Physically, what is the difference between systems of ODEs and PDEs?

[zheng89120]

What kind of systems do ODEs describe?

What kind of systems do PDEs describe?

 

[JJacquelin]

What kind of systems do ODEs describe?
What kind of systems do PDEs describe?

ODEs are used in case of a physical phenomena which can be modeled with functions of a variable only.
PDEs are used in case of a physical phenomena which can be modeled with functions of not only a variable, but several.

 

[Eynstone]

PDEs occur usually in dimension >1.

 

[Lavabug]

ODE's: problems with one independent variable (like time). Classic 1-dimensional F = ma problems (x depends on t), harmonic oscillators, transient response in a circuit (I depends on t), radioactive decay.

PDE's: problems where there is more than 1 independent variable. Temperature distribution on a plate (depends on x, y, and time), waves on a string, plate or sphere (the amplitude can depend on x,y,z and t), electric potential on a conductor... you get the idea.

 

[jostpuur]

If

$$u:[0,1]\times\mathbb{R}\to\mathbb{R}$$

is some function that is supposed to satisfy

$$\partial_t u(x,t) = -\partial_x^2 u(x,t)$$

that is a PDE.

If

$$u:\{1,2,\ldots, N\}\times\mathbb{R}\to\mathbb{R}$$

is some function that is supposed to satisfy

$$\partial_t u(n,t) = -\frac{1}{N^2}\big(u(n+1,t) - 2u(n,t) + u(n-1,t)\big),\quad\quad (u(0,t)=0,\; u(N+1,t)=0)$$

that is an ODE system.

Physically, what is the difference between systems of ODEs and PDEs?

What kind of systems do ODEs describe?

What kind of systems do PDEs describe?

When the amount of degrees of freedom is very large, physically they describe the same thing, because physically there is no difference in having infinite degrees of freedom or a very large but finite amount of degrees of freedom. You can describe the same physical systems with different mathematical models such that some are ODEs and some are PDEs.