(1) Solve the boundary value problem Δ u = uxx + uyy = 0 in...

(1) Solve the boundary value problem $\Delta u = u_{xx} + u_{yy} = 0$ in $(0,1) \times (0,1)$, $u_x(0,y) = u_x(1,y) = 0$ for $y \in (0,1)$, $u(x,0) = x, u(x,1) = 0$ for $x \in (0,1)$.

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Transcript

00:01 Okay, this is a boundary value problem.

00:04 So the thing about our original drawing is that it makes more sense if we try to shift the origin to the middle of the ring.

00:18 The reason we want to do that is because the boundary conditions written in the original coordinates are just awful.

00:27 And so we want to make the boundary conditions simpler.

00:31 So i'm going to redefine coordinates.

00:34 So x and y are the original coordinates.

00:37 I'm redefining them to x prime and y prime.

00:40 Because with x prime and y prime, the origin is going to be at the center of the ring.

00:47 Okay.

00:48 And when i take the derivatives and calculate how the partial derivatives transform, they look like this.

00:56 And then my laplocyan and the new coordinates looks like this.

01:00 And cartesian coordinates.

01:04 And then i'm going to change that to cylindrical coordinates, so that row squared is x prime squared plus y prime squared, and theta is the inverse tangent of y over x.

01:17 And my new boundary conditions are that phi equals zero when row equals one, and phi equals 10 when row equals two.

01:28 Because of symmetry, my potential function can't depend on, on angle because the whole thing doesn't depend on angle at all...

Question

(1) Solve the boundary value problem $\Delta u = u_{xx} + u_{yy} = 0$ in $(0,1) \times (0,1)$, $u_x(0,y) = u_x(1,y) = 0$ for $y \in (0,1)$, $u(x,0) = x, u(x,1) = 0$ for $x \in (0,1)$.

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