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Help solving heat equation with Neumann Boundary Conditions with different domain

  1. Nov 16, 2011 #1
    Hi guys!

    I'm to find the solution to

    [tex]\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} [/tex]

    Subject to an initial condition

    [tex] u(x,0) = u_0(x) = a \exp(- \frac{x^2}{2c^2}) [/tex]

    And Neumann boundary conditions

    [tex] \frac{\partial u}{\partial x} (-1,t) = \frac{\partial u}{\partial x} (1,t) = 0 [/tex]

    I can usually do this no problem assuming the domain is for instance [0,L], but I get stuck with this one :

    Using separation of variables :

    [tex] u(x,t) = f(x)g(t) [/tex]

    This yields:

    [tex] \frac{1}{g} \frac{dg}{dt} = \frac{1}{f} \frac{d^2f}{dx^2} = -\lambda [/tex]


    Spatial Part:

    [tex] \frac{1}{f} \frac{d^2f}{dx^2} = -\lambda [/tex]

    [tex] \frac{d^2f}{dx^2} + \lambda f = 0 [/tex]

    Therefore :

    [tex] f(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x) [/tex]

    And since I'm considering Neumann Boundary conditions I get the derivative of this

    [tex] f \prime (x) = -A \sqrt{\lambda}\sin(\sqrt{\lambda} x) + B \sqrt{\lambda} \cos(\sqrt{\lambda} x) [/tex]


    So, [tex] f \prime (-1) = 0 [/tex]

    This gives:

    [tex] A \sin(\sqrt{\lambda}) + B \cos(\sqrt{\lambda}) = 0[/tex]

    And for [tex] f \prime (1) = 0 [/tex]

    I get :

    [tex] -A \sin(\sqrt{\lambda}) + B \cos(\sqrt{\lambda}) = 0[/tex]

    So from these two equations I can conclude that:

    Firstly by just adding the two equations

    [tex] B \cos(\sqrt{\lambda}) = 0 [/tex]

    So either [itex] B = 0 [/itex] or [itex] \cos(\sqrt{\lambda}) = 0 [/itex]

    Now substituting [itex] B \cos(\sqrt{\lambda}) = 0 [/itex] back into [itex] A \sin(\sqrt{\lambda}) + B \cos(\sqrt{\lambda}) = 0[/itex]

    I also get
    [tex] A \sin(\sqrt{\lambda}) = 0 [/tex]

    So either [itex] A = 0 [/itex] or [itex] \sin(\sqrt{\lambda}) = 0 [/itex]

    Obviously [itex] \sin(\sqrt{\lambda}) [/itex] and [itex] \cos(\sqrt{\lambda}) [/itex] can't both equal zero, so how do I approach this...

    Apologies if this is a stupid question..
    Any help would be greatly appreciated
    Max
     
  2. jcsd
  3. Nov 20, 2011 #2

    hunt_mat

    User Avatar
    Homework Helper

    Have you tried a transform method? I might be tempted to try a Laplace transform on the t variable. I think that this problem requires a transform solution rather than a separation of variable solution.
     
  4. Nov 21, 2011 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    [itex]sin(\sqrt{\lambda})[/itex] and [itex]cos(\sqrt{\lambda})[/itex] don't both have to be 0: one of A and B can be 0 and still give you a non-trivial solution.
     
  5. Nov 24, 2011 #4
    Ah ok I see! Thanks for your help.

    So I can choose for example in case 1: A=0 , B=B . case2: B=0 , A=A
     
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