2.1: Examples of PDE
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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
\[{\nabla}^2 = \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2}+\ldots \nonumber \]
- The wave equation: \[{\nabla}^2 u= \dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2} \nonumber \] 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, \[ {\nabla}^2 u= \dfrac{1}{k} \dfrac{\partial u}{\partial t} \nonumber \] 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: \[{\nabla}^2 u= 0 \nonumber \]
- Helmholtz’s equation: \[{\nabla}^2 u + \lambda u = 0 \nonumber \] This occurs for waves in wave guides, when searching for eigenmodes (resonances).
- Poisson’s equation: \[{\nabla}^2 u = f( x, y, \ldots) \nonumber \] The equation for the gravitational field inside a gravitational body, or the electric field inside a charged sphere.
- Time-independent Schrödinger equation: \[{\nabla}^2 u= \dfrac{2 m}{\hbar^2}[ E− V( x, y,\ldots)] u= 0 \nonumber \] \(| u|^2\) has a probability interpretation.
- Klein-Gordon equation \[{\nabla}^2 u − \dfrac{1}{c^2}\dfrac{\partial^2 u}{\partial t^2}+\lambda^2 u= 0 \nonumber \] 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).