Math Problems and solutions
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11.5.2.31 Fourier method
7.43 $
11.5.2.32 Fourier method
5.45 $
11.5.2.33 Fourier method
8.67 $
11.5.2.34 Fourier method
6.19 $
11.5.2.35 Fourier method
11.5.2.36 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 \( \theta(\varphi)=\sin 2 \varphi \).](/storage/uploads/task_mls/1188/en/11.5.2.33.png){kind=link}
![Problem: Find the functions \( u(\rho, \varphi) \), that are harmonic outside the circle of radius \( \rho=a \) and satisfying the boundary condition (first exterior boundary value problem for a circle): \[ \left.u\right|_{\rho=a}=A \text {. } \]](/storage/uploads/task_mls/1189/en/11.5.2.34.png){kind=link}
![Problem: Find the solution of the exterior boundary value problem for the Laplace equation, if on the boundary of the circle the conditions are given: \[ \left.u\right|_{\rho-a}=A \sin ^{3} \varphi+B \text {. } \]](/storage/uploads/task_mls/1190/en/11.5.2.35.png){kind=link}
![Problem: Find the functions \( u(r, \varphi) \), harmonic outside the circle radius \( r=a \) and satisfying the boundary condition (the first exterior boundary-value for the circle): \[ \left.\frac{\partial u}{\partial r}\right|_{r=a}=A \cos 3 \varphi \text {. } \]](/storage/uploads/task_mls/1191/en/11.5.2.36.png){kind=link}