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Mathematics · Advanced university

Decay of a heat-equation mode

For u_t=κu_xx on 0<x<π with zero endpoint values and initial sine mode.

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Method and conditions

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 examples

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

For u_t=0.4u_xx on 0<x<π with zero boundary values and u(x,0)=12sin(4x), 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 −16sin(4x). Its amplitude solves A′ = −6.4A.

0.9277

Try it

  1. For u_t=0.5u_xx on 0<x<π with zero boundary values and u(x,0)=3sin(4x), find the amplitude of this mode at t=0.2. Round the final result to 4 decimal places if needed.

  2. For u_t=0.3u_xx on 0<x<π with zero boundary values and u(x,0)=9sin(1x), find the amplitude of this mode at t=0.5. Round the final result to 4 decimal places if needed.

Check your work

Answers and marking notes
  1. 0.6057

    The spatial mode has second derivative −16sin(4x). Its amplitude solves A′ = −8A.

  2. 7.7464

    The spatial mode has second derivative −1sin(1x). Its amplitude solves A′ = −0.3A.

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