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1st order quasi-linear pde

  • Thread starter sapiental
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1. Homework Statement


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. Homework 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

a(x(t),y(t),z(t))
b(x(t),y(t),z(t))
c(x(t),y(t),z(t))

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:

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