Mathematics · Advanced university
Decay of a heat-equation mode · formula reference
For u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
Formula
For u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.
u(x,t)=A exp(−κn²t)sin(nx)
Worked example
For u_t=0.5u_xx on 0<x<π with zero boundary values and u(x,0)=13sin(2x), find the amplitude of this mode at t=0.4. Round the final result to 4 decimal places if needed.
The spatial mode has second derivative −4sin(2x). Its amplitude solves A′ = −2A.
5.8413
Prerequisites and checking
Algebra; calculus or linear algebra as appropriate to the topic. Keep units and rounding consistent. State the model conditions before applying the formula.
Further reading and review references
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