# Separation of variables and generalised fourier series

• catcherintherye
In summary, the problem requires showing that u(x,y) = \frac{x}{2\pi} + \frac{cos2ysinh2x}{2sinh2\pi} is the solution to the given equation with the specified boundary conditions. The attempt at a solution involves using a general solution derived from a similar problem, but the solution does not match the required form. It is suggested to show that the given expression satisfies the equation and the boundary conditions to prove that it is the solution.

## Homework Statement

if $$\nabla^2u = 0$$ in $$0 \leq x \leq \pi, 0\leq y \leq \pi,$$

boundary conditions u(0,y)=0, $$u(\pi,y)=cos^2y, u_y(x,0) = u_y(x,\pi)=0$$

## Homework Equations

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

## The Attempt at a Solution

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

$$\nabla^2u = 0$$ in $$0 \leq x \leq a, 0\leq y \leq b,$$

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 $$u(x,y) = \sum_{n=0}^\infty D_nsin\frac{n\pix}{a}sinh\frac{n\piy}{a}$$

with $$D_n =\frac{2}{asinh\frac{n\pib}{a}}\int_{0}^{a}f(x)sin\frac{n\pix}{a}dx$$ 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

$$X(\pi)= Bsinp\pi=cos^2y$$

$$Y'(0) = F=0, Y'(\pi)=0 so Esinhp\pi=0$$ 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
$$X(\pi)= Bsinhp\pi =cos^2(y)$$

Y'(0)=E=0

$$Y'(pi)= -Fsinppi=0$$

when p = 1,2,3

so u(x,y) = $$\sum_{p=0}^\infty F_nsinpxsinhpy$$

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

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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?