Sun Aug 18 16:52:12 2002 run on Windows
%*********************************************************************;
% This is a test file for the CHANGEVAR package. ;
% Make sure that before you attempt to run it the ;
% MATRIX package and CHANGEVAR is loaded. ;
%*********************************************************************;
algebraic;
%*********************************************************************;
% ON DISPJACOBIAN; % To get the Jacobians printed, remove the... ;
% ... percentage sign before the word ON ;
%*********************************************************************;
% ;
% *** First test problem *** ;
% ;
% Here are two Euler type of differential equations, ;
% ;
% 3 2 ;
% 2 x y''' + 3 x y'' - y = 0 ;
% ;
% ;
% 2 ;
% 5 x y'' - x y' + 7 y = 0 ;
% ;
% ;
% An Euler equation can be converted into a (linear) equation with ;
% constant coefficients by making change of independent variable: ;
% ;
% u ;
% x = e ;
% ;
% The resulting equations will be ;
% ;
% ;
% 2 y''' - 3 y'' + y' - y = 0 ;
% ;
% and ;
% ;
% 5 y'' - 6 y' + 7 y = 0 ;
% ;
% ;
% Where, now (prime) denotes differentiation with respect to the new ;
% independent variable: u ;
% How this change of variable is done using CHANGEVAR follows. ;
% ;
%*********************************************************************;
operator y;
changevar(y, u, x=e**u, { 2*x**3*df(y(x),x,3)+3*x**2*df(y(x),x,2)-y(x),
5*x**2*df(y(x),x,2)-x*df(y(x),x)+7*y(x) } ) ;
{2*df(y(u),u,3) - 3*df(y(u),u,2) + df(y(u),u) - y(u),
5*df(y(u),u,2) - 6*df(y(u),u) + 7*y(u)}
%*********************************************************************;
% *** Second test problem *** ;
% ;
% Now, the problem is to obtain the polar coordinate form of Laplace's;
% equation: ;
% ;
% 2 2 ;
% d u d u ;
% ------ + ------ = 0 ;
% 2 2 ;
% d x d y ;
% ;
% (The differentiations are partial) ;
% ;
% For polar coordinates the change of variables are : ;
% ;
% x = r cos(theta) , y = r sin(theta) ;
% ;
% As known, the result is : ;
% ;
% ;
% 2 2 ;
% d u 1 d u 1 d u ;
% ------ + --- ------ + --- ---------- = 0 ;
% 2 r d r 2 2 ;
% d r r d theta ;
% ;
% How this change of variable is done using CHANGEVAR follows. ;
% ;
% 2 2 ;
% (To get rid of the boring sin + cos terms we introduce a LET ;
% statement) ;
% ;
%*********************************************************************;
operator u;
let sin theta**2 = 1 - cos theta**2 ;
changevar(u, { r , theta }, { x=r*cos theta, y=r*sin theta },
df(u(x,y),x,2)+df(u(x,y),y,2) ) ;
2
df(u(r,theta),r,2)*r + df(u(r,theta),r)*r + df(u(r,theta),theta,2)
---------------------------------------------------------------------
2
r
end;
Time for test: 221 ms