course | course_year | question_number | tags | title | year | |||
---|---|---|---|---|---|---|---|---|
Methods |
IB |
53 |
|
Paper 4, Section II, D |
2012 |
Let
$$G_{2}=G_{2}\left(\mathbf{r}, \mathbf{r}{0}\right)=\frac{1}{2 \pi} \log \left|\mathbf{r}-\mathbf{r}{0}\right|$$
where
derive Green's third identity
$$u\left(\mathbf{r}{0}\right)=\int{D} G_{2} \nabla^{2} u d a+\int_{S}\left(u \frac{\partial G_{2}}{\partial n}-G_{2} \frac{\partial u}{\partial n}\right) d \ell$$
[Here
Consider the Dirichlet problem on the unit
Show that, with an appropriate function
$$u\left(\mathbf{r}{0}\right)=\int{S_{1}} f(\mathbf{r}) \frac{\partial}{\partial n} G\left(\mathbf{r}, \mathbf{r}_{0}\right) d \ell$$
State the boundary conditions on
For
$$\left|\frac{\mathbf{r}}{|\mathbf{r}|}-\mathbf{r}{0}\right| \mathbf{r}||=\left|\frac{\mathbf{r}{0}}{\left|\mathbf{r}{0}\right|}-\mathbf{r}\right| \mathbf{r}{0}|| \text {, }$$
and deduce that if the point
$$\left|\mathbf{r}-\mathbf{r}{0}\right|=\left|\mathbf{r}{0}\right|\left|\mathbf{r}-\mathbf{r}{0}^{*}\right|, \text { where } \mathbf{r}{0}^{*}=\frac{\mathbf{r}{0}}{\left|\mathbf{r}{0}\right|^{2}}$$
Hence, using the method of images, or otherwise, find an expression for the function