Section 26.5

Solving the Heat Equation

Putting it all together: Separation of Variables + Fourier Series.

1

Introduction

Problem:
BCs:
IC:

2

The Solution Steps

  1. Separate variables to get and .
  2. Use BCs to find eigenvalues and eigenfunctions .
  3. Solve time equation: .
  4. Superposition: .
  5. Use IC and Fourier Sine Series to find .
3

Worked Examples

Example 1: Single Mode

on .

Already a sine mode for .

, all other .

.

Example 2: Constant Initial Temp

. Ends suddenly cooled to 0.

Find Fourier Sine series for 100.

.

.

Example 3: Visual Decay

Higher frequencies (n=3, 5...) decay much faster due to the in the exponent.

4

Practice Quiz

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