Laplace's equation: △u =1r∂u∂r+∂2u∂r2+1r2∂2u∂ϕ2

[Caracalla]

Laplace's equation: △u =1r∂u∂r+∂2u∂r2+1r2∂2u∂ϕ2

Hey guys I need some help with this particular task. I have to discretize the Laplace's equation: △u=0 between two circles. They both have a center in (0, 0), circle 1 (I will further refer it to as a C1) has a radius of 2, circle 2 (I will further refer it to as a C2) has a radius of 1. I am solving this using polar coordinates, where △u

[FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]u[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]u[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]1[/FONT][FONT=MathJax_Math]r[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]u[/FONT][FONT=MathJax_Main]∂[/FONT][FONT=MathJax_Math]ϕ[/FONT][FONT=MathJax_Main]2[/FONT]​

Now I have to discretize the field (reffered as D) in polar coordinates so I get a system of linear equations in order to get approximate solutions matrixes and MatLab. Any help or explanation of the problem that I admittedly do not understand completely will be really helpful.

 

[HallsofIvy]

You could, at the very least, have put "/" to indicate fractions and derivatives: \(\displaystyle \left(1/r\right)\partial u/\partial r+ \partial^2 u/\partial^2r+ \left(1/r^2\right)\partial^2u/\partial \phi^2= 0\) or since you are using Latex, used actual fractions: \(\displaystyle \left(\frac{1}{r}\right)\frac{\partial u}{\partial r}+ \frac{\partial^2 u}{\partial r^2}+ \left(\frac{1}{r^2}\right)\frac{\partial^2 u}{\partial \phi^2}= 0\).

In any case, to "discretize" this, divide the annulus between r= 1 and r= 2 into n sub-rings with \(\displaystyle r_i= 1+\frac{i}{n}\) with i going from 0 to n and each sector divided into m sub-sectors with \(\displaystyle \theta_j= \frac{2\pi j}{m}\). For each region, take "r" to be the distance from the origin to some point inside that region, take \(\displaystyle u_{ij}\) to be the value of u at that point, take \(\displaystyle \frac{\partial u}{\partial r}\) to be \(\displaystyle \frac{u_{(i+1)j}- u_{ij}}{n}\), take \(\displaystyle \frac{\partial^2 u}{\partial r^2}\) to be \(\displaystyle \frac{u_{(i+1)j}- 2u_{ij}+ u_{(i-1)j}}{n^2}\) and take \(\displaystyle \frac{\partial^2u}{\partial\theta^2}\) to be \(\displaystyle \frac{u_{i(j+1)}- 2u_{ij}+ u_{i(j-1)}}{m^2}\).