Need help to interpret unusual Maxima notation for aderivative i.e. (d^{y+1}/dx^{y+1}) rho

Vishal,

The notation in question means the (y+1)-th derivative of rho with respect
to x.

Presumably, it is a result of terms like this in your code:

diff(%rho, x, y)

which means the y-th derivative of %rho with respect to x. I suspect that
what you meant to write is

diff(%rho,x,1,y,1)

or, equivalently,

diff(diff(%rho,x),y).


Viktor
 

-----Original Message-----
From: maxima-bounces at math.utexas.edu [mailto:maxima-bounces at math.utexas.edu]
On Behalf Of Vishal Ramnath
Sent: Monday, June 08, 2009 11:50 AM
To: maxima at math.utexas.edu
Subject: Need help to interpret unusual Maxima notation for
aderivative i.e. (d^{y+1}/dx^{y+1}) rho

Hello,
 
I need some help in understanding what this term in a PDE that I set up in
wxMaxima 0.8.2 Windows XP actually is viz.
 
   y+1
d      
-------- rho
   y+1
dx
 
(LaTeX code is $\frac{d^{y+1}}{dx^{y+1}} \rho$)
 
It doesn't quite make sense if interpreted as
 
(d/dy) (d/dy) (d/dx) rho 
 
i.e. (y+1)-th derivative of the (d/dx) derivative of rho, for example in
Maxima (d/dy)(d/dx rho) would be written as (d^y rho)/(dx^y) which is
straightforward and understandable.
 
I attach the code below.
 
Thank you for any assistance.
 
Regards,
 
Vishal
 
ps. I use the Greek font symbols
 
"Burnett Equations - 8 June 2009"$
"This program sets up the Burnett equations (mass, momentum, energy) as
explicit PDEs"$
 
"set \rho(x,y) u(x,y) v(x,y)"$
depends([%rho, u, v], [x, y])$
 
"first order components for stress tensor"$
%sigma[11, 1]: -%mu*((4/3)*diff(u, x) - (2/3)*diff(v, y));
%sigma[22, 1]: -%mu*((4/3)*diff(v, y) - (2/3)*diff(u, x));
 
%sigma[12, 1]: -%mu*(diff(u, y) + diff(v, x));
%sigma[21, 1]: -%mu*(diff(u, y) + diff(v, x));
 
"heat transfer components to first order"$
q[1, 1]: -%kappa*diff(T, x);
q[2, 1]: -%kappa*diff(T, y);
 
"set up Navier-Stokes equations i.e. Burnett equation truncated to first
order"$
eq[1, 1]: diff(%rho*u, x) + diff(%rho*v, y);
eq[2, 1]: diff(%rho*u^2 + %sigma[11, 1], x) + diff(%rho*u*v + %sigma[21, 1],
y);
eq[3, 1]: diff(%rho*u*v + %sigma[12, 1], x) + diff(%rho*v^2 + %sigma[22, 1],
y);
 
"ideal gas equation of state"$
"Comment: I want to solve for u(x,y,t), v(x,y,t), \rho(x,y,t)"$
p: %rho*R*T;
 
%sigma[11, 2]: %sigma[11, 1] 
    + (%mu^2/p)*(%alpha[1]*(diff(u, x))^2 
              + %alpha[2]*(diff(u, y))^2 
              + %alpha[3]*(diff(v, x))^2 
              + %alpha[4]*(diff(v, y))^2 
              + %alpha[5]*(diff(u, x))*(diff(v, y))
              + %alpha[6]*(diff(u, y))*(diff(v, x))
        + %alpha[7]*R*(diff(T, x, 2))
        + %alpha[8]*R*(diff(T, y, 2))
        + %alpha[9]*(R*T/%rho)*(diff(%rho, x, 2))
        + %alpha[10]*(R*T/%rho)*(diff(%rho, y, 2))
        + %alpha[11]*(R*T/%rho^2)*((diff(%rho, x))^2)
        + %alpha[12]*(R*T/%rho^2)*((diff(%rho, y))^2)
        + %alpha[13]*(R/T)*((diff(T, x))^2)
        + %alpha[14]*(R/T)*((diff(T, y))^2)
        + %alpha[15]*(R/%rho)*(diff(T, x))*(diff(%rho, x))
        + %alpha[16]*(R/%rho)*(diff(T, y))*(diff(%rho, y))
        );
     
%sigma[22, 2]: %sigma[22, 1] 
    + (%mu^2/p)*(%alpha[1]*(diff(v, y))^2
        + %alpha[2]*(diff(v, x))^2
        + %alpha[3]*(diff(u, y))^2
        + %alpha[4]*(diff(u, x))^2
        + %alpha[5]*(diff(u, x))*(diff(v, y))
        + %alpha[6]*(diff(u, y))*(diff(v, x))
        + %alpha[7]*R*(diff(T, y, 2))
        + %alpha[8]*R*(diff(T, x, 2))
        + %alpha[9]*(R*T/%rho)*(diff(%rho, y, 2))
        + %alpha[10]*(R*T/%rho)*(diff(%rho, x, 2))
        + %alpha[11]*(R*T/%rho^2)*((diff(%rho, y))^2)
        + %alpha[12]*(R*T/%rho^2)*((diff(%rho, x))^2)
        + %alpha[13]*(R/T)*((diff(T, y))^2)
        + %alpha[14]*(R/T)*((diff(T, x))^2)
        + %alpha[15]*(R/%rho)*(diff(T, y))*(diff(%rho, y))
        + %alpha[16]*(R/%rho)*(diff(T, x))*(diff(%rho, x))
        );
        
%sigma[12, 2]: %sigma[12, 1] + (%mu^2/p)*(?ta[1]*(diff(u, x))*(diff(u, y))
           + ?ta[2]*(diff(v, x))*(diff(v, y))
           + ?ta[3]*(diff(u, x))*(diff(v, x))
           + ?ta[4]*(diff(u, y))*(diff(v, y))
           + ?ta[5]*R*(diff(T,x,y))
           + ?ta[6]*(R*T/%rho)*(diff(%rho, x, y))
           + ?ta[7]*(R/T)*(diff(T, x))*(diff(T, y))
           + ?ta[8]*(R*T/%rho^2)*(diff(%rho, x))*(diff(%rho, y))
           + ?ta[9]*(R/%rho)*(diff(%rho, x))*(diff(T, y))
           + ?ta[10]*(R/%rho)*(diff(T, x))*(diff(%rho, y))
           );
q[1, 2]: q[1, 1] + (%mu^2/%rho)*(%gamma[1]*(1/T)*(diff(T, x))*(diff(u, x))
         + %gamma[2]*(1/T)*(diff(T, x))*(diff(v, y))
         + %gamma[3]*(1/T)*(diff(T, y))*(diff(v, x))
         + %gamma[4]*(1/T)*(diff(T, y))*(diff(u, y))
         + %gamma[5]*(diff(u, x, 2))
         + %gamma[6]*(diff(u, y, 2))
         + %gamma[7]*(diff(v, x, y))
         + %gamma[8]*(1/%rho)*(diff(%rho, x))*(diff(u, x))
         + %gamma[9]*(1/%rho)*(diff(%rho, x))*(diff(v, y))
         + %gamma[10]*(1/%rho)*(diff(%rho, y))*(diff(v, x))
         + %gamma[11]*(1/%rho)*(diff(%rho, y))*(diff(u, y))
         );
           
           
eq[1, 2]: diff(%rho*u, x) + diff(%rho*v, y);
eq[2, 2]: diff(%rho*u^2 + %sigma[11, 2], x) + diff(%rho*u*v + %sigma[21, 2],
y);
eq[3, 2]: diff(%rho*u*v + %sigma[12, 2], x) + diff(%rho*v^2 + %sigma[22, 2],
y);
 
"Equations are built up now!"$
 
"Discretize first order equations using finite differences..."$