course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Methods |
IB |
49 |
|
Paper 3, Section I, C |
2013 |
The solution to the Dirichlet problem on the half-space
is given by the formula
$$u\left(\mathbf{x}{0}\right)=u\left(x{0}, y_{0}, z_{0}\right)=\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} h(x, y) \frac{\partial}{\partial n} G\left(\mathbf{x}, \mathbf{x}_{0}\right) d x d y$$
where
State the boundary conditions on
$$G_{3}\left(\mathbf{x}, \mathbf{x}{0}\right)=-\frac{1}{4 \pi} \frac{1}{\left|\mathbf{x}-\mathbf{x}{0}\right|}$$
is the fundamental solution to the Laplace equation in three dimensions.
Using the method of images find an explicit expression for the function