2.1: Examples of PDE

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).