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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

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

• 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).