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Teaching video
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Graduate chapters and video availability01 · Read and understand
What you will learn
- Solve a heat equation through its Fourier eigenmodes.
- Justify the conclusion "u(x,t)=exp(−νt)sin x+exp(−9νt)sin(3x)" using the stated assumptions.
Before you start
Fourier series and linear PDEs.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
On the 2π-periodic line, solve uₜ=νuₓₓ with u(x,0)=sin x+sin 3x, ν>0.
Why this math matters
Solve a heat equation through its Fourier eigenmodes. This worked micro-lesson connects a precise mathematical condition to a conclusion you can check. The transfer task asks you to change the setting and decide which parts of the reasoning still apply.
Set up the model
A useful answer starts with clear assumptions:
- The domain is periodic with period 2π.
- ν is a positive constant.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Watch diffusion damp high spatial frequencies faster
PausedQuestion: Start with the question. Paused.
Question
Start with the question
On the 2π-periodic line, solve uₜ=νuₓₓ with u(x,0)=sin x+sin 3x, ν>0.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Build the model
∂ₓₓsin(kx)=−k²sin(kx)
Each Fourier mode is an eigenfunction of the Laplacian.
Work through the mathematics
Its coefficient satisfies a′ₖ=−νk²aₖ
The PDE separates into scalar decay equations.
Check the conclusion
u(x,t)=exp(−νt)sin x+exp(−9νt)sin(3x)
The shorter-wavelength mode decays nine times faster in its exponential rate.
The result
u(x,t)=exp(−νt)sin x+exp(−9νt)sin(3x)
The shorter-wavelength mode decays nine times faster in its exponential rate.
Common mistakes to catch
- A decay rate is not the same as a wave's oscillation frequency.
- Forward smoothing does not imply stable inversion.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
What happens to a constant initial mode?
Show a hint
Its second derivative is zero.
Reveal answer and explanation
It remains constant
The zero-frequency mode has decay rate zero.
Practice 2
Why is reversing heat flow unstable?
Show a hint
Reverse the exponential factors.
Reveal answer and explanation
High frequencies would grow extremely rapidly
Tiny fine-scale errors are multiplied by e^(νk²t).
Take the idea with you
Relate diffusion smoothing to the loss of fine-scale information.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Derive an equation from stationary energy
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