Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

There is plenty on time-dependent or on special cases where various things
are constant and allow

analytical solution but would appreciate pointers on how to code this system
of PDEs.

Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

snip

let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).

Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
Assume we can divide (*) by fp(u), then we get by plugging (*)
into (**) :

d/dx (u(x,y(x))= (-gp(u)/fp(u) + yp) * u_y = (**)

Now, if we solve the ODE yp(x)= gp(u(x,y(x))) / fp(u(x,y(x)),
then (***) = 0, or u(x,y(x)) = constant.

I.e., along the so-called characteristic curve (x,y(x))
the solution u is constant,

Phil Scadden wrote:
Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

snip
let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).

Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
I'm having trouble with the notation. Can you describe further what you

let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).
Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
Assume we can divide (*) by fp(u), then we get by plugging (*)
into (**) :

d/dx (u(x,y(x))= (-gp(u)/fp(u) + yp) * u_y = (**)

Now, if we solve the ODE yp(x)= gp(u(x,y(x))) / fp(u(x,y(x)),
then (***) = 0, or u(x,y(x)) = constant.

I.e., along the so-called characteristic curve (x,y(x))
the solution u is constant,

Thanks very much for taking the time to comment.
The method of characteristics works well for a system of 2 PDEs and 2
unknown functions where it is easy to identify the so called Riemann
Invariants but what if you have 4 of each? This is this issue that
struggling with here.

"Helmut Jarausch" <jarausch@igpm.rwth-aachen.de> wrote in message
news:54qj34F2237qnU1@mid.dfncis.de...
Phil Scadden wrote:
Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

snip
let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).

Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
I'm having trouble with the notation. Can you describe further what you
mean with "characteristic?" "Ansatz" makes more sense.

'u_x' I think wants to be "u sub x" but I can't divine what 'u_y*yp' might
be. Gruss, LS

Lane Straatman wrote:
"Helmut Jarausch" <jarausch@igpm.rwth-aachen.de> wrote in message
news:54qj34F2237qnU1@mid.dfncis.de...
Phil Scadden wrote:
Looking to solve time-independent conservation law.
[f(u)]_x + [g(u)]_y =0

with initial condtions u(x_0(t),y_0(t))=u_0

and boundary conditions u(x_0),y) and u(x,y_0)

snip
let's rewrite your equation to
(*) fp(u)*u_x + gp(u)*u_y=0 where fp(u) is the derivate of f at
u(x,y).

Now make the ansatz (characteristic) y= y(x), then

(**) d/dx (u(x,y(x))= u_x + u_y*yp where yp= d/dx y .
I'm having trouble with the notation. Can you describe further what you
mean with "characteristic?" "Ansatz" makes more sense.

Characteristic curves (that's a technical term) are curves on which the
solution is constant or given by the solution of an ordinary differential
equation.
Gosh, I'm looking at my text from school that covered diff EQ. It calls the

'u_x' I think wants to be "u sub x" but I can't divine what 'u_y*yp'
might be. Gruss, LS

u_x and u_y denote the partial derivative w.r.t. x (y, resp.).
The characteristic curve is assumed to have the form (x,y(x)); here yp is
an abbreviation for the derivative of that y(x) w.r.t. x (x being a scalar
variable).
Warum hiesse denn "characteristic" "Ansatz?" LS

Lane Straatman wrote:
Warum hiesse denn "characteristic" "Ansatz?"

The German word "ansatz" means here we try to find curves on which the
solution is constant or given by an ODE. What comes out are curves
which are well-known as characteristic curves or stream-lines.
I'm missing something here, but I don't want to beat it into the ground.

"Helmut Jarausch" <jarausch@skynet.be> wrote in message
news:45edbdf9$0$2938$ba620e4c@news.skynet.be...
Lane Straatman wrote:
Warum hiesse denn "characteristic" "Ansatz?"
The German word "ansatz" means here we try to find curves on which the
solution is constant or given by an ODE. What comes out are curves
which are well-known as characteristic curves or stream-lines.
I'm missing something here, but I don't want to beat it into the ground.
Would not the above be equivalent with separability?

I've set a goal to calculate Bessel functions in fortran, which means that
I'm going to want to post something with derivatives. The notation you had
going for this problem looked like it was pretty standard for usenet. Would
I be consistent with your notation if I wrote:
x^2 * y'' + x * y' + (x^2 - p^2) * y = 0