Week 13

Tutorial 13: Heat & Wave Equations

1. Solve the heat equation
subject to
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Solution
Suppose
Case 1:
Case 2: ()
Case 3: ()
Applying the boundary conditions: ,
Case 1:
Case 2
Case 3
For non-trivial solutions, let ⇒ ⇒
Eigenvalues:
Eigenvectors:
Applying initial conditions: ,
For ,
For
Solution exists only when . Thus,

2. Solve the heat equation
subject to
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Solution
Suppose
Case 1:
⇒ This is a trivial solution.
Case 2:
Since
⇒ This is a trivial solution
Case 3:
To obtain non-trivial solution,
Eigenvalue:
Eigenfunction:
By superposition principle,
Given

3. Solve the wave equation
subject to the given conditions
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Solution
Suppose
For , and
Case 1:
Case 2:
⇒ Only trivial solution
Case 3:
⇒ Only trivial solution
For
From (1),
Hence,
Thus,