# Thread: PDE trouble - I'm a little rusty

1. ## PDE trouble - I'm a little rusty

Hello,
Its been a long time since I've done PDE's or even ODE's for that matter, and I'm quite rusty. Could anyone please help me with the following?

Solve the following PDE and explain why inverting the equations for $x(s, t)$ and $y(s,t)$ may cause problems:

$u_x + u_y = u$ subject to $u(x, ax) = F(x)$

Find the Jacobian of the coordinate transform to determine when the transformation is invertible.

2. If we let

$u_s = u_x x_s + u_y y_s$

then choosing

$x_s = 1, y_s = 1\;\;(1)$ gives $u_s = u_x + u_y = u \;\;(2)$ (your PDE)

The boundary conditions. If we let $y = ax$ be $s = 0$ and $x = r$ then we have

On $s = 0, x = r, y = ar$ and $u = F(r)\;\;(3)$

Integrating (1) and (2) gives

$x = s + A(r),\;\;\;y = s + B(r),\;\;\;u = C(r)e^s$

where $A, B$ and $C$ are arbitrary. Imposing the BCs (3) gives

$x = s + r,\;\;\; y = s + ar,\;\;\;u=F(r)e^s,\;\;(4)$

noting Jacobian of the transformation for $x$ and $y$ vanishes if $a = 1.$. Now solving the first two of (4) for r and s and substituting into the third of (4) gives your solution.