# PDE:Method of characteristics

solution $u_{t}+au_{x}=0,u(x,0)=f(x)\,$

solution $u_{t}+uu_{x}=0,u(x,0)=x\,$

solution $y^{{-1}}u_{x}+u_{y}=0,u(x,1)=x^{2}\,$

solution $u_{x}+2u_{y}=u^{2},u(x,0)=h(x)\,$

solution $u_{x}+xu_{y}=u^{2}\,$

solution $u_{x}+xu_{y}-u_{z}=u\,$, $u(x,y,1)=x+y\,$

solution $xu_{x}+u_{y}=y,u(x,0)=x^{2}\,$

solution $xu_{x}+yu_{y}+u_{z}=u,u(x,y,0)=h(x,y)\,$

solution $u_{x}+u_{y}+u=e^{{x+2y}}\,$, $u(x,0)=0\,$

solution Show that if $z=u(x,y)\,$ is an integral surface of $V=\,$ containing a point $P\,$, then the surface contains the characteristic curve $\chi \,$ passing through $P\,$. (Assume the vector field $V\,$ is $C^{1}\,$).

solution If $S_{1}\,$ and $S_{2}\,$ are two graphs $\left[S_{i}=u_{i}(x,y),i=1,2\right]\,$ that are integral surfaces of $V=\,$ and intersect in a curve $\chi \,$, show that $\chi \,$ is a characteristic curve.

solution $(x+u)u_{x}+(y+u)u_{y}=0\,$

solution $u_{t}+uu_{x}=0,u(x,0)={\begin{cases}1,&x\leq 0\\1-x,&01\end{cases}}\,$

solution Solve the initial value problem $a(u)u_{x}+u_{y}=0\,$ with $u(x,0)=h(x)\,$ and show the solution becomes singular for some $y>0\,$ unless $a(h(s))\,$ is a nondecreasing function of $s\,$.

solution Consider $uu_{x}+u_{y}=0\,$ with the IC $u(x,0)=h(x)={\begin{cases}u_{0}>0,&x\leq 0\\u_{0}(1-x),&0

Show that a shock develops at a finite time and describe the weak solution.

solution Consider $uu_{x}+u_{t}=0\,$ with the IC $u(x,0)=h(x)={\begin{cases}0,&x<0\\u_{0}(x-1),&x>0\end{cases}}\,$

Find the weak solution.

solution Consider the problem $u_{x}+u_{y}+u=1\,$ with condition: $u=\sin(x)\,$ on $y=x^{2}+x\,$