MathProblemsBank

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 \( \theta(\varphi)=\sin 2 \varphi \).