Week 13

Week 13

Tutorial 13: Heat & Wave Equations



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