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Partial Differential Equation with initial conditions

  1. Jun 25, 2014 #1
    Hello! This is my first post to this excellent forum! I would like some help with this exercise:

    [itex] u_{xx} (x,y) + u_{yy} (x,y) = 0[/itex], with [itex] 0 < x < 2 \pi [/itex], [itex] 0 < y < 4 \pi [/itex]
    [itex] u_x (0,y) = 0, \, u_x(2 \pi, y) = 0, \, 0< y < 4 \pi [/itex]
    [itex] u(x,0) = a \cos(2x), \, u(x, 4 \pi) = a \cos^3(x), \, 0<x<2\pi[/itex]

    I think that the first step is to set [itex] u(x,y) = X(x) Y(y) [/itex], from which the first equation becomes [itex] X''(x) Y(y) + X(x) Y''(y) = 0 [/itex]. And by dividing with [itex] X(x) Y(y) [/itex] with obtain [itex] \frac{X''(x)}{X(x)} = \frac{Y''(y)}{Y(y)} = - \lambda [/itex]

    Now, be obtain two ordinary PDEs, [itex] X''(x) + \lambda X(x) = 0 [/itex], and [itex] Y''(y) - \lambda Y(y) = 0 [/itex].

    I don't know how to continue from now one. Especially what to do with the non-homogeneous initial conditions.

    Thank you very much in advance!
  2. jcsd
  3. Jun 25, 2014 #2


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    Without analyzing the equations in detail, consider:

    (a) λ is a constant (parameter) shared by both equations;
    (b) the ODE's can be solved in the "ordinary" way; don't stop until you have the general solutions for both!
    (c) the boundary conditions will not be applied until you have found the general solutions; you may end up having to "patch together" pieces drawn from the two sets of general solutions which are continuous where the patches join.
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