{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "read(\"d:/gem31.txt\"):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%doGEM~v.031.28.6~~<>~Copyright(C)~Alexei~F. ~Cheviakov,~2004-2009G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 " #conservation laws of the 3D static MHD equations\n#Six scalar, plus o ne infinite family (in the current multiplier ansatz)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ind:=x,y,z;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\$ind G6%%\"xG%\"yG%\"zG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "gem_d ecl_vars(indeps=[ind], deps=[B1(ind),B2(ind),B3(ind),P(ind)]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#B1G6%%\"xG%\"yG%\"zG\"\"\"%9will~ now~be~displayed~asGF*F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%#B2G 6%%\"xG%\"yG%\"zG\"\"\"%9will~now~be~displayed~asGF*F%F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#*(-%#B3G6%%\"xG%\"yG%\"zG\"\"\"%9will~now~be~dis played~asGF*F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(-%\"PG6%%\"xG%\" yG%\"zG\"\"\"%9will~now~be~displayed~asGF*F%F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%AIndependent~variables:~[x,~y,~z]G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%EDependent~variables:~[B1,~B2,~B3,~P]G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%3Free~functions:~[]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%3Free~constants:~[]G" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#%XDefinition~of~variables~succesful;~~~GeM~is~initializedG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gP:=grad(P(ind),[ind]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#gPGK%'vectorG6#7%&%\"PG6#%\"xG&F*6# %\"yG&F*6#%\"zGQ(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "B_vect:=[B1(ind),B2(ind),B3(ind)];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'B_vectG7%%#B1G%#B2G%#B3G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "divB:=diverge(B_vect,[ind]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%divBG,(&%#B1G6#%\"xG\"\"\"&%#B2G6#%\"yGF*&%#B3G6#%\" zGF*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "curlB:=curl(B_vect, [ind]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&curlBGK%'vectorG6#7%,&&% #B3G6#%\"yG\"\"\"&%#B2G6#%\"zG!\"\",&&%#B1GF1F.&F+6#%\"xGF3,&&F0F8F.&F 6F,F3Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "cuBxB: =crossprod(curlB,B_vect);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&cuBxBG K%'vectorG6#7%,&*&,&&%#B1G6#%\"zG\"\"\"&%#B3G6#%\"xG!\"\"F0F2F0F0*&,&& %#B2GF3F0&F-6#%\"yGF5F0F9F0F5,&*&F7F0F-F0F0*&,&&F2F;F0&F9F.F5F0F2F0F5, &*&F@F0F9F0F0*&F+F0F-F0F5Q(pprint06\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "gem_de cl_eqs(\n[\ncuBxB[1]=gP[1],\ncuBxB[2]=gP[2],\ncuBxB[3]=gP[3],\ndivB=0 \n],\nsolve_for=[\ndiff(P(ind),x),\ndiff(P(ind),y),\ndiff(P(ind),z),\n diff(B1(ind),x)\n]\n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~1 ~~(Order=1):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%#B3G\"\"\"%\$B1z GF'F'*&F&F'%\$B3xGF'!\"\"*&%#B2GF'%\$B2xGF'F+*&F-F'%\$B1yGF'F'%#PxGF+\"\" !" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~2~~(Order=1):G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%#B1G\"\"\"%\$B2xGF'F'*&F&F'%\$B1yG F'!\"\"*&%#B3GF'%\$B3yGF'F+*&F-F'%\$B2zGF'F'%#PyGF+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~3~~(Order=1):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,,*&%#B2G\"\"\"%\$B3yGF'F'*&F&F'%\$B2zGF'!\"\"*&%#B1GF'% \$B1zGF'F+*&F-F'%\$B3xGF'F'%#PzGF+\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Equation~4~~(Order=1):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(%\$B 1xG\"\"\"%\$B2yGF&%\$B3zGF&\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%UNo ~differential~consequences~required~on~this~stage.G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%?Equation~definition~successfulG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "det_eqs:=gem_conslaw_det_eqs(\n[\nind, B1( ind),B2(ind),B3(ind),P(ind)\n]):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Y Generating~determining~equtions~for~conservation~laws...G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "CL_multipliers:=gem_conslaw_multipl iers();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%/CL_multipliersG7&-%(Lamb da1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG-%(Lambda2GF(-%(Lambda3GF(-%( Lambda4GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "simplified_eq s:=DEtools[rifsimp](det_eqs, CL_multipliers, mindim=1);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#>%/simplified_eqsGK%&TABLEG6#7\$/%*dimensionG%)in finityG/%'SolvedG7@/&%(Lambda4G6\$%\"yGF3\"\"!/&F16\$F3%\"zGF4/&F16\$%\"P GF3F4/&%(Lambda2G6\$F8F8F4/&F1F@F4/&F16\$F " 0 "" {MPLTEXT 1 0 49 "multipliers_sol:=pdsolve(simplified_eqs[Solved]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%0multipliers_solG<&/-%(Lambda1G6)%\"xG%\"yG%\"zG %#B1G%#B2G%#B3G%\"PG,**&F+\"\"\"%\$_C2GF3F3*&F,F3%\$_C3GF3F3%\$_C4GF3*&&% %_F10G6#F0F3F-F3!\"\"/-%(Lambda2GF),**&F,F3%\$_C1GF3F<*&F*F3F4F3F<%\$_C5 GF3*&F9F3F.F3F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "#Infinite set of cons. laws, depending on an arbitrary function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "#Since the multipliers involve arbitrary function, we can only us e Anco's scaling method. \n#Luckily, the PDE system has an obvious sca ling symmetry X=Bi D_Bi + 2P D_P" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "gem_ge t_CL_fluxes(multipliers_sol, method=\"Scaling\", symmetry=\{eta_B1=B1, eta_B2=B2,eta_B3=B3,eta_P=2*P\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %'MethodGQ(Scaling6\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#%^^lMultiplie rs:~[Lambda1(x,y,z,B1,B2,B3,P)~=~y*_C2+z*_C3+_C4-diff(_F10(P),P)*B1,~L ambda2(x,y,z,B1,B2,B3,P)~=~-z*_C1-x*_C2+_C5-diff(_F10(P),P)*B2,~Lambda 3(x,y,z,B1,B2,B3,P)~=~_C1*y+_C6-x*_C3-diff(_F10(P),P)*B3,~Lambda4(x,y, z,B1,B2,B3,P)~=~_F10(P)+(y*_C2+z*_C3+_C4)*B1+(-z*_C1-x*_C2+_C5)*B2+(_C 1*y+_C6-x*_C3)*B3]G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6\$Q-Symmetries:~6 \"K%'vectorG6#7&%#B1G%#B2G%#B3G,\$*&\"\"#\"\"\"%\"PGF/F/Q(pprint0F\$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%7----------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%coVerifying~div(fluxes)=0~using~solutions~and~ differential~consequences...G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7Ind eed,~div(fluxes)=0!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%7------------ ----------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%ho-----~Full~Fluxes~(m aybe~involving~arbitrary~const.~and/or~functions):~------G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%'Flux_xG,.*&)%#B2G\"\"#\"\"\",**&%\"yGF*%\$ _C2GF*F**&%\"zGF*%\$_C3GF*F*%\$_C4GF**&&%%_F10G6#%\"PGF*%#B1GF*!\"\"F*F9 *&)%#B3GF)F*F+F*F9*(F)F*F7F*F+F*F9*(F(F*,**&F0F*%\$_C1GF*F9*&%\"xGF*F.F *F9%\$_C5GF**&F4F*F(F*F9F*F8F*F**(FF6F@F(F(F7F (F(*&,(FHF(FIF(FJF6F(FDF(F(F(F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%' Flux_zG,.*(%#B1G\"\"\",**&%\"yGF(%\$_C2GF(F(*&%\"zGF(%\$_C3GF(F(%\$_C4GF( *&&%%_F10G6#%\"PGF(F'F(!\"\"F(%#B3GF(F(*(%#B2GF(,**&F.F(%\$_C1GF(F6*&% \"xGF(F,F(F6%\$_C5GF(*&F2F(F9F(F6F(F7F(F(*&)F'\"\"#F(,**&FF(F/F(F6*&F2F(F7F(F6F(F6*&)F9FCF(FDF(F6*(FCF(F5F(FDF(F6*&F7F(, *-F3F4F(*&,(F*F(F-F(F0F(F(F'F(F(*&,(F;F6F=F6F?F(F(F9F(F(*&,(FEF(FFF(FG F6F(F7F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%@------------------ -------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%SCases~separated~w. r.t.~free~constants~/~functions:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% @-------------------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#% D------------Case~C1~---------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&/-%(Lambda1G6)%\"xG%\"yG%\"zG%#B1G%#B2G%#B3G%\"PG\"\"!/-%(Lambda2GF &,\$F)!\"\"/-%(Lambda3GF&F(/-%(Lambda4GF&,&*&F)\"\"\"F+F " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {MARK "3 0 0" 122 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }