A differential equation is a relation between a mathematical function and
its derivates. If the function depend on more than one variable, the
differential equation is said to be **partial**. These are a lot harder to
solve analytically and numerically than the ordinary differential equations. The
theories for PDEs has been developed mainly for the field of theoretical physics,
where they play a very important part, especially in quantum mechanics. Below
I list the most important PDEs in physics and how you would solve them.

(An excuse goes out to all Netscape users who may not be able to see the
equations below.)

One of the most important PDEs in physics is ** Poisson's partial
differential equation**, which is what mechanical problems usually reduces to:

n ∂^{2}u
Δu = ∑ ----- = f (1)
j=1 ∂x^{2}_{j}

Where *u* is the sought function, *f* is a known function, and *x*
is a coordinate in space. In physics, n usually equals 3, for our 3 dimensional
world. When f=0 the equation is called Laplace's equation. The Newton
potential gives solutions to this as:

1 f(y)
u(x) = ---- ∫∫∫------- dy_{1}dy_{2}dy_{3}
4π |x-y|

In physical applications of this, you usually search for solutions within a
domain Ω
where you have a Dirichlet boundary condition u = φ on the boundary ∂Ω.
There are many methods to find this solution analytically, and usually they are
focused on finding upper and lower values of the function . This PDE can also be
solved for other boundary conditions such as Neumann boundary condition.
Equation (1) is and example of a elliptical differential equation. The term
"elliptical" refers to the characteristic polynomial of the
equation, which is a way to classify PDEs.

Another common PDE is the ** heat equation, or diffusion equation:**

∂u
--- - Δu = f (2)
∂t

where u is a function both in the room (x_{1}, x_{2}, x_{3}... x_{n}) and time (t). For
n=3 this equation describes the distribution of heat in an homogenous material
or some other process of diffusion in the physical world. If we have the
initial value of u, u(t=0) = u_{0}, the solutions for t > 0 can be derived from

u(t,x) = (4πt)^{-π/2}∫e^{-|x-y|2/4t}·u_{0}(y)dy

For solutions within a domain Ω you can also apply boundary conditions
which makes the PDE solvable. Equation (2) is a hypo elliptical differential
equation. In quantum mechanics, the ** Schrödinger equation ** is an important PDE,
and it is fairly similar to the heat equation. The difference is that it has a
imaginary factor:

∂u
i--- - Δu + Vu = 0 (3)
∂t

This equation is not easily solved, but great progress has been made during
the last decades. In physics, where V is the same potential function as in the
N-body problem, and u = u(x,y,z), this is the equation for the quantum
mechanical, non-relativistic N-body problem. For each
φ where ∫|φ(x)|^{2}dx
< there is a solution *u* to (3) which is equal to φ for t = 0
and ∫|u(t,x)|^{2}dx is independent of
t.

The** wave equation**, which describes the movement of light and electromagnetic
waves, is also a PDE:

∂^{2}u
----- - Δu = f (4)
∂t^{2}

Maxwell's equations reduce to the above. For u = u(x,y,z) a solution is
given by

f(t-|y|,x-y)
u(t.x)= ∫∫∫------------ dy_{1}dy_{2}dy_{3}
4π|y|

The wave equation (4) is an example of a hyperbolical differential
equation. Cauchy's problem, to find a solution for u=φ and ∂u/∂t=Ψ,
can easily be derived from this last equation. Other problems lead to more
complex calculations. For instance, the Dirichlet problem leads to a complicated
study of singularities, which is related to the study of refraction and
diffraction in optics.

All of the above are examples of **linear partial differential equations**.
However, many of the fundamental equations in physics are non-linear, such as
Navier-Stokes equations in fluid dynamics. These are a bit harder to solve analytically
and are usually the subject to numerical methods. Usually one has to study the
linear solution to the non-linear problem, and then based upon this adjust the
solution in the surrounding you are interested in.

*Reference: ne.se*