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1st order quasi-linear pde

  1. Apr 9, 2008 #1
    1. The problem statement, all variables and given/known data

    u*u_x + y*u_y = x

    Initial condition:

    u = 2*s on the parametric curve given by x = s, y = s, s is any real number.

    2. Relevant equations

    Given the equation:

    a(x,y,z)*u_x + b(x,y,z)*u_y = c(x,y,z)

    Here, u(x,y) is an unknown function which we're trying to find.

    a(x,y,z),b(x,y,z),c(x,y,z) are given functions of 3 variables.

    The initial condition is a requirement that the unknown function u has a prescribed value when restricted to the parametric curve:

    u(x_o(s),y_o(s)) = u_o(s), u_o(s) is a given function.

    3. The attempt at a solution

    My attempt crumbles early on.

    From looking at the equation I get:

    a(x,y,z) = u

    b(x,y,z) = y

    c(x,y,z) = x

    We alos have x(0) = s, y(0) = s, and u(x(0),y(0)) = u(s,s) = 2s

    My professor mentioned we need to introduce a 2nd variable, t, for the following relations

    dx(t)/dt = a(x(t),y(t),z(t))
    dy(t)/dt = b(x(t),y(t),z(t))
    dz(t)/dt = c(x(t),y(t),z(t))

    1.dx/dt = u , x(0) = s

    2.dy/dt = y , y(0) = s

    3.dz/dt = x , u(x(0),y(0)) = 2s

    I have a feeling that this might be wrong, and my other hypothesis is the following set up:

    dz(t)/dt = a(x(t),y(t),z(t))
    dy(t)/dt = b(x(t),y(t),z(t))
    dx(t)/dt = c(x(t),y(t),z(t))

    *note the swap between z and x.

    my main problem is relating a,b,c to the equation. For example, are


    fixed with respect to the partial derivattives u_x and u_y, or can I switch them around like I did in my second hypothesis?

    This clarification will allow me to finish the problem. Any input is appreciated. Thank you!
    Last edited: Apr 9, 2008
  2. jcsd
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