PDE - Review 1

Here are some problems to play with for the exam on Wednesday.

Consider the first order equation $u_{x} - u_{t} = 2 x - t$ .
1. Verify that $u (x, t) = f (2 x + t) - x t$ solves the equation sufficiently smooth function $f$ .
2. Use part (a) to solve the initial value problem $u_{x} - u_{t} = 2 x - t$ subject to $u (x, 0) = e^{2 x}$ .
Express the first order equation $u_{x} - u_{y} = u + x y$ in terms of the new variables $ξ = 2 x + y$ and $ψ = x - 2 y$ .
A string of length one is held tightly at either end, making a straight line from $x = 0$ to $x = 1$ . At time $t = 0$ , we slap the string imparting an initial velocity of $v (x) = - \sin (x)$ at each point $x$ along the string. Write down a PDE description of the subsequent motion of the string, assuming that the density and tightness yield a string constant of $c = 1$ .
The initial temperature of a bar of length one and diffusivity one is set to $u_{0} (x) = \sqrt{x}$ . The left end held temperature zero and the right end point is set to $t / (1 + t)$ .
1. Write down a PDE description of this problem.
2. Are the boundary conditions consistent with the initial condition?
3. Sketch the solution $u (x, t)$ as a function of $x$ for $t = 0$ , $t = 0.01$ , $t = 0.1$ , and $t = 10^{10}$ .
Consider the equation $2 u_{x x} + u_{x y} + 2 u_{y y} + 4 u_{x} u_{y} + 2 u = 2$ .
1. Is this equation linear or non-linear?
2. Is this equation quasi-linear?
3. Is this equation parabolic, hyperbolic, or elliptic?
Solve the heat problem $u_{t} = 2 u_{x x}$ on $R$ subject to $u (x, 0) = {\begin{cases} 1 - x^{2} & | x | < 1 \\ 0 & | x | > 1. \end{cases}$
We consider a heat conduction problem on the rectangle $- 2 \leq x \leq 2$ , $- 1 \leq y \leq 1$ . The initial temperature distribution is identically zero throughout. At time $t = 0$ , the temperature of the vertices are instantly set to either $1$ or $- 1$ and the edges are either insulated or held at the constant temperature obtained via linear interpolation as indicated in Figure 1. Write down a PDE description of this problem.

Note: The shading in the figure illustrates the ultimate steady state temperature distribution. The problem is dynamic, however, so your solution will involve $t$ .

Figure 1: A 2D steady state temperature distribution