Skip to main content
\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)
[ "article:topic", "authorname:nwalet", "license:ccbyncsa", "showtoc:no" ]
Mathematics LibreTexts

2.1: Examples of PDE

  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    Partial differential equations occur in many different areas of physics, chemistry and engineering. Let me give a few examples, with their physical context. Here, as is common practice, I shall write \({\nabla}^2\) to denote the sum

    \[{\nabla}^2 = \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\ldots\]

    • The wave equation: \[{\nabla}^2 u= \dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}\] This can be used to describes the motion of a string or drumhead (\(u\) is vertical displacement), as well as a variety of other waves (sound, light, ...). The quantity \(c\) is the speed of wave propagation.
    • The heat or diffusion equation, \[ {\nabla}^2 u= \dfrac{1}{k} \dfrac{\partial u}{\partial t}\] This can be used to describe the change in temperature (\(u\)) in a system conducting heat, or the diffusion of one substance in another (\(u\) is concentration). The quantity \(k\), sometimes replaced by \(a^2\), is the diffusion constant, or the heat capacity. Notice the irreversible nature: If \(t→ −t\) the wave equation turns into itself, but not the diffusion equation.
    • Laplace’s equation: \[{\nabla}^2 u= 0\]
    • Helmholtz’s equation: \[{\nabla}^2 u + \lambda u = 0\] This occurs for waves in wave guides, when searching for eigenmodes (resonances).
    • Poisson’s equation: \[{\nabla}^2 u = f( x, y, \ldots)\] The equation for the gravitational field inside a gravitational body, or the electric field inside a charged sphere.
    • Time-independent Schrödinger equation: \[{\nabla}^2 u= \dfrac{2 m}{\hbar^2}[ E− V( x, y,\ldots)] u= 0\] \(| u|^2\) has a probability interpretation.
    • Klein-Gordon equation \[{\nabla}^2 u − \dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}+\lambda^2 u= 0\] Relativistic quantum particles,\(|u|^2\) has a probability interpretation.

    These are all second order differential equations. (Remember that the order is defined as the highest derivative appearing in the equation).