# PDE - Review 1

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

1. Consider the first order equation $$u_x-u_t=2x-t$$.

1. Verify that $$u(x,t)=f(2x+t)-x t$$ solves the equation sufficiently smooth function $$f$$.

2. Use part (a) to solve the initial value problem $$u_x-u_t=2x-t$$ subject to $$u(x,0)=e^{2x}$$.

2. Express the first order equation $$u_x-u_y=u+x y$$ in terms of the new variables $$\xi =2x+y$$ and $$\psi =x-2y$$.

3. 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 $$c=1$$.

4. 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 $$t=10^{10}$$.

5. Consider the equation $$2u_{x x}+u_{x y}+2u_{y y}+4u_xu_y+2u=2$$.

1. Is this equation linear or non-linear?

2. Is this equation quasi-linear?

3. Is this equation parabolic, hyperbolic, or elliptic?

6. Solve the heat problem $$u_{t} = 2u_{xx}$$ on $$\mathbb R$$ subject to $u(x,0) = \begin{cases} 1-x^2 & |x| < 1 \\ 0 & |x| > 1. \end{cases}$

7. 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 $$t$$.