EC-3.4 The Wave and the Diffusion Equation
You can tell a wave equation from a diffusion equation by inspection, solve either by separating variables, and name where each one appears in electronics.
Before:nothing requiredUnlocks:EC-4. Transforms, Complex Variables and Numerical Methods
Two partial differential equations account for a surprising amount of electronics, and both are met here rather than being derived twice inside other subjects. The wave equation is what a transmission line and a plane wave obey; the diffusion equation is what charge carriers in a semiconductor obey. They differ in one derivative and behave completely differently because of it — one carries a shape along undistorted, the other smears it out — and seeing that contrast once is worth more than either derivation alone.
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Partial against ordinary: what changes when there are two independent variables
With both position and time present, a derivative has to say which variable it is with respect to, and the arbitrary constants of an ordinary equation become arbitrary functions. That is why boundary conditions matter far more here.
NPTEL: Differential Equations for Engineers · CourseThe wave equation, and its two travelling solutions
A second derivative in time equal to a constant times a second derivative in position is satisfied by any shape moving left and any shape moving right at one fixed speed. The shape is preserved, which is the defining property of a wave.
The diffusion equation, and why it smooths instead of travelling
A first derivative in time against a second in position describes something that flows from where there is more to where there is less. Sharp features vanish quickly, and nothing travels at a fixed speed.
Separation of variables, the one method worth learning here
Assuming the answer is a function of position multiplied by a function of time splits one partial equation into two ordinary ones. It works for both equations in this topic and it is the method most later courses assume you have.
MIT OpenCourseWare 18.03: Differential Equations · CourseBoundary conditions, and how they select the answer from the family
The separated solutions form an infinite family and the boundary conditions decide which combination of them fits, usually by matching a series at the edges. This is where the Fourier series first becomes necessary rather than decorative.
Where each one appears: a transmission line, and carriers in a semiconductor
Voltage along a line satisfies the wave equation, which is why a pulse arrives at the far end with its shape intact. Carrier concentration in a doped region satisfies the diffusion equation, which is why an injected excess spreads and decays.
NPTEL: Differential Equations for Engineers · Course
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