2.4.1 Inviscid Burgers \(u_x + u u_y = 0\)

problem number 84

Taken from Mathematica Symbolic PDE document

quasilinear first-order PDE, scalar conservation law

Solve for \(u(x,y)\) \[ u_x + u u_y = 0 \]


\[\left \{\left \{u(x,t)\to x^4-12 t^2\right \}\right \}\] Implicit solution


\[x u \left (x , y\right )-y +\textit {\_F1} \left (u \left (x , y\right )\right ) = 0\]

Hand solution

Solve for \(u\left (x,y\right ) \) in \(u_{x}+u\ u_{y}=0.\) Using the Lagrange-Charpit method, the characteristic equations are\[ \frac {dx}{1}=\frac {dy}{u}=\frac {du}{0}\] From the first pair of equation we obtain\[ u=\frac {dy}{dx}\] But \(du=0\) or \(u=C_{2}\). Hence the above becomes

\begin {align*} \frac {dy}{dx} & =C_{2}\\ y & =xC_{2}+C_{1}\\ C_{1} & =y-xC_{2} \end {align*}

Since \(C_{2}=F\left (C_{1}\right ) \) where \(F\) is arbitrary function, then \[ u\left (x,y\right ) =F\left (y-ux\right ) \]