
11.5.2.33 Fourier method
Problem:
The Fourier method for the Laplace equation outside the circle.
- Find the representation of the solution of the twodimensional Laplace equation \( u_{x x}+u_{y y}=0 \) in the given region, in the form of a series, performing separation of variables in polar coordinates;
- sum the series, present the solution in the form of an integral;
- calculate the solution for the boundary values, shown in the table.
The region is outside the circle \( r \geq R \Rightarrow\left(r=\sqrt{x^{2}+y^{2}}\right) \Rightarrow \)
\[
\Rightarrow\left\{\begin{array}{l}
\Delta u=0, \quad r \geq R \\
\left.\left(u_{r}-u\right)\right|_{r=R}=\theta(\varphi)
\end{array}\right.
\]
the boundary conditon is \( \theta(\varphi)=\sin 2 \varphi \).