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[Maxima-commits] CVS: maxima/share/contrib/diffequations contrib_ode.usg,1.4,1.5 ode1_lie.mac,1.5,1.6
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From: David B. <bil...@us...> - 2006-05-18 12:45:42
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Update of /cvsroot/maxima/maxima/share/contrib/diffequations In directory sc8-pr-cvs7.sourceforge.net:/tmp/cvs-serv5811 Modified Files: contrib_ode.usg ode1_lie.mac Log Message: Remove tariling white space from lines. Add arxiv.org URL for one reference. Index: contrib_ode.usg =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/contrib_ode.usg,v retrieving revision 1.4 retrieving revision 1.5 diff -u -d -r1.4 -r1.5 --- contrib_ode.usg 8 Dec 2004 22:22:33 -0000 1.4 +++ contrib_ode.usg 18 May 2006 12:45:33 -0000 1.5 @@ -8,18 +8,18 @@ The calling convention for contrib_ode is identical to ODE2. It takes three arguments: an ODE (only the left hand side need be given if the right hand side is 0), the dependent variable, and the independent -variable. When successful, it returns a list of solutions. +variable. When successful, it returns a list of solutions. A solution can have a number of forms: - o an explicit solution for the dependent variable, - o an implicit solution for the dependent variable, - o a parametric solution in terms of variable %t, or + o an explicit solution for the dependent variable, + o an implicit solution for the dependent variable, + o a parametric solution in terms of variable %t, or o a tranformation into another ODE in variable %u. - -%c is used to represent the constant in the case of first order equations, -and %k1 and %k2 the constants for second order equations. If contrib_ode -cannot obtain a solution for whatever reason, it returns false, after -perhaps printing out an error message. + +%c is used to represent the constant in the case of first order equations, +and %k1 and %k2 the constants for second order equations. If contrib_ode +cannot obtain a solution for whatever reason, it returns false, after +perhaps printing out an error message. (C1) eqn:x*'diff(y,x)^2-(1+x*y)*'diff(y,x)+y=0; @@ -77,7 +77,7 @@ (D9) lagrange The following Riccati equation is transformed into a linear -second order ODE in the variable %u. maxima is unable to +second order ODE in the variable %u. maxima is unable to solve the new ODE. (%i7) eqn:'diff(y,x)+1-x=(x+y)*y; @@ -102,9 +102,9 @@ For first order ODEs contrib_ode calls ode2. It then tries the following methods: factorization, Clairault, Lagrange, Riccati, -Abel and Lie symmetry methods. The Lie method is not attempted -on Abel equations if the Abel method fails, but it is tried -if the Riccati method returns an unsolved second order ODE. +Abel and Lie symmetry methods. The Lie method is not attempted +on Abel equations if the Abel method fails, but it is tried +if the Riccati method returns an unsolved second order ODE. For second order ODEs contrib_ode calls ode2. No other methods are implemented yet. @@ -172,16 +172,17 @@ [4] F Schwarz, Symmetry Analysis of Abel's Equation, Studies in Applied Mathematics, 100:269-294 (1998) -[5] F Schwarz, Algorithmic Solution of Abel's Equation, +[5] F Schwarz, Algorithmic Solution of Abel's Equation, Computing 61, 39-49 (1998) -[6] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order - ODE Patterns, Computer Physics Communications 113 (1998), p 239. +[6] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order + ODE Patterns, Computer Physics Communications 113 (1998), p 239. (http://lie.uwaterloo.ca/papers/ode_vii.pdf) - -[7] E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs, - Symmetries and Linear Transformations, European Journal of + +[7] E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs, + Symmetries and Linear Transformations, European Journal of Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). + (http://arxiv.org/abs/math-ph/0007023) (http://lie.uwaterloo.ca/papers/ode_iv.pdf) [8] G W Bluman, S C Anco, Symmetry and Integration Methods for Index: ode1_lie.mac =================================================================== RCS file: /cvsroot/maxima/maxima/share/contrib/diffequations/ode1_lie.mac,v retrieving revision 1.5 retrieving revision 1.6 diff -u -d -r1.5 -r1.6 --- ode1_lie.mac 31 Jan 2004 12:40:55 -0000 1.5 +++ ode1_lie.mac 18 May 2006 12:45:33 -0000 1.6 @@ -1,37 +1,38 @@ /* ode1_lie.mac Solution of first order ODEs using Lie symmetry methods - - References: - [1] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order - ODE Patterns, Computer Physics Communications 113 (1998), p 239. + References: + + [1] E. S. Cheb-Terrab, A. D. Roche, Symmetries and First Order + ODE Patterns, Computer Physics Communications 113 (1998), p 239. (http://lie.uwaterloo.ca/papers/ode_vii.pdf) - - [2] E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs, - Symmetries and Linear Transformations, European Journal of - Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). + + [2] E. S. Cheb-Terrab, T. Koloknikov, First Order ODEs, + Symmetries and Linear Transformations, European Journal of + Applied Mathematics, Vol. 14, No. 2, pp. 231-246 (2003). + (http://arxiv.org/abs/math-ph/0404014) (http://lie.uwaterloo.ca/papers/ode_iv.pdf) [3] G W Bluman, S C Anco, Symmetry and Integration Methods for Differential Equations, Springer, (2002) - Copyright (C) 2004 David Billinghurst - - This program is free software; you can redistribute it and/or modify - it under the terms of the GNU General Public License as published by - the Free Software Foundation; either version 2 of the License, or - (at your option) any later version. - - This program is distributed in the hope that it will be useful, - but WITHOUT ANY WARRANTY; without even the implied warranty of - MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the - GNU General Public License for more details. - + Copyright (C) 2004 David Billinghurst + + This program is free software; you can redistribute it and/or modify + it under the terms of the GNU General Public License as published by + the Free Software Foundation; either version 2 of the License, or + (at your option) any later version. + + This program is distributed in the hope that it will be useful, + but WITHOUT ANY WARRANTY; without even the implied warranty of + MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the + GNU General Public License for more details. + You should have received a copy of the GNU General Public License - along with this program; if not, write to the Free Software + along with this program; if not, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */ @@ -58,7 +59,7 @@ xi:sym[1], eta:sym[2], ode_disp2("In ode1_lie: xi = ",xi), ode_disp2("In ode1_lie: eta = ",eta), - + /* Check symmetry generators */ r:symtest(phi,xi,eta,y,x), ode_disp2("In ode1_lie: symtest = ",r), @@ -94,14 +95,14 @@ [ %A, %Ay, %Ayy, %Ayyy, %phi_yy, %phi_yyy ], ode_disp("In ode1_a ..."), ode_disp2(" phi = ",phi), -/* +/* Using [2], eq (13),(14) -*/ +*/ /* Confirm that diff(phi,y,3)#0 */ %phi_yy: ratsimp(diff(phi,y,2)), %phi_yyy: ratsimp(diff(%phi_yy,y)), - if is(%phi_yyy=0) then - ( + if is(%phi_yyy=0) then + ( /* %phi_yyy=0 - Not covered in [2] */ ode_disp(" %phi_yyy = 0"), return(lie_FxHx(phi,y,x)) @@ -111,7 +112,7 @@ %Ax: ratsimp(diff(%A,x)), if is(%Ay=0) then - /* Case 1: diff(A,y)=0 + /* Case 1: diff(A,y)=0 Change variables using y=A*u (15) */ block( [u], @@ -121,7 +122,7 @@ /* Here neweq is RHS only */ neweq: ratsimp((subst(%A*u,y,phi)-%Ax*u)/%A), ode_disp2(" transformed equation is ",neweq), - + /* Symmetry generators of u equation */ ode_disp(" Finding symmetries of transformed equation"), gen:radcan(lie_FxHx(neweq,u,x)), @@ -129,14 +130,14 @@ if (gen=false) then false - else block( + else block( /* transform [F,H] to [xi,eta] */ [_F:gen[1],_H:gen[2],dydt:%Ax*y/%A,dudt:%A], [_F, ratsimp(_F*dydt+_H*dudt)] ) ) else if is((%Ayy:ratsimp(diff(%Ay,y)))=0) then - /* Case 2: diff(A,y,2)=0 and diff(A,y)#0 + /* Case 2: diff(A,y,2)=0 and diff(A,y)#0 Change variables using u=ln(A) (17) */ block( [neweq,sub,u], @@ -158,7 +159,7 @@ /* transform [F,H] to [xi,eta] */ [_F:gen[1],_H:gen[2],dydt,dudt,_xi,_eta], _xi:_F, - dydt:diff(rhs(sub),x), + dydt:diff(rhs(sub),x), dudt:diff(rhs(sub),u), _eta: _F*dydt+_H*dudt, /* Now eliminate u from eta */ @@ -167,19 +168,19 @@ [ _xi, _eta ] ) ) - else + else block( [%Axy,%eye,%Iy,%p,%px], %Axy: ratsimp(diff(%Ay,x)), %eye: ratsimp(%Axy/%Ayy), - if linear2(y,%eye) then - /* Case 3: diff(A,y,2)#0 and I=Axx/Ayy linear in y + if linear2(y,%eye) then + /* Case 3: diff(A,y,2)#0 and I=Axx/Ayy linear in y p=exp(integrate(Iy,x)) Substitute y = u/p */ block( [ _F, _H, _xi, _eta, u ], ode_disp(" ..... Case 3 true"), - if get('contrib_ode,'verbose) then + if get('contrib_ode,'verbose) then print(" A is ",%A), %Iy: diff(%eye,y), %Iy:ratsimp(%Iy), @@ -199,17 +200,17 @@ ode_disp(" Finding symmetries of transformed equation"), gen:lie_FxHx(neweq,u,x), ode_disp2(" [xi,eta] = ",gen), - + if (gen=false) then false else ( - /* Now we need to transform xi and eta back to original - equation. Have + /* Now we need to transform xi and eta back to original + equation. Have F(x).(d/dx) + H(x).(d/du) y = u/%p */ - _F:radcan(gen[1]), + _F:radcan(gen[1]), _H:radcan(gen[2]), - _xi:_F, + _xi:_F, ode_disp2(" _xi = ",_xi), _eta: ratsimp(( _H - _F*%px*y )/%p), ode_disp2(" _eta = ",_eta), @@ -226,8 +227,8 @@ /* Determine if ODE y' = Phi(x,y) has linear symmetry of form xi = F(x), eta = H(x) - This uses the notation from [1] E. S. Cheb-Terrab, A. D. Roche, - Symmetries and First Order ODE Patterns + This uses the notation from [1] E. S. Cheb-Terrab, A. D. Roche, + Symmetries and First Order ODE Patterns */ lie_FxHx(Phi,y,x) := block( [Phi_y, Phi_yy, Q, Q_y,_a,_b,_c,gen], @@ -276,7 +277,7 @@ block( [_a,_b,_c,_f,_h], /* In this case Q is constant and Phi must have the form - y' = Phi(x,y) = A(x) + B(x)*exp(y/Q) + y' = Phi(x,y) = A(x) + B(x)*exp(y/Q) and F(x), H(x) are simple functions of A and B */ _c: bothcoef(expand(Phi),exp(y/Q)), _a: second(_c), @@ -294,16 +295,16 @@ of first order ode y'=phi(x,y) Returns 0 when the symmetry generators are OK. If the result is - non-zero, then further simplification may reduce it to 0. + non-zero, then further simplification may reduce it to 0. */ -symtest(phi,xi,eta,y,x) := +symtest(phi,xi,eta,y,x) := ratsimp( diff(eta,x) + (diff(eta,y)-diff(xi,x))*phi - diff(xi,y)*phi^2 - xi*diff(phi,x) - eta*diff(phi,y) )$ -/* Determine integrating factor of first order ode y'=phi(x,y) +/* Determine integrating factor of first order ode y'=phi(x,y) given symmetry generators [ xi(x,y), eta(x,y) ] */ -lie_integrating_factor(phi,xi,eta) := +lie_integrating_factor(phi,xi,eta) := 1/ratsimp(eta-xi*phi)$ /* Solve DE given integrating factor */ @@ -317,7 +318,7 @@ -/* from ode2.mac +/* from ode2.mac EXACT(M,N):=BLOCK([A,ynew,%c], INTFACTOR: SUBST(YOLD,YNEW,%q%), A: INTEGRATE(RATSIMP(M),X), @@ -326,14 +327,14 @@ */ /* Find symmetry generators [ xi, eta ] of separable - first order ode 'diff(y,x) = phi(x,y) = f(x)*g(y) + first order ode 'diff(y,x) = phi(x,y) = f(x)*g(y) Can find generator [ xi(x), 0 ] = [ 1/f, 0 ] */ lie_symgen_separable(f,g,y,x) := block(return([1/f,0]))$ -/* Find symmetry generators [ xi, eta ] of linear - first order ode 'diff(y,x) = phi(x,y) = f(x)*y + g(x) +/* Find symmetry generators [ xi, eta ] of linear + first order ode 'diff(y,x) = phi(x,y) = f(x)*y + g(x) Can find generator [ xi(x), eta(x) ] */ |