UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 5

Teori för differentialekvationer 1998 02 01

5.1. Determine all characteristic curves to the equation

\frac{∂^{2}u}{∂x^{2}} − y^{2} \frac{∂^{2}u}{∂y^{2}} = 0.

Transform the equation to normal form in the open set y ≠ 0. 5.2. Determine the characteristic curves to the equation

\frac{∂^{2}u}{∂x^{2}} − 9x^{4} \frac{∂^{2}u}{∂y^{2}} − 6xu = 0.

Transform the equation to normal form in the domain x > 0. 5.3. Determine the characteristic curves to the equation

x^{2} \frac{∂^{2}u}{∂x^{2}} − 2x\frac{∂^{2}u}{∂x∂y} + \frac{3}{4} \frac{∂^{2}u}{∂y^{2}} + \frac{1}{2} \frac{∂u}{∂y} = 0.

Transform the equation to normal form in all of R^{2}. Find the general solution. 5.4. Solve the differential equation

\frac{∂^{2}u}{∂t^{2}} = \frac{∂^{2}u}{∂x^{2}} + \frac{∂u}{∂x} + \frac{∂u}{∂t}

with the initial-value conditions u(x, 0) = x, u_{t}(x, 0) = 0. Hint: Experiment with exponential functions multiplied by solutions to the ordinary wave equation.

5.5. Solve the equation

\frac{∂^{2}u}{∂x^{2}} − \frac{∂^{2}u}{∂t^{2}} = 0

in the domain t > 0, x > 0 with the boundary conditions

u(x, 0) = x^{2},  \frac{∂u}{∂t}(x, 0) = 0,  u(0, t) = 0.

Hint: Think about even and odd functions. 5.6. Solve the initial-value problem

\frac{∂^{2}u}{∂t^{2}} = c^{2}(\frac{∂^{2}u}{∂x_{1}^{2}} + \frac{∂^{2}u}{∂x_{2}^{2}} + \frac{∂^{2}u}{∂x_{3}^{2}})

with the conditions

u(x, 0) = 0,
\frac{∂u}{∂t}(x, 0) = \{ \begin{matrix}1, |x| ⩽ a \\ 0, |x| > a. \end{matrix}

Discuss the behavior of the solution at the point (0, a/c); it is not continuous there, a fact which depends on the discontinuities in the initial conditions on the sphere t = 0, |x| = a. The phenomenon is called the focusing effect. Hint: Show first that u is a function of (|x|, t) = (r, t) and that v = ru solves the wave equation in the variables (r, t). (This is special for three space variables.)