1. Not finding help here? Sign up for a free 30min 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!

Separation of variables and generalised fourier series

  1. Mar 20, 2007 #1
    1. The problem statement, all variables and given/known data

    if [tex]\nabla^2u = 0[/tex] in [tex] 0 \leq x \leq \pi, 0\leq y \leq \pi, [/tex]

    boundary conditions u(0,y)=0, [tex] u(\pi,y)=cos^2y, u_y(x,0) = u_y(x,\pi)=0 [/tex]


    2. Relevant equations

    I am required to show that [tex] u(x,y) = \frac{x}{2\pi} + \frac{cos2ysinh2x}{2sinh2\pi} [/tex]

    3. The attempt at a solution


    I have done similar questions before for example consider then following problem:

    [tex]\nabla^2u = 0[/tex] in [tex] 0 \leq x \leq a, 0\leq y \leq b, [/tex]

    but with boundary conditions:

    u(0,y)=0, u(a,y)=0, u(x,0)=0, u(x,b)=f(x)


    I derived the general solution to be [tex] u(x,y) = \sum_{n=0}^\infty D_nsin\frac{n\pix}{a}sinh\frac{n\piy}{a} [/tex]

    with [tex] D_n =\frac{2}{asinh\frac{n\pib}{a}}\int_{0}^{a}f(x)sin\frac{n\pix}{a}dx [/tex] n=1,2,3....



    .....This example I understand but in the first example I am confused so far I have done the following,

    let u(x,y) =X(x)Y(y)

    X(x) =Acospx + Bsinpx
    Y(y) =Ccoshpy + Dsinhpy
    then Y'(y) =Esinhpy + Fcoshpy

    B.C's ->

    X(0) =A=0

    [tex] X(\pi)= Bsinp\pi=cos^2y [/tex]


    [tex]Y'(0) = F=0, Y'(\pi)=0

    so Esinhp\pi=0 [/tex] this seems to tell me nothing about E I have tried also type three solutions viz.

    X(x)=Acoshpx + Bsinhpx

    Y(y)= Ccospy + D sinpy
    Y'(y)=Ecospy-Fsinpy

    X(0)=A=0
    [tex] X(\pi)= Bsinhp\pi =cos^2(y) [/tex]

    Y'(0)=E=0

    [tex] Y'(pi)= -Fsinppi=0 [/tex]

    when p = 1,2,3

    so u(x,y) = [tex] \sum_{p=0}^\infty F_nsinpxsinhpy [/tex]

    which is markedly different to the form of solution required, what am I doing wrong??













    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Mar 20, 2007
  2. jcsd
  3. Mar 20, 2007 #2

    cristo

    User Avatar
    Staff Emeritus
    Science Advisor

    If you only need show that the expression you give is the solution, can you not just show it satisfies the equation and that it satisfies the boundary conditions?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Separation of variables and generalised fourier series
Loading...