A third order differential equation for the temperature appears in the heat transport theory of materials contradicting the "fading memory paradigm".

See the attached pdf.

There are some examples in theory of electrical circuits.

Here are some references that may be useful --

https://www.researchgate.net/search.Search.html?type=publication&query=third%20order%20linear%20differential%20equations%20applications

https://link.springer.com/chapter/10.1007/978-94-009-3715-4_4?no-access=true

See also

https://math.stackexchange.com/questions/2167292/are-there-examples-of-third-or-higher-order-linear-differential-equations-in-p

Hi Bahram,

Although most natural phenomena we know of are observed to have behaviors described by at most second order ordinary/partial differential equations, but there are behaviors that require third order differential equations. One example is in motion and it is called jerk. Jerk is the third order derivative of the position/displacement function of a moving object, where the first and second order derivatives represent velocity and acceleration.

Regards,

You can start from KdV equation: U_t+UU_x+U_xxx=0. It is nonlinear PDE, derived in hydrodynamics, but it has analytical solution. If we linearize it, it will contain 3-rd derivative, which comes from dispersive effects.

In general, I would recommend this book of Whitham, which has many DE for different problems in physics and applied mathematics. See https://archive.org/details/LinearAndNonlinearWaves

You can find few applications here:

http://www.sciencedirect.com/science/article/pii/S001600320900043X

And the following master thesis:

Highly accurate difference schemes for the numerical solution of third order ordinary and partial differential equations.

Dear Bahram

This is a nice question.

There are some applications in which third order ODEs are formed. They are in the momentum tranport of viscoelastic fluids in porous channels, plates etc.

regards

Ali

Shear in beams and beams on Elastic foundations.

Sure, there are lots of them in astrodynamics: the Clohessy-Wiltsjire equations (relative motion about a circular orbit), the Tschauner-Hempel equations (relative motion about an ellipse), the two-body equation after the substitution y=1/r. There are others,

Dear Bahram,

Colleagues have already pointed a lot of processes that can be modelled through 3rd order differential equations, ordinary and partial. Let me add one PDE example, emerging in porous media flows. So-called non-equilibrium models involving dynamic capillarity and/or play-type hysteresis, lead to equations where the time derivative of the unknown quantity (saturation) is combined with second order derivatives in space.

The resulting model is a nonlinear version of the one mentioned above by Roman Poznanski, and of the Benjamin–Bona–Mahony equation.

Good luck,

Sorin

Another example in PDE: KdV equation ( linearized version for linear).

I have two books about the differential equation of third order:

1- Theory of Third-Order Differential Equations

2- Third Order Linear Differential Equations

A third order differential equation for the temperature appears in the heat transport theory of materials contradicting the "fading memory paradigm".

See the attached pdf.

Good application offered by L. Herrera.

Luca Parisi offered some good insights.

Here is an example from MEMS:

http://congressline.hu/enoc2017/abstracts/77.pdf

Analysis of a simplified MEMS oscillator

Richard H. Rand, Alan T. Zehnder and B. Shayak

Physical examples are: the Abraham–Lorentz force (electron self-force) and the Jerk for parabolic curves in the roads.

Physical examples like bio-heat equations (in Cartesian and Polar coordinates)

As I understood from the question. You are requesting third order equations for applications.

There are many equations of the third order such as

1. KDV equation 2. RLW equation 3. BBM equation, ....

Very rarely do they crop up in Mechanics, the Abraham Lorenz electron theory

an exception, as stated above. You dont normally go beyond acceleration, which requires a second derivative.

For

water waves and soliton theory, maybe you need (ie. KDV)

Of course in math you do anything you like.

The theory of linear differential equations works for all orders, easy to solve.

Only if it is non linear, then you really need to struggle sometimes. Solitons strive on nonlinear, when there is lack of superposition.

Unfortunately, the example proposed by L.Herrera for the temperature is physically inconsistent, in that it has three characteristic roots: one real negative root and two complex conjugate roots with POSITIVE real part. Hence, the corresponding solutions blow up.

@Giorgi, The equation (3) in my answer follows from the standard continuity equation. This is a rough approximation because as we know, the continuity equation depends on the transport equation, and it might happen that it is incompatible, with the proposed transport equation. The point is that the kernel (2) leads to a third order differential equation for the temperature. Probably some how different from (3), but still a third order equation.

you can see an exemple in Ref: 10.1142/S0217751X1950115X

Specific applications to material systems contravening the fading memory paradigm, may be found in: L. Herrera. Causal heat conduction contravening the fading memory paradigm. Entropy 21, 950, (2019).

PHYS. REV. D 99, 044029 (2019)

Please see Gupta Jasim Two step Method

The Moore–Gibson–Thompson equation is third-order in time. If you want a linear equation, linearize the nonlinear terms....