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### Important PDEs in Physics

On this page you will find a list of most of the important PDEs in physics with their names. Notice that some of the equation have no time dependence, some have a first order time derivative, and some have a second order time derivative. This difference is the foundation of an important classification scheme. Also notice that the spatial dependence always comes in the form of the laplacian $\nabla^2$. This particular spatial dependence occurs because space is rotationally invariant.

**Laplace's Equation:** Many time-independent problems are described by *Laplace's equation*. This is defined for $\psi=\psi(x,y,z)$ by:

$$ \nabla^2\psi={\partial^2\psi\over \partial x^2}+{\partial^2\psi\over \partial y^2}+{\partial^2\psi\over\partial z^2}=0. $$

The differential operator, $\nabla^2$, defined by eq.(1) is called the *Laplacian* operator, or just the *Laplacian* for short. Some examples of Laplace's equation are the electrostatic potential in a charge-free region, the gravitational potential in a matter-free region, the steady-state temperature in a region with no heat source, the velocity potential for an incompressible fluid in a region with no vortices and no sources or sinks.

**Poisson's Equation:**

*Poisson's
equation* is like Laplace's equation except that it allows an inhomogeneous term, $f(x,y,z)$, known as the source density. It has the form:

$$ \nabla^2\psi=f(x,y,z) $$.

**Schrödinger's Equation:**

A great deal of quantum mechanics is devoted to the study of the solutions to the *time-dependent Schrödinger equation*:

$$ -{\hbar^2\over 2m}\nabla^2\psi+V(x,y,z)\psi=i\hbar{\partial\psi\over\partial t}. $$

This equation governs the time dependence of the wave-function of a particle moving in a given potential, $V(x,y,z)$. A special role is played by solutions to (4) that have the simple form: $\psi=\phi(x,y,z)\exp(-iEt/\hbar)$. The function $\phi$ satisfies the *time-independent Schrödinger equation* or, more correctly, the *Schrödinger eigenvalue equation*:

$$ -{\hbar^2\over 2m}\nabla^2\phi+V(x,y,z)\phi=E\phi. $$

In both of these equations $\hbar$ and $m$ represent real constants. $E$ is a constant that emerges during the separation of variables procedure. $i$, as usual, satisfies $i^2=-1$.

**The Diffusion Equation:**

The pde governing the concentration of a diffusing substance or the non-steady-state temperature in a region with no heat sources is the *diffusion equation*:

$$ {\partial\psi\over\partial t}-\kappa\nabla^2\psi=0. $$

$\kappa$ is a real constant called the *diffusivity*.

**The Wave Equation:**

Wave propagation, including waves on strings or membranes, pressure waves in gasses, liquids or solids, electromagnetic waves and gravitational waves, and the current or potential along a transmission line all satisfy the following *wave equation*:

$$ -{1\over v^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi=0. $$

The real constant $v$ can be interpreted as the speed of the corresponding wave.

**The Klein-Gordon Equation:**

Disturbances traveling through fields that mediate forces with a finite range, satisfy a modification of the wave equation called the *Klein-Gordon
equation*. It is given by:

$$ -{1\over c^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi+{m^2c^2\over\hbar^2} \psi=0. $$

The coefficients $c$, $\hbar$ and $m$ all represent constants.

**Helmholtz's Equation:**

The equation:

$$\nabla^2\psi+k^2\psi=0$$

is known as *Helmholtz's equation* and arises as the time-independent part of the diffusion or wave equations. $k$ is a constant that emerges during the separation of variables procedure.