- #1

lriuui0x0

- 101

- 25

- Homework Statement
- Solve the laplace equation ##u_{xx} + u_{yy} = 0##, with ##x \ge 0, y \ge 0##, using Fourier transform.

Subject to the boundary conditions:

$$

\begin{aligned}

u(x, 0) &= \begin{cases}1 & 0 < x < 1 \\ 0 & \text{otherwise}\end{cases} \\

u(0, y) &= \lim_{x\to\infty} u(x,y) = \lim_{y\to\infty} u(x,y) = 0

\end{aligned}

$$

- Relevant Equations
- Fourier transform and inverse transform

I have tried to Fourier transform in ##x## and get the result in the transformed coordinates, please check my result:

$$

\tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky}

$$

However, I'm having some problems with the inverse transform:

$$

\frac{1}{2\pi}\int_{-\infty}^\infty \frac{1-e^{-ik}}{ik}e^{-yk}dk

$$

Not sure how to do this integral. The solution says it's

$$

\frac{4xy}{\pi}\int_0^1 \frac{vdv}{[(x-v)^2+y^2][(x+v)^2+y^2]}

$$

$$

\tilde{u}(k, y) = \frac{1-e^{-ik}}{ik}e^{-ky}

$$

However, I'm having some problems with the inverse transform:

$$

\frac{1}{2\pi}\int_{-\infty}^\infty \frac{1-e^{-ik}}{ik}e^{-yk}dk

$$

Not sure how to do this integral. The solution says it's

$$

\frac{4xy}{\pi}\int_0^1 \frac{vdv}{[(x-v)^2+y^2][(x+v)^2+y^2]}

$$