1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: PDE, heat equation with mixed boundary conditions

  1. Nov 4, 2013 #1
    1. The problem statement, all variables and given/known data
    solve the heat equation over the interval [0,1] with the following initial data and mixed boundary conditions.

    2. Relevant equations
    [itex]\partial _{t}u=2\partial _{x}^{2}u[/itex]
    [itex]u(0,t)=0, \frac{\partial u}{\partial x}(1,t)=0[/itex]

    with B.C

    where f is piecewise with values:
    [itex]0, 0<x\leq \frac{1}{2}[/itex]
    [itex]3, \frac{1}{2}<x<1[/itex]

    3. The attempt at a solution

    after separation of variables where [itex]u(x,t)=h(x)\phi (t)[/itex]:

    [itex]h''(x)=-\frac{\lambda }{2}h(x)[/itex]
    [itex]\phi'(t)=-\lambda \phi(t)[/itex]

    gen. solution to h is
    [itex]h(x)=a\sin \sqrt{\frac{\lambda }{2}}[/itex] the constant with the cos term is 0 from initial value

    I'm stuck trying to find the eigenvalue
    [itex]h'(1)=\frac{\lambda }{2}a\cos\sqrt{\frac{\lambda }{2}}=0[/itex]
    [itex]\sqrt\frac{\lambda }{2}=\arccos 0[/itex]

    I'm not sure what expression with n to use for arccos of 0. npi/2 won't work, or (n+1)pi/2, is this the right procedure though?

    Any help is greatly appreciated!


    I'm trying [itex]\frac{\pi }{2}+n\pi[/itex] now to solve for the eigenvalue
  2. jcsd
  3. Nov 4, 2013 #2
    So for the general solution of u I have [itex]u(x,t)=\sum_{n=1}^{\infty}A_{n}\sin [\frac{\pi}{2}(1+2n)x]\exp -2t[\frac{\pi}{2}(1+2n)]^2[/itex]

    and the coefficient A_n given by

    [itex]A_{n}=\frac{12}{\pi(1+2n)}\cos \pi(1+2n)[/itex]

    There was another cosine term with pi/2 in the argument that was always zero for any n, but is this an acceptable way to leave the coefficient expression?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted