# PDE:Method of characteristics

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find the weak solution.

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