Mathematics · Advanced university
PDE modes, boundaries and initial data · revision notes
A sine mode for u_t=κu_xx decays as exp(−κk²t).
Key ideas and conditions
Heat equation
A sine mode for u_t=κu_xx decays as exp(−κk²t).
Wave equation
A fixed-end string supports discrete sine modes with angular frequencies cnπ/L.
Boundary data
The spatial domain and endpoint conditions select admissible modes.
Uniqueness
An equation alone is not a complete problem; initial and boundary data matter.
Explain without looking
- Explain why a sine mode is natural for zero displacement at both ends of a finite interval.
- Why is replacing a boundary-value problem by only its differential equation incomplete?
Further reading and review references
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