PDE:Method of characteristics

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

solution u_t + u u_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=<a,b,c>\, 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=<a,b,c>\, 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\le 0 \\ 1-x, & 0<x\le 1\\ 0, & x>1 \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\le 0 \\ u_0(1-x), & 0<x<1\\ 0, & x\ge 1 \end{cases}\,

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\,

Main Page : Partial Differential Equations

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