# What is the essential difference between ordinary differential equations and partial differential equations?

So many years after my undergraduate study and so many years after dealing with various concrete ODEs and PDEs, I still cannot tell the essential difference between them.

What specific belongs to PDEs but not to ODEs? What conclusion for ODEs cannot be generalized to PDEs?

At the moment, my understanding is simply that PDEs have more than one variables.

• In PDEs the geometry if often far more subtle. Aug 24 '15 at 21:16
• What I don't see in any of the answers: while for ODE the initial value problem and some boundary value problems have unique solutions (up to some constants at least), for PDE, even linear ones, there can be infinitely many completely different solutions, for example time dependent Schrodinger equation for some potentials admits a lot of mathematically valid, but unphysical solutions. Usually in applications one restricts themselves to a particular function space. Mar 10 '20 at 10:42

Both are differential equations (equations that involve derivatives). ODEs involve derivatives in only one variable, whereas PDEs involve derivatives in multiple variables. Therefore all ODEs can be viewed as PDEs.

PDEs are generally more difficult to understand the solutions to than ODEs. Basically every big theorem about ODEs does not apply to PDEs. It's more than just the basic reason that there are more variables. For an ODE, we can often view the single independent variable as a time variable, so that ODEs govern a motion or flow of an object in time. The idea of ODEs governing "motion" allows us to use many mathematical results that have analogues in physics (for example empirical behavior regarding Newton's law) and allow us to understand the solutions much more precisely.

Well, given a linear ODE, the set of solutions form a vector space with finite dimension. However, a linear PDE (like the heat equations) has a set of solution that form a vector space with infinitely many dimensions.

To see that, one may consider the ODE

$$y'=-ay(t),$$ with solution, $$y(t)=e^{-at}y_0,$$ (so, the vector space is one dimensional) Then the heat equation with periodic boundary conditions $$u_t=u_{xx},$$ has solution (use Fourier series/separation of variables) $$u(t)=\sum_{k\in\mathbb{Z}}e^{ikx}\hat{u}(k).$$ We see that the linear combination has infinitely many terms, all them linearly independent, so, the vector space has infinitely many dimensions.

Linear PDEs may not have solutions. Hans Lewy constructed such an example about sixty years ago, and it probably surprised just about everyone in the field at the time. ODEs are much nicer in that regard. http://www.jstor.org/stable/1970121?&seq=1#page_scan_tab_contents

ODE has one Independent variable, say $$x$$. Solution is $$y(x)$$.

PDE has more than one independent variables say $$(x_1,x_2,...,x_n)$$: solution is $$y(x_1,x_2,..x_n)$$. Partial derivatives are in the equation. A partial derivative differentiates with respect to one independent variable (say $$x_3$$)while holding the other independent variables constant.

Perhaps I'm missing something about your question (if so, please forgive my stupidity), but ISTM the essential difference between ODEs and PDEs == what specific[ally] belongs to PDEs but not to ODEs == ∂. Period. If the equation involves derivatives, and at least one is partial, you have a PDE. If you have a differential equation with no partial derivatives (i.e., all the equation's derivatives are total), you have an ODE. (Annoyingly for this terminology, one can also refer to total differential equations, and {TDEs} ≠ {ODEs}: rather, {TDEs} ⊆ {ODEs}.)

In addition to having more variables, PDEs also have more complex solutions than ODEs. In general, for an ODE we can often picture the solution as some sort of flow that varies with time. PDEs have more complex interpretations with the heat equation as an instance. In the heat equation the solution is such that it depends on both the position in space as well as time, and its more difficult to visualise the flow of heat as a function.

If we replace continuous derivative by its finite analog $(f(x+h)-f(x))/h$ then, in some sense, there is no difference between linear PDEs and ODEs, both with non-constant coefficients. Finite PDEs and finite ODEs generate the same $C^*$-algebra, namely universal UHF algebra $\mathbb{C}^{1\times1}\otimes\mathbb{C}^{2\times2}\otimes\mathbb{C}^{3\times3}\otimes...$ https://arxiv.org/abs/1807.09327