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Section 13.1 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 the spatial derivatives always comes in the form of the laplacian \(\nabla^2\text{.}\) This particular spatial dependence occurs in physics PDEs because space is rotationally invariant.

Also notice that some of the equation have no time derivatives, 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 (see Section 13.2) and also affects what kinds of initial conditions are appropriate to guarantee that a unique solution exists (see Section 13.3).

Laplace's Equation:

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

\begin{equation} \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\text{.}\tag{13.1.1} \end{equation}

The differential operator, \(\nabla^2\text{,}\) 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, \(f(x,y,z)\text{,}\) known as the source density. It has the form:

\begin{equation} \nabla^2\psi=f(x,y,z)\tag{13.1.2} \end{equation}

Schrödinger's Equation:

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

\begin{equation} -{\hbar^2\over 2m}\nabla^2\psi+V(x,y,z)\psi=i\hbar{\partial\psi\over\partial t}\text{.}\tag{13.1.3} \end{equation}

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

\begin{equation} -{\hbar^2\over 2m}\nabla^2\phi+V(x,y,z)\phi=E\phi\text{.}\tag{13.1.4} \end{equation}

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

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:

\begin{equation} {\partial\psi\over\partial t}-\kappa\nabla^2\psi=0\text{.}\tag{13.1.5} \end{equation}

\(\kappa\) 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:

\begin{equation} -{1\over v^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi=0\text{.}\tag{13.1.6} \end{equation}

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:

\begin{equation} -{1\over c^2}{\partial^2\psi\over\partial t^2}+\nabla^2\psi+{m^2c^2\over\hbar^2} \psi=0\text{.}\tag{13.1.7} \end{equation}

The coefficients \(c\text{,}\) \(\hbar\) and \(m\) all represent constants.

Helmholtz's Equation:

The equation:

\begin{equation} \nabla^2\psi+k^2\psi=0\tag{13.1.8} \end{equation}

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.