Initial & Boundary Value Problem-Fourier

[mathmari]

Hey!

We have the initial and boundary value problem $$u_{xx}(x,y)+u_{yy}(x,y)=0 , x^2+y^2<1 \\ u(x,y)=0 \\ u(1, \theta)=\sin{\theta}, 0< \theta< \pi$$

$$U_{\rho \rho}(\rho, \theta)+ \frac{1}{\rho^2} U_{\theta \theta}=0, 0< \rho<1, 0< \theta< \pi$$

We are looking for solutions of the form $R(\rho) \Theta(\theta)$

$R(\rho) \Theta(0)=0 \Rightarrow \Theta(0)=0\\ R(\rho) \Theta(\pi)=0 \Rightarrow \Theta(\pi)=0$

$R''(\rho) \Theta(\theta)+ \frac{1}{\rho^2} R(\rho) \Theta''(\theta)=0\\ \frac{\Theta''(\theta)}{\Theta(\theta)}=-\rho^2 \frac{R''(\rho)}{R(\rho)}=-\lambda $

$\Theta''(\theta)+ \lambda \Theta(\theta)=0 \\ \Theta(0)=\Theta(\pi)=0$$\rho^2 R''(\rho)-\lambda R(\rho)=0$

$\lambda_k=k^2$

$\Theta_k(\theta)=\sin{k \theta}$

$\rho^2 R_k''(\rho)-k^2 R_k(\rho)=0$

We are looking for a solution of the form $R(\rho)=\rho^m \Rightarrow m(m-1) \rho^m-k^2 \rho^m=0$

$m^2-m-k^2=0$

$\Delta=1+4k^2$

$m=\frac{1 \pm \sqrt{1+4k^2}}{2}$

For $m>0$, the solution is unbounded.

$R_k(\rho)=\rho^{\frac{1+ \sqrt{1+4k^2}}{2}}$

$U_k(\rho, \theta)=\rho^{\frac{1+ \sqrt{1+4k^2}}{2}} \sin{k \theta}$

$U(\rho, \theta)=\sum c_k $We determine $c_k$$U(1, \theta)=\sum_{k=1}^{\theta} c_k \sin{k \theta}=\sin{\theta}$

$c_1=1\\c_k=0, k=2, \dots$

$$$$

Why is the problem in polar coordinates $$U_{\rho \rho}+\frac{1}{\rho^2}U_{\theta \theta}=0$$ and not $$U_{\rho \rho}+\frac{1}{\rho}U_{\rho}+\frac{1}{\rho^2}U_{\rho \rho}=0$$ ??

 

[jvicens]

Hello there!

The reason the problem is in the form $U_{\rho \rho}+\frac{1}{\rho^2}U_{\theta \theta}=0$ is because we are dealing with a Laplace equation in polar coordinates. In polar coordinates, the distance from the origin, $\rho$, is a key variable and is used to define the geometry of the problem. The term $\frac{1}{\rho^2}$ in the equation is known as the "scale factor" and is necessary to take into account the changing distance from the origin.

If we were to use the form $U_{\rho \rho}+\frac{1}{\rho}U_{\rho}+\frac{1}{\rho^2}U_{\rho \rho}=0$, we would be incorporating an extra term that does not accurately represent the geometry of the problem. This would lead to incorrect solutions and would not accurately describe the physical system.

I hope this helps clarify the use of polar coordinates in this problem. Let me know if you have any further questions.