Separation of variables and generalised fourier series

  • #1

Homework Statement



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]


Homework Equations



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

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??













Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 
Last edited:

Answers and Replies

  • #2
cristo
Staff Emeritus
Science Advisor
8,107
73
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?
 

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