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 {\mathop{∇}}^{2} to denote the sum

• The wave equation, {\mathop{∇}}^{2}u = {1\over { c}^{2}} {{∂}^{2}u\over ∂{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, {\mathop{∇}}^{2}u = {1\over k} {∂u\over ∂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 {\mathop{∇}}^{2}u = 0.
• Helmholtz’s equation {\mathop{∇}}^{2}u + λu = 0.
  This occurs for waves in wave guides, when searching for eigenmodes (resonances).
• Poisson’s equation {\mathop{∇}}^{2}u = f(x,y,\mathop{\mathop{…}}).
  The equation for the gravitational field inside a gravitational body, or the electric field inside a charged sphere.
• Time-independent Schrödinger equation, {\mathop{∇}}^{2}u = {2m\over {ℏ}^{2}} [E − V (x,y,\mathop{\mathop{…}})]u = 0.
  |u{|}^{2} has a probability interpretation.
• Klein-Gordon equation {\mathop{∇}}^{2}u − {1\over { c}^{2}} {{∂}^{2}u\over ∂{t}^{2}} + {λ}^{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.)