# What exactly are partial differential equations?

What exactly are partial differential equations? I know what differential equations are but I want to know what a PDE is since the Schrödinger equation for example is a PDE too. Also, is there a good video or book explaining PDE's for beginners?

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They are the multivariable version of differential equations. –  Andrew Oct 9 '12 at 6:57
Did you try Wikipedia? –  Qiaochu Yuan Oct 9 '12 at 7:06
While @Andrew is certainly right, in a narrow technical sense, the field of PDEs has a completely different flavour than the field of ODEs. –  Harald Hanche-Olsen Oct 9 '12 at 7:14
@ Qiaochu Yuan, I find wikipedia's definition too 'mathematically formal'. Before I understand that definition, I need a more 'intuitive' understanding. –  ZafarS Oct 9 '12 at 7:39

Since the OP said that the Wiki definition is too mathematically formal, let me give some intuition of the partial differential equation starting from the first order case.

First consider a first order ordinary differential equation $$\frac{\mathrm{d}}{\mathrm{d}t} X = F(t,X)$$ where $X$ takes values in, say, $\mathbb{R}^n$ and $F(t,X)$ is some Lipschitz continuous function. (In other words, this is a dynamical system.) What it says is that it tells us how $X$ ought to change, at an instant in time $t$, based on the time $t$ and the current value of $X$. This is what we call an "evolutionary point of view".

The analogue of a partial differential equation that is "evolutionary" is an equation for $X$, which now depends on not only the time $t$ but also some spatial coordinates $(x_1, \ldots, x_N)$ would be something like

$$\frac{\partial}{\partial t} X = F\left(t,x_1, \ldots, x_N, X, \frac{\partial}{\partial x_1} X, \ldots, \frac{\partial}{\partial x_N} X\right)$$

Now we have that how $X$ ought to change at an instant in time $t$ and at position $(x_1,\ldots, x_N)$, is based on a function of not only the coordinate values of $t$ and $(x_1, \ldots, x_N)$, but also the value of $X$ at that space-time point, and also the value of its spatial directional derivatives at that space-time point.

There is a different point of view, however, for ordinary differential equations. This is the "constraint point of view". For this we consider the equation $$X'' = F(s,X)$$ and try to solve it while prescribing boundary conditions $X(0) = f_1$ and $X(1) = f_2$. What we should think of is that the differential equation describe some "compatibility condition" for a certain physical system in stasis. For example, the above equation can be used to describe the distribution of temperature along a rod that is kept at temperature $f_1$ at one end and temperature $f_2$ at the other. The equation says that the second derivative of the temperature function depends on the physical characteristic of the rod at the point $s$ as well as the current temperature at that point $x$. In other words the laws of nature constrains what temperature profiles are possible.

From this point of view, we also get a type of partial differential equations that describes a constraint. In this case, the PDE is usually written as an analytic expression relating the various partial derivatives of a function. What this says is that for the question we are considering, not all functions are admissible as solutions. That some law (most frequently a physical law) requires that the only admissible functions describing the situation (this is a constraint) obey certain relationships imposed upon their Taylor coefficients up to some order $k$ at every point. In other words, the function is not allowed to wiggle willy-nilly. Its rates of changes between the various different directions are tied together.

Intuition aside, the mathematical formulation of a PDE can be stated relatively simply.

A partial differential equation is a equation which expresses an equality between expressions involving partial derivatives of a given function. More precisely, taking one of the simpler cases, a partial differential equation on a scalar function $u$ defined on some subset $U\subseteq \mathbb{R}^N$ is the equation $$F(x,u,\nabla u, \nabla^2 u, \ldots , \nabla^k u) = 0$$ where $x\in U\subseteq \mathbb{R}^N$ are the independent variables, $\nabla^ju$ are the tensors representing the $j$-th fold partial derivatives of $u$ ($\nabla^2 u$ is the Hessian matrix, $\nabla u$ is the gradient vector), and $F$ is some function $$F: U \times \mathbb{R} \times \mathbb{R}^N \times \mathbb{R}^{N^2} \times \cdots \times \mathbb{R}^{N^k} \to \mathbb{R}$$ The number $k$, the maximum order of the derivatives involved in the equation, is called the "order" of the equation.

For some simple examples:

• The transport equation (or linear advection equation) are cases where $k = 1$, and where $$F(x,u,p) = V(x)\cdot p$$ where $p\in \mathbb{R}^N$ and $V(x)$ is some vector field on $U$.

• The Laplace equation is when $k = 2$ and $$F(x,u,p,q) = \operatorname{trace} q$$ where $p\in\mathbb{R}^N$ and $q\in \mathbb{R}^{N^2}$ is interpreted as an $N\times N$ matrix.

• The wave equation is when $k = 2$ and $$F(x,u,p,q) = \operatorname{trace} q - T^\dagger q T$$ where $\dagger$ is the matrix transpose, $T$ is a vector with $\|T\|^2 > 1$

• The linear Schroedinger equation is also when $k = 2$ and $$F(x,u,p,q) = \operatorname{trace} q - T^\dagger q T - i T\cdot p$$ where $T$ is a vector with $\|T\|^2 = 1$. If we remove the imaginary $i$ from the equation, we end up with the linear heat equation instead. Note that necessarily for Schrodinger's equation we need $u$ to take values in the complex number $\mathbb{C}$, and so its gradient and Hessian will be complex-valued vector and complex-valued matrix.

And now, for an extremely high-brow definition (which is a bit beyond the "beginner's scope" asked by the original poster, but nonetheless interesting):

A partial differential relation (of which a partial differential equation is a special type) for a fibre-bundle $F$ over some smooth manifold $M$ is a subset $\mathcal{R}\subseteq F^{(r)}$ of the $r$-th jet bundle of $F$ over $M$. A partial differential equation is one where $\mathcal{R}$ has co-dimension 1. To bring it back to the simplest case defined above the cut: a class of simple fibre-bundles are the trivial bundles $F = M\times N$. Here $M$ is the domain of independent variables (what is $U$ in the definition above). $N$ is the domain of dependent variables (what is $\mathbb{R}$ or $\mathbb{C}$ above, but we can also think of vector valued dependent variables taking values in, say, $\mathbb{R}^n$ or $\mathbb{C}^n$, then we get what are sometimes called systems of partial differential equations). The $k$-th jet bundle is, roughly speaking, the set of all possible $k$-th order Taylor expansions; in other words, it represents the space $\mathbb{R}\times \mathbb{R}^N \times \mathbb{R}^{N^2}\times \cdots \times \mathbb{R}^{N^k}$ of the value of the function and all its (partial) derivatives up to order $k$.

Then the single equation $F(x,u,p,q,r,\ldots,s) = 0$, the partial differential equation, should carve out a codimension 1 subset of $U \times \mathbb{R} \times \cdots \times\mathbb{R}^{N^k}$. (See my question on MO for some tangentially related discussions.)

• Sergiu Klainerman's essay, an abridged version of which appeared in the Princeton Companion to Mathematics. It assume a little bit more than absolute beginner, but not too much more.

• Jürgen Jost's Partial Differential Equations textbook, while on the whole may be a bit too advanced for the OP, has a short introductory chapter titled "What are Partial Differential Equations?", which should also give some intuition.

• Ka Kit Tung's Partial differential equations and Fourier analysis - A short introduction is a textbook aimed at students who have had one year of calculus and one course of ordinary differential equations. It has a decent first chapter reviewing ODEs, and a second chapter explaining the physical origins of partial differential equations while comparing and contrasting them to ordinary differential equations which the OP understood better. This may be a reasonable first book for the OP to consult.

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Beautiful, magnificent answer! –  ZafarS Oct 9 '12 at 12:02