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Partial Differentiation Laplace Equation Question

  1. Nov 30, 2016 #1
    1. The problem statement, all variables and given/known data
    Consider the Laplace Equation of a semi-infinite strip such that 0<x< π and y>0, with the following boundary conditions:
    \begin{equation}
    \frac{\partial u}{\partial x} (0, y) = \frac{\partial u}{\partial x} (0,\pi) = 0
    \end{equation}

    \begin{equation}
    u(x,0) = cos(x) \text{for}\ 0<x<\pi
    \end{equation}

    \begin{equation}
    u(x,y) = \to 0 \ \text{as} \ y \to 0 \ \text{for}\ 0<x<\pi
    \end{equation}


    2. Relevant equations
    General Solution:
    \begin{equation}
    \begin{split}
    u(x,y) = \frac{a_0}{2L}(L-y) + \sum_{n = 1}^\infty \alpha_n sinh(\frac{n\pi}{L})(L-y)cos(\frac{n\pi x}{L}) \\
    \text{where} \ \alpha_n = \frac{a_n}{sinh(n\pi} \\
    \text{where} \ a_n = \frac{2}{L} \int_0^L cos(\frac{n\pi x}{L}) f(x) dx
    \end{split}
    \end{equation}

    3. The attempt at a solution
    I am asking if I setup the problem right so far.
    The visualization (i.e my picture of how I visualize the problem) The boundary conditions.
    diagram1.png

    My attempt so far:
    \begin{equation}
    u_{xx} + u_{yy} = 0
    \end{equation}

    Assume:
    \begin{equation}
    u = F(x)G(y)
    \end{equation}

    Thus:
    \begin{equation}
    \begin{split}
    u_{xx} = F''(x)G(y) \\
    u_{yy} = F(x)G''(y) \\
    F''(x)G(y) + F(x)G''(y) = 0 \\
    \end{split}
    \end{equation}
    From this, we can define the following:
    \begin{equation}
    \frac{F''(x)}{F(x)} = \frac{G''(y)}{G(y)} = K \ \text{some constant}
    \end{equation}
    If u(x,0) = cos(x)
    \begin{equation}
    F(x)G(0) = cos(x) \\
    \end{equation}
    Can I assume that G(y) is equal to sin(y)?
     
    Last edited: Nov 30, 2016
  2. jcsd
  3. Nov 30, 2016 #2

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    No: you said that ##u_x(0,y) = 0## and ##u_x(0,\pi) = 0##: I assume you want ##u_x(0,y) = u_x(\pi,y) = 0##. If so, your boundary conditions in the diagram are incorrect: you do not need to have ##u = 0## on the vertical sides; you need ##u_x = 0## on the sides.

    Furthermore, you say that ##u(x,0) = \cos x## for ##0 < x < \pi##, and you also say ##u(x,y) \to 0## as ##y \to 0##. These contradict each other; which do you mean?
     
  4. Nov 30, 2016 #3
    Yes, I want ##u_x (\pi,y) = 0,## sorry for the confusion. And also, I meant to write as ##y \to \infty##
     
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