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Is it possible to replace a large-dimensional system of differential equations with one partial differential equation?

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    $\begingroup$ The answer obviously depends on the details of a specific case. Would you care to elaborate on yours? Note that the reverse operation is a routine aspect of numerical analysis of PDEs. However the resulting ODEs have special "sparse" structure. $\endgroup$
    – Igor Khavkine
    Commented Mar 26, 2013 at 17:27
  • $\begingroup$ Clearly the heat equation is an example, in that the motion of the molecules of a heated material is described by a huge system of ODEs, but in a certain limit the material behaves like a continuum. But this sort of ``rescaling'' is really the domain of expertise of physicists. Maybe ask on physics.stackexchange.com. $\endgroup$
    – Ben McKay
    Commented Mar 26, 2013 at 17:41
  • $\begingroup$ Same answer for vibrating string. That was how it was actually presented in a physics class I took. $\endgroup$
    – Will Jagy
    Commented Mar 26, 2013 at 19:31
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    $\begingroup$ "Hamilton-Jacobi" may be a useful keyword. Many Hamiltonian systems can be described in terms of either individual particle trajectories (ODE) or propagation of the corresponding wavefront (PDE). $\endgroup$
    – TerronaBell
    Commented May 25, 2013 at 22:27
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    $\begingroup$ In what sense replace? $\endgroup$
    – András Bátkai
    Commented Jun 8, 2013 at 19:12

2 Answers 2

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Let me single out a situation which goes the other way around: how a system of ODE is describing the propagation of singularities for a principal type PDE.

Take a linear (pseudo)differential operator of real principal type with smooth coefficients: the principal symbol $p(x,\xi)$ is real-valued and $dp\wedge \xi\cdot dx\not=0$ (verified for the wave equation or a non-vanishing real vector field). Then the singularities are moving along the bicharacteristic curves, which are the integral curves of the Hamiltonian vector field of $p$, $$ H_p=\frac{\partial p}{\partial \xi}\cdot \frac{\partial }{\partial x}- \frac{\partial p}{\partial x}\frac{\partial }{\partial \xi}.\quad $$ Solving the system of ODE, $\dot \Gamma=H_p(\Gamma)$ is enough to understand the propagation of singularities: if $p(x,D) u\in C^\infty$ the the wave-front-set of $u$ is invariant by the flow of the Hamiltonian vector field. There is no need to solve the PDE if you are only interested in singularities.

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This idea is studied in the field "kinetic PDE." See these lecture notes by Clément Mouhot:

http://cmouhot.wordpress.com/1900/10/25/mathematical-topics-in-kinetic-theory-part-iii-course/

Ch 2 in particular should be of interest.

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