# PDE:Method of characteristics

solution $\displaystyle u_t + a u_x = 0, u(x,0)=f(x)\,$

solution $\displaystyle u_t + u u_x = 0, u(x,0)=x\,$

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 $\displaystyle u_t+uu_x=0, u(x,0)=\begin{cases} 1, & x\le 0 \\ 1-x, & 01 \end{cases}\,$

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\le 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 $\displaystyle y=x^2+x\,$