Partial Differential Equations

Assignment 1

1. [5 points] Find and sketch some sample characteristic curves of the PDE:

   $$(x + 2)u_x + 2y u_y = 2u$$
   in the$x-y$plane. Write the ODE for$u$along a characteristic curve with$x$as the parameter and then solve the PDE with the initial condition$u(-1, y) = \sqrt{|y|}, y>0$.

2. [15 points] Consider the PDE:

   $$x u_x + y u_y = 2u, x > 0, y > 0.$$
   Plot the characteristic curves and solve the equation with the following initial conditions in the domain given above:

   1. $u = 1$on the hyperbola$xy = 1$.

   2. $u = 1$on the circle$x^2 + y^2 = 1$.

   3. Can you solve the equation, in general, if certain initial data is prescribed on the initial curve$y = e^x$? Justify with reasons.

3. [20 points] Sketch the characteristic curve, the initial curve, and solve the following problems:

   1. $x u_x + y u_y = k u, \; x \in \mathbb{R}, \; y \geq \alpha > 0; \; u(x, \alpha) = F(x)$, where$k$,$\alpha$are fixed and$F$is a given smooth function.

   2. $y u_x - x u_y = 0; \; u(x, 0) = x^2$.

   3. $x^2 u_x - y^2 u_y = 0; \; u(1, y) = F(y)$.

   4. $y u_x + x u_y = 0; \; u(0, y) = e^{-y^2}$.

4. [10 points] Solve the quasi-linear problem and verify transversality conditions:

   1. $u u_x + u_y = 0, \; u(x, 0) = x$.
   2. $u u_x + u_y = 1, \; u(x, x) = x/2, \; x \in (0, 1]$.