16 October 2007

Dan Lev

Math Interview with Phil Holmes

I strongly recommend taking Holmes as an interviewer. He sees the interview as more than just a way to examine the student but also as a way to strengthen the student’s knowledge on topics where the student is weak. He asked me what I wanted him to ask me about and I asked to be mainly examined on non-homogenous PDEs and Green’s function in 2-D. I was weak in both.
The whole interview took 3 hours.

  • Write the heat equation.

    What is the meaning of ?
    It is the rate in which the heat decays. He really wanted to hear the word diffusion.
    What are the dimensions of ?

    What will happen to this equation when ?
    It will dissipate to the boundary conditions.
    Solve the non-homogenous boundary problem

    You have to separate the solution to a steady state and transient solutions

    will take care of the non-homogenous boundary conditions so that will have homogenous boundary conditions. Be aware to the fact that the initial conditions change. .
    How can you find the coefficients in the final solution ?
    Use the property of orthogonality much like when finding the coefficients of the Fourier series.
    Can we reverse time ? (look at negative values of t)
    Physically it doesn’t have meaning but when time is reversed the solution will tend to blow up. The problem will no longer be well posed since the boundary conditions will be insignificant in comparison with the huge values that the solution will take. It’s in the notes.
    Can we reverse time for the wave equation ? (look at negative values of t)
    It will not change anything in the equation since we differentiate time twice so we will multiply by a minus sign twice.
    Lets say we have a “hot spot” on a one dimensional rod, where would the spot move ? (see exam for MAE502 2007)
    It will move to the left since for lower the n the more weight the function has. For full explanation ask me.
  • Solve the non-homogenous heat equation by using Green’s function

    The process is long and is in the notes. To be short, first prove the identity and apply the divergence theorem. Also explain the divergence theorem.
    Assign and write the condition .
    Plug it in and obtain an expression for that includes Green’s function.
    Expression contains derivatives of both and so you can decide on two boundary conditions for .
    You have to explain how you chose them and how boundary conditions on (Dirichle and Newman) affected your choice.
    Then solve the homogenous heat equation to find .
    This step led to Laplace equation.
  • What will the solution look like ?
    He referred to Bessel’s equation. I drew the functions qualitatively because no sane person remembers the actual function form.
  • Solve the Laplace equation on a disk
    Write down the PDE in polar coordinates (I didn’t remember it) and use separation of variables.
    Let’s say you get to an equation that looks like this: what is its name and how would you solve it ?
    This is the Euler equation and it is solved by assuming solutions of the form
    The problem with this equation is that you get with a multiplicity of 2.
    The solution for this one is either a constant () or .
  • Sturm Liouville operator
    Write it down. Write down the boundary conditions.
    What are the conditions on ?
    The first and the last are positive. All of them are real valued.
    Show that it is self-adjoint
    Like in the notes
    What can you say on the eigenvalues of the operator ?
    The are infinite and blow up
    According to the spectral theorem the eigenvalues have to converge to zero. Why doesn’t it happen here ?
    The spectral theorem does not apply for this operator since it isn’t compact.
    It is important to mention that the operator still have an infinite orthogonal set of eigenfunctions.
    Prove that it is not a compactoperator
    The differentiation operator is not compact since you can take a bounded function such as the Heavyside function and see that it maps to an unbounded function – the delta function.
  • Inner product
    Give a property of an inner product
    Can an inner product induce a norm ?
    yes,
    Write an example of a weighted inner product

    What are the conditions for ?
    It has to be positive
  • Liner operator
    What is the definition of a linear operator ?

  • Some basic questions about eigenvalues for matrices and algebraic/geometric multiplicity
    Diagonalize a matrix A. What if A is defected ?
    . If A is defected then we can use Jordan’s form (I didn’t have to prove it).