{e}^{i\,\pi/2} \, , \\ & \omega_3 = \left(\frac{k}{m}\right)^{2/3} \text{e}^{i \, 5\pi/6} \, , \\ & \omega_4 = 0 \, . \end{align}

Then the most general solution to your problem is

$$x(t) = C_1 \text{e}^{i\omega_1 t} + C_2 \text{e}^{i\omega_2 t} + C_3 \text{e}^{i\omega_3 t} + C_4\, .$$

$C_i$ are integration constants. Explicitly one finds,

$$x(t) = C_1 \, \text{e}^{i \,t \, (k/m)^{2/3} \sqrt{3}/2} \, \text{e}^{-(k/m)^{2/3} t/2} + C_2 \, \text{e}^{- (k/m)^{2/3} t} + C_3 \, \text{e}^{-i \, t \, (k/m)^{2/3} \sqrt{3}/2} \, \text{e}^{-(k/m)^{2/3} t/2} + C_4 \, .$$

We find three solutions that decay exponentially and one constant. Two of the decaying solutions oscillate as well.

In order to make a plot, I set C_4 = 0 because this amounts to a simple vertical shift of $x(t)$, I choose $C_1 = C_2^* = (c_1 + i c_2)/2$ and $C_3 = C_3^*$ so that $x(t)$ is real, I rescale the time according to $\tau = t \, (k/m)^{2/3}/2$ and rescale $x(t)$ according to $y = x \, C_3$. Then my solution becomes,

$$y(\tau) = \text{e}^{-\tau} \left[ c_1 \, \cos(\sqrt{3} \, \tau) + c_2 \sin(\sqrt{3} \, \tau) \right] + \text{e}^{-2 \tau} \, .$$

Here is a plot of $y(\tau)$ for different values of $c_{1,2}$: • Sorry for the multiple algebra mistakes. I checked it and re-checked. I hope that it's Ok now. Please tell me if you find something strange. Oct 8, 2014 at 23:32
• How does this generalization of derivatives to fractional derivatives compare to the definition JamalS uses? Oct 16, 2014 at 1:44
• I think this was an interesting approach. But from comments in Tobias's answer, this seems to be a different generalization of derivatives, since the fractional derivatives would commute now. Oct 16, 2014 at 5:17

You can simply take the semi-derivative of your equation again, which yields

\begin{align} m\frac{d^2}{dt^2}\underbrace{\frac{d^{\tfrac12}x}{dt^{\tfrac12}}}_{=-\frac mk\frac{d^2x}{dt^2}} &= -k\frac{dx}{dt} \\\Rightarrow m^2\frac{d^4x}{dt^4} &= k^2\frac{dx}{dt} \tag{*} \end{align}

and then solve that ODE. But, similarly to squaring an algebraic equation to eliminate roots, you (probably) have to discard halve of the solutions to satisfy the original equation.

As hinted at by user121330's comment, I assumed commutativity of $D^2$ (where $D:=d/dt$) and $D^{\tfrac12}$ which is not granted in general. I suspect that basically correlates with my above statement that some solutions of $(*)$ must actually be discarded. I'll try and think about this some more, but for now please be aware that there may be flaws or even uselessness in this answer...

• Any thoughts on initial conditions? This seems to require more conditions than usual, but then again the half-derivative is not time-local. Oct 15, 2014 at 12:02
• @EmilioPisanty I honestly don't have much experience with fractional derivatives, but my guess is that similarly to squaring an equation (or taking derivatives of normal ODEs) you get more solutions than are valid for the original equation, i.e. two initial conditions should suffice. But as you state, the non-locality may yield curious effects. However, JamalS' answer (and others) clearly suggest two initial conditions just like for a usual 2nd degree ODE are required. Oct 15, 2014 at 12:06
• I get a quite similar answer to Steven when I solve that, but I don't know what to discard as I don't know what restrictions I really have. Oct 15, 2014 at 12:07
• $m D^{1/2} D^2 x \neq m D^2 D^{1/2} x$ I desperately wanted an easy solution like this, but that commutation is pretty non-trivial. Oct 15, 2014 at 16:37
• I think this and Steven's answer are probably wrong, but I still found them very helpful for thinking about the problem. Maybe just add a note about why this approach doesn't work and keep it? It is still useful information. +1 for an interesting idea and realizing the mistake. Oct 16, 2014 at 5:13

One way to try to solve the equation is transforming it in an ODE. Apply the fractional derivative $D^{1/2}$ again to the equation to find $$D^{1/2}[D^2x(t)]=D^{5/2}x(t)-C_1t^{-3/2}-C_2t^{-5/2}-C_3t^{-7/2},$$ and $$D^{1/2}[D^{1/2}x(t)]=Dx(t)-C_4t^{-3/2}$$ Hence we got $$D^{5/2}x(t)=-\frac{k}{m}Dx(t)+C_1t^{-3/2}+C_2t^{-5/2}+C_3t^{-7/2}$$ But, we also have that $$D^{5/2}x(t)=D^2[D^{1/2}x(t)]=-\frac{m}{k}D^4x(t)$$ Hence we get the following ODE: $$x^{(4)}(t)-\omega^3 x'(t)=C_1t^{-3/2}+C_2t^{-5/2}+C_3t^{-7/2},$$ where $\omega^3=\dfrac{k^2}{m^2}$ Let $x'(t)=v(t)$ to get $$v'''(t)-\omega^3 v(t)=C_1t^{-3/2}+C_2t^{-5/2}+C_3t^{-7/2}$$ The general solution of the homogeneous equation for $v(t)$ is $$v_0(t)=c_1 e^{\omega t}+e^{-\omega t/2}\left(c_2\cos\dfrac{\sqrt3}{2}\omega t+c_3\sin\dfrac{\sqrt3}{2}\omega t\right)$$ The general solution of the homogeneous equation $x(t)$ is then $$x_0(t)=c_1 e^{\omega t}+e^{-\omega t/2}\left(c_2\cos\dfrac{\sqrt3}{2}\omega t+c_3\sin\dfrac{\sqrt3}{2}\omega t\right)+c_4,$$ where the constants are different in the two last equations. So, the general solution is of the form $$x(t)=x_0(t)+x_p(t),$$ where $x_p(t)$ is the particular solution.

• How did you get the first two equations? Oct 16, 2014 at 5:21
• By the rules of composition of fractional derivatives. The constants $C_i$ depend on the initial values of the function and fractional derivatives. The rules can be seen here in section 3.5.3: vutbr.cz/www_base/zav_prace_soubor_verejne.php?file_id=10060 Oct 16, 2014 at 11:57

So, what you have here is, as others have mentioned, a fractional order differential equation. Since others have provided graphs, there seems to be little point to adding one. Also, you seem more interested in the qualitative aspect than actual analytical solutions, as you've mentioned.

In essence, what you have here is some second order system with some element that behaves like something between a spring and a damper. There is some dissipative behavior that will come into play here, but also some oscillatory behavior, as you see from the other graphs.

A special result of this being a fractional order system is that that specific derivative is also non-local, meaning that the current dynamics of the system are not only dependent on the current state, but also past state. This is called historicity, or a system with memory.

There are many ways to represent that fractional order derivative mathematically, if you want to get into that. Most of the people here seem to have used some integral definition (whether explicitly, or in the case of the Laplace transform, implicitly), but you can also represent it as a series.

If you wanted a full on analytical solution, you might run into some non-elementary functions, but you can derive series representations from them using some of the methods that Odibat as developed as a result of his work on generalized Taylor series expansions.

If you have any more questions, let me know.

An attempt for a more explicit solution. Using the definition of the half-derivative given by JamalS, one can transform the differential equation using the Laplace transform and get $$s^2X(s)-sx(0)-x'(0)=-\gamma^3{\sqrt s}\;X(s)$$ where $\gamma=\sqrt{\frac km}$ is a positive constant. Solving for $X$ gives $$X(s)=\frac{sx(0)+\dot x(0)}{s^2+\gamma^3{\sqrt s}}.$$ Let us call $G(s)=1/(s^{3}+\gamma^3)$. The Laplace inverse of $s^{-1/2}G(s^{1/2})$ is given by (Erdelyi, Table of integral transforms, chapter IV, equation 4.1.33) $$z(t)=\frac{1}{\sqrt{\pi t}}\int_0^\infty g(u)\mathrm e^{-u^2/4t}\mathrm du.$$ Therefore we have $x(t)=\dot x(0)z(t)+x(0)\dot z(t)$.

Computation of $z(t)$ Let us write $\mathrm j=\mathrm e^{\mathrm i 2\pi/3}$. The inverse $g(t)$ of $G(s)$ is a simple Laplace inversion and yields $$g(t)=\frac1{3\gamma^2}\left[\mathrm e^{-\gamma t}+\mathrm e^{-\mathrm j\gamma t}+\mathrm e^{-\mathrm j^2\gamma t}\right]$$ Let us now write $$\Phi(a,t)=\frac{1}{\sqrt{\pi t}}\int_0^\infty\mathrm e^{-a u}\mathrm e^{-u^2/(4 t)}\mathrm du=\mathrm e^{a^2 t}\mathrm{erfc}(a\sqrt t)=\varphi(a^2t).$$

EDIT

A remarkable property of $\varphi$ is $\varphi'(x)=\varphi(x)-\frac1{\sqrt{\pi x}}$. Reporting in the definition of $z$ we find $$z(t)=\frac1{3\gamma^2}\left[\varphi(\gamma^2 t)+\varphi(\mathrm j\gamma^2t)+\varphi(\mathrm j^2\gamma^2 t)\right].$$ $$\dot z(t)=\frac13\left[\varphi(\gamma^2 t)+\mathrm j\varphi(\mathrm j\gamma^2 t)+\mathrm j^2\varphi(\mathrm j^2\gamma^2t)\right]-\frac2{3\sqrt{\pi\gamma^2t}}$$

Mathematica made the following plots of $z(t)$ and $\dot z(t)$ for $\gamma=1$, showing that $z$ is quite severely damped. • It's somewhat curious that you obtain something radically different than what JamalS and Steven Mathey obtained. Oct 15, 2014 at 14:07
• @KyleKanos. The function ensemble in which the equations are solved differ, this is why, the final solutions are different. JamalS had troubles with the integration, the method I propose also uses a difficult integration but as the functions are analytical in the half complex plane instead of the imaginary line, it is possible to perform it. We actually used the same method, only with different initial conditions. Steven Mathey's answer is different, because he solved the problem without boundary conditions. Also keep in mind that $z(t)$ is not the solution, it is useful to build the solution. Oct 16, 2014 at 7:30
• @KyleKanos. I have gone back to Mathematica to plot $\dot z$ together with $z$ and I did not get the same plot. I checked carefully and I am quite sure the new version is the correct one. There must have been a typo in the last version. The functions actually oscillate in my solution too. Oct 16, 2014 at 8:05