Partial Differential Equations

Assignment 6

1. [5 points] Prove the characteristic parallelogram property for wave equations.

2. [5 points] Solve the problem with two different characteristic speeds$c_1$and$c_2$:

   $$\begin{equation*} \left(\frac{\partial}{\partial t} - c_1 \frac{\partial}{\partial x}\right)\left(\frac{\partial}{\partial t} - c_2 \frac{\partial}{\partial x}\right) u = 0, \quad \text{in } \mathbb{R} \times (0,\infty). \end{equation*}$$

   Analyze the cases$c_1 \neq c_2$and$c_1 = c_2$, and derive D’Alembert’s formula as a special case. Discuss any potential loss of regularity in the one-dimensional case.

3. [5 points] Integrate the wave equation$u_{tt} - u_{xx} = f(x,t)$in the characteristic triangle$P(x,t)$,$Q(x-ct,0)$,$R(x+ct,0)$to derive a formula for the solution. (Hint: Use the identity$u_{tt} - u_{xx} = (u_t)_t - (u_x)_x$.)

4. [5 points] Solve the wave equation in the first quadrant with non-homogeneous Dirichlet boundary condition:

   $$\begin{equation*} u_{tt} - u_{xx} = 0, \quad \text{in } (0,\infty) \times (0,\infty) \end{equation*}$$

   with initial and boundary conditions:

   $$\begin{equation*} u(x,0) = f(x), \quad u_t(x,0) = g(x), \quad u(0,t) = h(t), \quad t > 0. \end{equation*}$$

   Derive the formula for$x < ct$using the parallelogram identity.

5. [5 points] Solve the above equation with the Neumann non-homogeneous boundary condition where$u(0,t) = h(t)$is replaced by$u_x(0,t) = h(t)$.

6. [5 points] Let$c_1, \dots, c_k$be distinct positive real numbers. Show that the solution of the equation:

   $$\begin{equation*} (\partial_t^2 - c_1^2 \partial_x^2) \dots (\partial_t^2 - c_k^2 \partial_x^2) u = 0, \end{equation*}$$

   can be written as:

   $$\begin{equation*} u(x,t) = \sum_{j=0}^{k} u_j(x,t), \end{equation*}$$

   where each$u_j$satisfies$\partial_t^2 u_j - c_j^2 \partial_x^2 u_j = 0$. This result also holds in higher dimensions.

7. [5 points] Consider the case$n=3$for:

   $$\begin{equation*} (\partial_t^2 - c^2 \partial_x^2)(\partial_t^2 - c^2 \partial_x^2) u = 0, \quad c > 0. \end{equation*}$$

   Given smooth initial data$\partial_t^j u(x,0) = f_j(x)$for$j = 0,1,2,3$, explicitly determine the solution.

8. [5 points] Consider the wave equation:

   $$\begin{equation*} u_{tt} - \Delta u = 0, \quad \text{in } \mathbb{R}^n \times (0,\infty), \end{equation*}$$
   with initial data:
   $$\begin{equation*} u(x,0) = \phi(x), \quad u_t(x,0) = \psi(x). \end{equation*}$$
   Verify that the solution is given by:
   $$\begin{equation*} u(x,t) = u_{\phi}(x,t) + \int_0^t u_{\psi}(x,s)ds, \end{equation*}$$
   and also by:
   $$\begin{equation*} u(x,t) = v_{\psi}(x,t) + \frac{\partial}{\partial t} v_{\phi}(x,t). \end{equation*}$$