Chapter 14

The equation $$ A(x, y) \partial f / \partial x+B(x, y) \hat f / \partial y=0 $$ may be written as $$ \mathbf{A} \cdot \nabla f=0 $$ Use vector methods to show that \(f\) is constant on curves which everywhere have the direction of the vector \(\mathbf{A}\).

Derive the equations of the characteristics of the differential equation \(t \partial f / \partial x+x^{3} \partial f / \partial t=0\) for the quadrant \(x>0, t>0 .\) Hence sketch the characteristics and solve the initial value problem $$ f(x, 0)=e^{-x} $$ What is the range of influence in this problem?

Derive the equations of the characteristics of the differential equation \(t \partial f / \partial x+x^{3} \partial f / \partial t=0\) for the quadrant \(x>0, t>0\). Hence sketch the characteristics and solve the initial value problem $$ f(x, 0)=e^{-x} $$ What is the range of influence in this problem?

Consider the initial value problem for one-dimensional subsonic gas flow in which $$ u(x, 0)= \begin{cases}U(1+x / l) & \text { for }-ll\end{cases} $$ and in which \(a(x, 0)\) is such that $$ s(x, 0)=a_{0} /(\gamma-1) $$ for all \(x\). Show that the characteristics are all straight, and sketch them. Draw graphs for \(u(x, t)\) for $$ t=0, \frac{1}{3} t_{s}, \frac{2}{3} t_{s}, t_{s} $$ where \(t_{s}\) is the time at which characteristics meet.