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Partial Differential Equation using separation of variables

  1. Jun 10, 2009 #1
    1. The problem statement, all variables and given/known data
    Solve the heat flow problem using the method of separation of variables:


    2. Relevant equations
    PDE:[tex]\frac{\partial u}{\partial t}=k\frac{\partial^{2} u}{\partial t^{2}}[/tex]
    for 0<x<L, 0<t<[tex]\infty[/tex]

    BC's:[tex]\frac{\partial u}{\partial x}(0,t)=0,\frac{\partial u}{\partial x}(L,t)=0[/tex]
    for 0<t<[tex]\infty[/tex]

    IC's: [tex]u(x,0)=[/tex]
    {0, [tex]0<x<L/4[/tex]
    {1, [tex]L/4<x<3L/4[/tex]
    {0, [tex]3L/4<x<L[/tex]
    (Piecewise IC)

    3. The attempt at a solution
    I have separated the variables, then applied the boundary conditions. I am stuck on applying the initial conditions.

    I have come up with a general product solution of [tex]u_{n}=F_{n}cos(\frac{n \pi x}{L}) e^{-k(\frac{n \pi x}{L})^{2}t}[/tex]

    Trying to combine all product solutions and match the initial data:
    [tex]u(x,0)=f(x)[/tex]
    [tex]\sum^{\infty}_{n=1}u_{n}(x,0)=f(x)[/tex]
    [tex]\sum^{\infty}_{n=1}F_{n}cos(\frac{n \pi x}{L})=f(x)[/tex]


    I don't know how to apply the piecewise initial condition, any help would be appreciated. Thank you
     
  2. jcsd
  3. Jun 10, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    Once you have
    [tex]\sum^{\infty}_{n=1}F_{n}cos(\frac{n \pi x}{L})=f(x)[/tex]
    you do it exactly the same way you would if f were not "piecewise". Find the Fourier cosine coefficients by doing the appropriate integrals for C0 and Cn for n> 0. The only difference "piecewise" makes is that instead of integrating a single formula from 0 to L, you integrate using the given formulas from 0 to L/4, L/4 to 3L/4, 3L/4 to L and adding those integrals. (Which I notice now is just integrating from L/4 to 3L/4 since outside that the function is 0. The difference between this and just doing the problem on the interval from L/4 to 3L/4 is that you use the whole interval, of length L in determining the "normalization" constant.)
     
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