Dirichlet Problem on a Disc - Transfomation

[lkh1986]

Homework Statement

This is a problem I found in my note. It says that from u xx(subscript) + u yy = 0, we can arrive at u rr + (1/r) u r + (1/r^2) u theta theta = 0 by using the transformation x = r cos theta, y = r sin theta, r = square root of (x^2+y^2), theta = arctan (y/x).

The problem is how to prove it?

Homework Equations

The strategy is to express u xx and u yy in terms of u rr , u r theta and u theta theta.

The Attempt at a Solution

I can differentiate r with respect to x an y. Also, I have found the derivative of theta with respect to x and y.

I have no problem of finding u x and u y. The problems arise when I try to find u xx and u yy. ;(

I use the search engine to search for the proof but can't find it anywhere. Does someone have the link for this proof? Thanks.

 

[mighty2000]

Hello,

The proof for this transformation can be found in many textbooks on partial differential equations or mathematical methods in physics. Here is a brief explanation of the proof:

Starting with the given equation u_xx + u_yy = 0, we can use the chain rule to express u_xx and u_yy in terms of u_rr, u_rtheta, and u_thetatheta.

For u_xx, we have:

u_xx = (d^2u/dx^2) = (d^2u/drdx)(dxdx) + (d^2u/dthetadx)(dx/dtheta)

= (d^2u/drdx)(cos(theta)) + (d^2u/dthetadx)(-r sin(theta))

= cos^2(theta) u_rr + cos(theta)(-r sin(theta)) u_rtheta

= (1/r^2) u_rr + (-1/r) u_rtheta

Similarly, for u_yy, we have:

u_yy = (d^2u/dy^2) = (d^2u/drdy)(dydx) + (d^2u/dthetady)(dy/dtheta)

= (d^2u/drdy)(sin(theta)) + (d^2u/dthetady)(r cos(theta))

= sin^2(theta) u_rr + sin(theta)(r cos(theta)) u_rtheta

= (1/r^2) u_rr + (1/r) u_rtheta

Substituting these expressions for u_xx and u_yy into the original equation, we get:

(1/r^2) u_rr + (-1/r) u_rtheta + (1/r^2) u_rr + (1/r) u_rtheta = 0

Simplifying, we get:

u_rr + (1/r) u_r + (1/r^2) u_thetatheta = 0

which is the desired result.

I hope this explanation helps. Good luck with your studies!