View source for PDE:Method of characteristics

[[PDEMOC1|solution]] <math>u_t + a u_x = 0, u(x,0)=f(x)\,</math>

[[PDEMOC2|solution]] <math>u_t + u u_x = 0, u(x,0)=x\,</math>

[[PDEMOC3|solution]] <math>y^{-1} u_x + u_y = 0, u(x,1)=x^2\,</math>

[[PDEMOC4|solution]] <math>u_x + 2u_y = u^2, u(x,0)=h(x)\,</math>

[[PDEMOC5|solution]] <math>u_x + xu_y = u^2\,</math>

[[PDEMOC6|solution]] <math>u_x + xu_y -u_z = u\,</math>, <math>u(x,y,1)=x+y\,</math>

[[PDEMOC7|solution]] <math>xu_x + u_y = y, u(x,0)=x^2\,</math>

[[PDEMOC8|solution]] <math>xu_x + yu_y + u_z = u, u(x,y,0) = h(x,y)\,</math>

[[PDEMOC16|solution]] <math>u_x + u_y + u = e^{x + 2y}\,</math>, <math>u(x,0) = 0\,</math>


[[PDEMOC9|solution]] Show that if <math>z=u(x,y)\,</math> is an [[integral surface]] of <math>V=<a,b,c>\,</math> containing a point <math>P\,</math>, then the surface contains the [[characteristic curve]] <math>\chi\,</math> passing through <math>P\,</math>. (Assume the vector field <math>V\,</math> is <math>C^1\,</math>).

[[PDEMOC10|solution]] If <math>S_1\,</math> and <math>S_2\,</math> are two graphs <math>\left[ S_i = u_i(x,y), i=1,2\right]\,</math> that are integral surfaces of <math>V=<a,b,c>\,</math> and intersect in a curve <math>\chi\,</math>, show that <math>\chi\,</math> is a characteristic curve.

[[PDEMOC11|solution]] <math>(x+u)u_x + (y+u)u_y = 0\,</math>

[[PDEMOC12|solution]] <math>u_t+uu_x=0, u(x,0)=\begin{cases} 1, & x\le 0 \\ 1-x, & 0<x\le 1\\ 0, & x>1 \end{cases}\,</math>

[[PDEMOC13|solution]] Solve the initial value problem <math>a(u)u_x+u_y=0\,</math> with <math>u(x,0)=h(x)\,</math> and show the solution becomes singular for some <math>y>0\,</math> unless <math>a(h(s))\,</math> is a nondecreasing function of <math>s\,</math>.

[[PDEMOC14|solution]] Consider <math>uu_x+u_y=0\,</math> with the IC <math>u(x,0)=h(x)=\begin{cases} u_0>0, & x\le 0 \\ u_0(1-x), & 0<x<1\\ 0, & x\ge 1 \end{cases}\,</math>

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

[[PDEMOC15|solution]] Consider <math>uu_x+u_t=0\,</math> with the IC <math>u(x,0)=h(x)=\begin{cases} 0, & x<0 \\ u_0(x-1), & x>0 \end{cases}\,</math>

Find the weak solution.

[[PDEMOC17|solution]] Consider the problem <math>u_x + u_y + u = 1\,</math> with condition: <math>u=\sin(x)\,</math> on <math>y=x^2+x\,</math>



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