Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics?

The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.

I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples?

Solution 1:

The Euler-Bernoulli equation for beams

$$(EIw'')'' = q$$

is another example of a linear 4. order equation. $w$ represents the deflection and $q$ the distributed load.

Solution 2:

One prominent example is the Abraham–Lorentz force, which depends on the derivative of the acceleration (AKA the jerk) of a charged particle. This leads to weird effects like pre-acceleration, since adding this into the force equation leads to a third-order equation, which can be integrated to show that the acceleration depends on the external force in a way that includes parts of it that are supposed to be in the particle's future.

Solution 3:

The Dirac equation is a first-order linear PDE taking values in $\mathbb{C}^{4}$. It can be recast as a second-order linear PDE taking values in $\mathbb{C}^{2}$, and yet again, it can be recast as a 4th-order PDE taking values in $\mathbb{R}$.

Feynman regarded the secord-order formulation of the Dirac equation as the "true" fundamental form. The point is that if you can map solutions of one lower-order PDE bijectively to solutions of a high-order PDE, it's not even clear which of these PDEs "occurs" in physics.

Solution 4:

Not a physical law, but appearing in an engineering and design context (that is physics, after all), minimising the third derivative (the jerk) in a transport vehicle improves quality. Intuitively, if a curve get its shape from a circle, the normal acceleration for a vehicle will change abruptly from zero to some value, making the jerk get a high value. So, it's better if its radius of curvature grows from zero to the suitable value in a continuous way.

Examples:

Track transition curve

Jerk

Bidirectional jerk