UPPSALA UNIVERSITET

Matematiska institutionen Christer Kiselman

ÖVNINGAR, Blad 7

Teori för differentialekvationer 1998 02 15

7.1. Rewrite the initial-value problem

u_{tt} = c^{2}u_{xx}; u(x, 0) = f(x), u_{t}(x, 0) = g(x) when x ∈ R,

as an initial-value problem for the vector-valued function (v_{1}, v_{2})^{T} = (u_{t}, u_{x})^{T}. Reduce it to the canonical form v_{t} + Bv_{x} = Cv + D with a diagonal matrix B.

7.2. Use the canonical form obtained in 7.1 to solve the mixed problem

u_{tt} = c^{2}u_{xx}, when x > 0, t > 0; u(x, 0) = f(x) and u_{t}(x, 0) = g(x) when x > 0,

u_{t}(0, t) + au_{x}(0, t) = h(t) when t > 0,

where a denotes a constant.

7.3. Consider the system

u_{t} + Bu_{x} = Cu + D,

where

u = ( \begin{matrix}u_{1} \\ u_{2} \end{matrix} ) and B = ( \begin{matrix}sin^{2} x sin t − sin x cos x & cos x sin x sin t + sin^{2} x \\ cos x sin x sin t − cos^{2} x & cos^{2} x sin t + cos x sin x \end{matrix} ) ,

and where C and D are matrices whose entries are smooth functions of x and t. Determine the eigenvalues for the system at every point (x, t) ∈ R^{2}. Determine for which points (x, t) ∈ R^{2} the system is hyperbolic.

7.4. Rewrite the equation

u_{tt} = (1 + u_{x})^{2}u_{xx}

as a first-order system for v = (u_{x}, u_{t})^{T}. At which points in (x, t, z)-space is it hyperbolic? Reduce it to the canonical form v_{t} +Bv_{x} = Cv+D with B diagonal.

7.5. Determine for which points (x, t) ∈ R^{2} the system

u_{t} + v_{x} = v + w
v_{t} + u_{x} = w
w_{t} + w_{x} sin x = u

is hyperbolic.