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"_cstyle86" -1 313 "T imes" 1 10 0 0 0 0 0 0 2 2 2 2 0 0 0 1 }{PSTYLE "_pstyle43" -1 246 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle44" -1 247 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 2 0 2 0 2 2 -1 1 }{PSTYLE "_pstyle45" -1 248 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 2 2 2 0 0 0 1 }0 0 0 -1 -1 -1 1 0 1 0 2 2 -1 1 }} {SECT 0 {PARA 233 "" 0 "" {TEXT 285 96 "diffalg - Package for differen tial elimination and analysis of differential systems (ODE or PDE)" }} {SECT 1 {PARA 234 "" 0 "" {TEXT 286 11 "Description" }}{PARA 235 "" 0 "" {TEXT 287 4 "The " }{TEXT 288 7 "diffalg" }{TEXT 287 321 " package \+ is a collection of routines to handle systems of polynomial differenti al equations and inequations. The functionalities include differential elimination, expansion of the solutions into formal power series and \+ analysis of singular solutions. The underlying theory and terminology \+ belongs to differential algebra." }}{PARA 236 "" 0 "" {TEXT 289 152 "B asic concepts in constructive differential algebra and their represent ation in the diffalg package are presented in a more substantial way i n the page " }{HYPERLNK 290 "diffalg[differential_algebra]" 2 "diffalg [differential_algebra]" "" }{TEXT 289 1 "." }}{PARA 236 "" 0 "" {TEXT 291 54 "This page gives an overview of the functionalities of " } {TEXT 292 7 "diffalg" }{TEXT 291 49 " illustrated by some basic exampl es. Pointers to " }{HYPERLNK 290 "diffalg[differential_algebra]" 2 "di ffalg[differential_algebra]" "" }{TEXT 291 40 " will be given for prec ise definitions. " }}{PARA 235 "" 0 "" {TEXT 287 296 "Everywhere in th is package differential equations are represented by the equivalent di fferential polynomials (i.e. the right hand sides - the left hand side s of the equations). We speak of a zero of a differential polynomial t o mean a meromorphic solution of the associated differential equation. " }}{PARA 236 "" 0 "" {TEXT 291 125 "The terms derivation variables an d differential indeterminates refer respectively to the independent an d dependent variables." }}{PARA 235 "" 0 "" {TEXT 287 39 "Central in t his package is the command " }{TEXT 288 18 "Rosenfeld_Groebner" } {TEXT 287 1 "." }}{PARA 236 "" 0 "" {TEXT 291 17 "In simple terms, " } {TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 226 " decomposes the zero set of a system of differential polynomials into the union of the non singular zeros of differential characteristic sets. Each of these dif ferential characteristic set defines a characterisable component. " }} {PARA 235 "" 0 "" {TEXT 287 66 "Differential characteristic sets, as t o be found in the output of " }{TEXT 288 18 "Rosenfeld_Groebner" } {TEXT 287 135 ", are kinds of canonical forms. Their non singular zero s are defined as the solutions of a set of equations plus a set of ine quations. " }}{PARA 236 "" 0 "" {TEXT 291 257 "A differential characte ristic set is minimal among the sets of differential polynomials admit ting the same non singular zero set. This entails that many structural properties of the non singular zero set can be read from the differen tial characteristic set." }}{PARA 235 "" 0 "" {TEXT 287 48 "For illust ration, the decomposition operated by " }{TEXT 288 18 "Rosenfeld_Groeb ner" }{TEXT 287 113 " on a single differential polynomial will split o ut the singular zeros, if there are any, from the general zero (" } {HYPERLNK 290 "example" 2 "" "singsol" }{TEXT 287 2 ")." }}{PARA 236 " " 0 "" {TEXT 291 42 "As for systems of differential equations, " } {TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 100 " will first detect i f the system bears inconsistency (contradiction) and therefore has no \+ solution (" }{HYPERLNK 290 "example" 2 "" "triviality" }{TEXT 291 3 ") . " }}{PARA 235 "" 0 "" {TEXT 287 2 "A " }{HYPERLNK 290 "ranking" 2 "d iffalg[differential_algebra]" "ranking" }{TEXT 287 207 " is an order o n the differential indeterminates and all their derivatives that is co mpatible with derivation. The computations and the differential charac teristic sets obtained depend on the chosen ranking. " }}{PARA 236 "" 0 "" {TEXT 291 134 "Various properties of the zero set of a system of \+ differential polynomials can be exhibited with an appropriate choice o f the ranking." }}{PARA 236 "" 0 "" {TEXT 291 70 "We can answer questi ons like: do the zeros of the given system satisfy" }}{PARA 236 "" 0 " " {TEXT 291 187 "- differential equations involving only a specific su bset of the differential indeterminates? Computing with an elimination ranking will uncover the minimal such equations if they exist (" } {HYPERLNK 290 "example" 2 "" "elimination_ranking" }{TEXT 291 2 ")." } }{PARA 236 "" 0 "" {TEXT 291 103 "- algebraic equations? Computing wit h an orderly ranking will uncover these constraints if they exist (" } {HYPERLNK 290 "example" 2 "" "orderly_ranking" }{TEXT 291 3 "). " }} {PARA 236 "" 0 "" {TEXT 291 165 "- an ordinary differential equation i n one of the derivation variable? Computing with a lexicographic ranki ng will uncover the minimal such equations if they exist (" } {HYPERLNK 290 "example" 2 "" "lexicographic_ranking" }{TEXT 291 2 ")." }}{PARA 236 "" 0 "" {TEXT 291 119 "The ranking to be used together wi th the derivation variables and the differential indeterminates are de fined with the " }{HYPERLNK 290 "differential_ring" 2 "diffalg[differe ntial_ring]" "" }{TEXT 291 44 " command. More examples are provided on its " }{HYPERLNK 290 "help page" 2 "diffalg[differential_ring]" "" } {TEXT 291 1 "." }}{PARA 235 "" 0 "" {TEXT 287 54 "An example of the an alysis that can be performed with " }{TEXT 288 20 "essential_component s" }{TEXT 287 61 " on the singular zeros of a differential polynomial \+ is given " }{HYPERLNK 290 "below" 2 "" "weierstrassp" }{TEXT 287 1 "." }}{PARA 235 "" 0 "" {TEXT 287 127 "Power series expansion of the non \+ singular zeros of a differential characteristic set can be performed w ith the functionnality " }{TEXT 288 21 "power_series_solution" }{TEXT 287 31 ". Examples are provided in its " }{HYPERLNK 290 "help page" 2 "diffalg[power_series_solution]" "" }{TEXT 287 1 "." }}}{SECT 1 {PARA 234 "" 0 "" {TEXT 286 48 "Typical uses and examples for Rosenfeld_Groe bner" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 14 "with(diffalg);" }}{PARA 238 "" 1 "" {XPPMATH 20 "6#7<%3Rosenfeld_GroebnerG%+belongs_to G%-delta_leaderG%1delta_polynomialG%'denoteG%,derivativesG%2differenti al_ringG%3differential_spremG%.differentiateG%*equationsG%5essential_c omponentsG%0field_extensionG%(greaterG%,inequationsG%(initialG%3initia l_conditionsG%.is_orthonomicG%'leaderG%6power_series_solutionG%7prepar ation_polynomialG%.print_rankingG%%rankG%(reducedG%-reduced_formG%.rew rite_rulesG%)separantG" }{TEXT 294 1 " " }{TEXT 294 1 " " }{TEXT 294 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 35 "Prior to any computations wit h the " }{TEXT 292 7 "diffalg" }{TEXT 291 187 " package, the appropria te differential indeterminates (dependent variables) and derivation va riables (independent variables) together with the ranking have to be d efined with the command " }{HYPERLNK 290 "differential_ring" 2 "diffal g[differential_ring]" "" }{TEXT 291 48 ". This command will issue a ta ble (appearing as " }{TEXT 292 8 "ODE_ring" }{TEXT 291 4 " or " } {TEXT 292 8 "PDE_ring" }{TEXT 291 65 ") that has to be used as a last \+ parameter to most other commands." }}{SECT 0 {PARA 234 "" 0 "" {TEXT 295 28 "Computing singular solutions" }}{PARA 236 "" 0 "" {TEXT 291 62 "Consider the following Clairaut partial differential equation:" }} {EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 74 "u(x,y) = x*diff(u(x,y),x )+y*diff(u(x,y),y)+diff(u(x,y),x)*diff(u(x,y),y) ;" }}{PARA 239 "" 1 " " {XPPMATH 20 "6#/-%\"uG6$%\"xG%\"yG,(*&F'\"\"\"-%%diffG6$F$F'F+F+*&F( F+-F-6$F$F(F+F+*&F,F+F0F+F+" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 47 "We represent it by the di fferential polynomial " }{TEXT 292 1 "p" }{TEXT 291 36 " with a more c ompact syntax, called " }{TEXT 292 3 "jet" }{TEXT 291 10 " notation." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 34 "p := -u[]+x*u[x]+y*u[y ]+u[x]*u[y];" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"pG,*&%\"uG6\"!\" \"*&%\"xG\"\"\"&F'6#F+F,F,*&%\"yGF,&F'6#F0F,F,*&F-F,F1F,F," }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 111 "Before any manipulation of this differential polynomial we define a differential polynomial ring it belongs to:" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 69 "R := differential_ring(ranking= [u], derivations=[x,y], notation=jet);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"RG%)PDE_ringG" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 92 "The Clairaut equation under c onsideration has a singular solution. This will be unveiled by " } {TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 16 ". The result of " } {TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 35 " is a list of tables \+ (appearing as " }{TEXT 292 15 "characterisable" }{TEXT 291 45 "), each defining a characterisable component." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 39 "Clairaut := Rosenfeld_Groebner(\{p\}, R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%)ClairautG7$%0characterisableGF&" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 108 "Each characterisable component is defined by a differen tial characteristic set that can be accessed via the " }{TEXT 292 9 "e quations" }{TEXT 291 181 " command. The non singular zeros of this dif ferential characteristic set are the zeros that do not make vanish som e additional differential polynomials that can be accessed via the " } {TEXT 292 11 "inequations" }{TEXT 291 9 " command." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 49 "equations(Clairaut[1]), inequations(C lairaut[1]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6$7#,*&%\"uG6\"!\"\"*&% \"xG\"\"\"&F&6#F*F+F+*&%\"yGF+&F&6#F/F+F+*&F,F+F0F+F+7#,&F*F+F0F+" } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 49 "equations(Clairaut[2]), inequations(Clair aut[2]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6$7#,&&%\"uG6\"\"\"\"*&%\"y GF(%\"xGF(F(7\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }} }{PARA 236 "" 0 "" {TEXT 291 153 "The second component represents the \+ singular solution of this Clairaut equation while the first represents the general solution of the Clairaut equation." }}}{SECT 0 {PARA 234 "" 0 "" {TEXT 295 53 "Detection of inconsistencies in a differential s ystem" }}{PARA 236 "" 0 "" {TEXT 291 50 "Consider the 3 following diff erential polynomials:" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 28 "p1:=diff(u(x,y),x)+v(x,y)-y;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%#p 1G,(-%%diffG6$-%\"uG6$%\"xG%\"yGF,\"\"\"-%\"vGF+F.F-!\"\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 26 "p2:=diff(u(x,y),y)+v(x,y);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%#p2G,&-%%diffG6$-%\"uG6$%\"xG%\"yGF-\"\"\"-%\"vGF+F." }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 34 "p3:=diff(v(x,y),x)-diff(v(x,y),y);" } }{PARA 239 "" 1 "" {XPPMATH 20 "6#>%#p3G,&-%%diffG6$-%\"vG6$%\"xG%\"yG F,\"\"\"-F'6$F)F-!\"\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 49 "They define the system of dif ferential equations " }{TEXT 292 26 "S : p1 = 0, p2 = 0, p3 = 0" } {TEXT 291 1 "." }}{PARA 236 "" 0 "" {TEXT 291 168 "Before any manipula tion of these differential polynomials we define a structure they belo ng to. We have the possibility to work with the Maple notation of deri vatives. " }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 72 "R := differ ential_ring(ranking=[u,v], derivations=[x,y], notation=diff);" }} {PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"RG%)PDE_ringG" }{TEXT 296 1 " " } {TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 32 "No te that we can convert to the " }{TEXT 292 3 "jet" }{TEXT 291 46 " not ation, introduced in the previous example:" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 65 "denote(p1, 'jet', R), denote(p2, 'jet', R), deno te(p3, 'jet', R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6%,(&%\"uG6#%\"xG \"\"\"&%\"vG6\"F(%\"yG!\"\",&&F%6#F,F(F)F(,&&F*F&F(&F*F0F-" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 10 "What does " }{TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 26 " tell us about the system " }{TEXT 292 1 "S" }{TEXT 291 2 "? " }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 34 "Rosenfeld_Groebner(\{ p1,p2,p3\}, R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 54 "This indicates that the system bears a contradiction: " }{TEXT 292 10 "p1, p2, p3" }{TEXT 291 21 " have no common zero." }}{PARA 236 "" 0 "" {TEXT 291 58 "In this case it is quite easy to detect the cont radiction." }}{PARA 236 "" 0 "" {TEXT 291 16 "Differentiating " } {TEXT 292 2 "p1" }{TEXT 291 14 " according to " }{TEXT 292 1 "y" } {TEXT 291 11 " we obtain:" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 24 "differentiate(p1, y, R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#, (-%%diffG6%-%\"uG6$%\"xG%\"yGF*F+\"\"\"-F%6$-%\"vGF)F+F,!\"\"F," } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 29 "The meromorphic solutions of " }{TEXT 292 1 "S" }{TEXT 291 131 ", if they exist, shall make this differential polynomial vani sh. They shall also make the following differential polynomial vanish \+ " }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 28 "differentiate(p2, x, R)- p3;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#,&-%%diffG6%-%\"uG6$%\"xG %\"yGF*F+\"\"\"-F%6$-%\"vGF)F+F," }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 66 "The two equations c an not be satisfied simultaneously: the system " }{TEXT 292 1 "S" } {TEXT 291 28 " bears here a contradiction " }{TEXT 292 3 "1=0" }{TEXT 291 34 " and therefore admits no solution." }}}{SECT 0 {PARA 234 "" 0 "" {TEXT 295 58 "Solving ordinary differential systems: Elimination ra nking" }}{PARA 236 "" 0 "" {TEXT 291 114 "To perform a chain resolutio n of a system of ordinary differential equations one shall use an elim ination ranking." }}{PARA 236 "" 0 "" {TEXT 291 62 "Consider the diffe rential system defined by the following set " }{TEXT 292 1 "S" }{TEXT 291 54 " of differential polynomials in the unknown functions " } {TEXT 292 14 "x(t),y(t),z(t)" }{TEXT 291 1 "." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 66 "S := [diff(x(t),t)-x(t)*(x(t)+y(t)),diff(y(t) ,t)+y(t)*(x(t)+y(t))," }{MPLTEXT 1 293 52 "\ndiff(x(t),t)^2+ diff(y(t) ,t)^2+ diff(z(t),t)^2-1 ];" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"SG7 %,&-%%diffG6$-%\"xG6#%\"tGF-\"\"\"*&F*F.,&F*F.-%\"yGF,F.F.!\"\",&-F(6$ F1F-F.*&F1F.F0F.F.,**$F'\"\"#F.*$F5F:F.*$-F(6$-%\"zGF,F-F:F.F3F." } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 35 "Using an elimination ranking where " }{TEXT 292 14 "z(t )>y(t)>x(t)" }{TEXT 291 89 " we will obtain differential characteristi c sets containing a differential polynomial in " }{TEXT 292 4 "x(t)" } {TEXT 291 46 " alone, a differential polynomial determining " }{TEXT 292 4 "y(t)" }{TEXT 291 13 " in terms of " }{TEXT 292 4 "x(t)" }{TEXT 291 51 " and finally a differential polynomial determining " }{TEXT 292 4 "z(t)" }{TEXT 291 13 " in terms of " }{TEXT 292 4 "y(t)" }{TEXT 291 5 " and " }{TEXT 292 4 "x(t)" }}{PARA 236 "" 0 "" {TEXT 291 88 "To see more clearly the dependencies of these differential polynomials, \+ one can use the " }{TEXT 292 13 "rewrite_rules" }{TEXT 291 41 " comman d, which is an alternative to the " }{TEXT 292 9 "equations" }{TEXT 291 9 " command." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 72 "R := differential_ring(ranking=[z,y,x], derivations=[t], notation=diff):" }{MPLTEXT 1 293 30 "\nG :=Rosenfeld_Groebner(S, R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"GG7$%0characterisableGF&" }{TEXT 296 1 " " } {TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 39 "rewrite_rules(G[1]); inequations(G[1]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/*$-%%diffG6$-%\"zG6#%\"tGF,\"\"#,$*&,**& -%\"xGF+\"\"%-F'6$F2F,F-F-*$F2F4!\"\"*$F5F4\"\"\"*&F5\"\"$F2F-!\"#F:F2 !\"%F8/-%\"yGF+*&,&F5F:*$F2F-F8F:F2F8/-F'6$F2-%\"$G6$F,F-,$*&F2F:F5F:F -" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7$-%\"xG6#%\"tG-%%diffG6$-%\"zGF&F'" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 39 "rewrite_rules(G[2]); inequations(G[2]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/*$-%%diffG6$-%\"zG6#%\"tGF,\"\"#,&\"\"\" F/*$-%\"yGF+\"\"%!\"\"/-F'6$F1F,,$*$F1F-F4/-%\"xGF+\"\"!" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7#-%%diffG6$-%\"zG6#%\"tGF*" }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 100 "We can perform a c hain resolution to both components of the result. Note that the singul ar solution " }{TEXT 292 6 "x(t)=0" }{TEXT 291 115 " has to be discard ed in the resolution of the first component, as indicated by the diffe rential polynomials of its " }{TEXT 292 11 "inequations" }{TEXT 291 37 ". The correct complete solution with " }{TEXT 292 6 "x(t)=0" } {TEXT 291 34 " is given by the second component." }}}{SECT 0 {PARA 234 "" 0 "" {TEXT 295 47 "Constrained systems: Orderly and mixed ranki ngs" }}{PARA 236 "" 0 "" {TEXT 291 100 "Consider the set of differenti al polynomials describing the motion of a pendulum, i.e. a point mass \+ " }{TEXT 292 1 "m" }{TEXT 291 39 " suspended at a massless rod of leng th " }{TEXT 292 1 "l" }{TEXT 291 32 " under the influence of gravity " }{TEXT 292 1 "g" }{TEXT 291 27 ", in Cartesian coordinates " }{TEXT 292 5 "(x,y)" }{TEXT 291 117 ". The Lagrangian formulation leads to tw o second order differential equations together with an algebraic const raint. " }{TEXT 292 1 "T" }{TEXT 291 31 " is the Lagrangian multiplier . " }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 83 "P := [m*diff(y(t), t,t)+T(t)*y(t)+g, m*diff(x(t),t,t)+T(t)*x(t), x(t)^2+y(t)^2-l^2];" }} {PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"PG7%,(*&%\"mG\"\"\"-%%diffG6$-%\" yG6#%\"tG-%\"$G6$F0\"\"#F)F)*&-%\"TGF/F)F-F)F)%\"gGF),&*&F(F)-F+6$-%\" xGF/F1F)F)*&F6F)F=F)F),(*$F=F4F)*$F-F4F)*$%\"lGF4!\"\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 90 "To be in a position to manipulate this system we have to declare t he additional constants " }{TEXT 292 7 "m, l, g" }{TEXT 291 65 " appea ring in the coefficients of these differential polynomials." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 54 "K := field_extension(transcende ntal_elements=[m,l,g]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"KG%-gr ound_fieldG" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}} {PARA 235 "" 0 "" {TEXT 288 15 "Orderly ranking" }}{PARA 236 "" 0 "" {TEXT 291 252 "If one wishes to find the lowest order differential pol ynomials vanishing on the zeros of the system modelling the pendulum a nd in particular all the algebraic constraints (i.e. the differential \+ polynomials of order 0) one shall use an orderly ranking." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 97 "RO := differential_ring(ranking =[[T,x,y]], derivations=[t], field_of_constants=K, notation=diff):" } {MPLTEXT 1 293 32 "\nGO:= Rosenfeld_Groebner(P, RO);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%#GOG7$%0characterisableGF&" }{TEXT 296 1 " " } {TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 41 "rewrite_rules(GO[1]); inequations(GO[1]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/-%%diffG6$-%\"TG6#%\"tGF+,$*(%\"gG\"\"\" -F&6$-%\"yGF*F+F/%\"lG!\"#!\"$/*$F0\"\"#*(,**&F2\"\"$F.F/!\"\"*(F2F/F. F/F4F9F/*(F(F/F4F9F2F9F>*&F(F/F4\"\"%F/F/%\"mGF>F4F5/*$-%\"xGF*F9,&*$F 2F9F>*$F4F9F/" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }} {PARA 239 "" 1 "" {XPPMATH 20 "6#7$-%\"xG6#%\"tG-%%diffG6$-%\"yGF&F'" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 41 "rewrite_rules(GO[2]); inequations(GO[ 2]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/-%\"TG6#%\"tG,$*(-%\"yGF' \"\"\"%\"gGF-%\"lG!\"#!\"\"/-%\"xGF'\"\"!/*$F+\"\"#*$F/F8" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7#-%\"yG6#%\"tG" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 38 "The second component in the out put of " }{TEXT 292 18 "Rosenfeld_Groebner" }{TEXT 291 254 " correspon ds to the equilibria of the pendulum. From the first component of the \+ output we see that the actual motion is described by two first order d ifferential equations together with a constraint: the solution depends on only two arbitrary constants. " }}{PARA 235 "" 0 "" {TEXT 288 13 " Mixed ranking" }}{PARA 236 "" 0 "" {TEXT 291 58 "If we wish to obtain \+ the differential system satisfied by " }{TEXT 292 1 "x" }{TEXT 291 5 " and " }{TEXT 292 1 "y" }{TEXT 291 27 " alone one shall eliminate " } {TEXT 292 1 "T" }{TEXT 291 1 ":" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 97 "RM := differential_ring(ranking=[T,[x,y]], derivati ons=[t], field_of_constants=K, notation=diff):" }{MPLTEXT 1 293 32 "\n GM:= Rosenfeld_Groebner(P, RM);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>% #GMG7$%0characterisableGF&" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 21 "rewrite_rul es(GM[1]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/-%\"TG6#%\"tG,$*(,(* (%\"mG\"\"\"-%%diffG6$-%\"yGF'F(\"\"#%\"lGF4!\"\"*&F2\"\"$%\"gGF.F6*(F 2F.F9F.F5F4F.F.F5!\"#,&*$F5F4F.*$F2F4F6F6F6/-F06$F2-%\"$G6$F(F4,$**,** &F2\"\"%F9F.F.*(F5F4F2F4F9F.F;*&F5FIF9F.F.**F2F.F-F.F/F4F5F4F.F.F-F6F5 F;F " 0 "" {MPLTEXT 1 293 21 "rewrite_rules(G M[2]);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7%/-%\"TG6#%\"tG,$*(-%\"yGF '\"\"\"%\"gGF-%\"lG!\"#!\"\"/-%\"xGF'\"\"!/*$F+\"\"#*$F/F8" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 78 "Note here that the motion is given by a second order dif ferential equation in " }{TEXT 292 4 "y(t)" }{TEXT 291 64 ". Then the \+ Lagrange multiplier is given explicitely in terms of " }{TEXT 292 4 "y (t)" }{TEXT 291 26 " and its first derivative." }}}{SECT 0 {PARA 234 " " 0 "" {TEXT 295 58 "Solving partial differential system: Lexicographi c ranking" }}{PARA 236 "" 0 "" {TEXT 291 232 "To solve overdetermined \+ systems of partial differential equations it is interesting to compute first the ordinary differential equations satisfied by the solutions \+ of the system. A lexicographic ranking has to be chosen to this end." }}{PARA 236 "" 0 "" {TEXT 291 33 "Consider the differential system " } {TEXT 292 1 "S" }{TEXT 291 86 " defining the infinitesimal generators \+ of the symmetry group of the Burgers equations:" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 69 "Burgers_pde := Diff(u(t,x),t)=Diff(u(t,x),x ,x)-u(t,x)*Diff(u(t,x),x);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%,Burg ers_pdeG/-%%DiffG6$-%\"uG6$%\"tG%\"xGF,,&-F'6$F)-%\"$G6$F-\"\"#\"\"\"* &F)F5-F'6$F)F-F5!\"\"" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 76 "with(liesymm): Bu rgersSymm := determine(Burgers_pde, `V`, u(t,x), [ux, ut]);" }}{PARA 240 "" 1 "" {TEXT 297 56 "Warning, the protected name close has been r edefined and" }{TEXT 297 12 "\nunprotected" }}{PARA 238 "" 1 "" {XPPMATH 20 "6#>%,BurgersSymmG<+/-%%DiffG6$-%#V3G6%%\"tG%\"xG%\"uG-%\" $G6$F/\"\"#,&-F(6%-%#V2GF,F/F.F3*&F/\"\"\"-F(6$F7F/F:F3/-F(6$-%#V1GF,F .\"\"!/-F(6$F@F/FB/-F(6$F7F0FB/-F(6$F@F0FB/-F(6%F@F/F.,$F;!\"\"/-F(6$F @F-,&-F(6$F@-F16$F.F3F:-F(6$F7F.F3/-F(6$F*FW,&-F(6$F*F-F:*&F/F:-F(6$F* F.F:F:/-F(6$F7FW,**&F/F:FYF:FP-F(6%F*F/F.F3-F(6$F7F-F:F*FP" }{TEXT 294 1 " " }{TEXT 294 1 " " }{TEXT 294 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 64 "Let us seek the ordinary differential equations with res pect to " }{TEXT 292 1 "t" }{TEXT 291 43 " satisfied by the solutions \+ of this system." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 84 "R := \+ differential_ring(ranking=[lex[V3,V2,V1]], derivations=[u,x,t], notati on=Diff);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"RG%)PDE_ringG" } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 64 "G := Rosenfeld_Groebner(map(eq->rhs(eq)-l hs(eq),BurgersSymm),R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"GG7#%0 characterisableG" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " } }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 20 "rewrite_rules(G[1]);" }}{PARA 238 "" 1 "" {XPPMATH 20 "6#7+/-%%DiffG6$-%#V3G6%%\"uG%\"xG%\"t GF+,$-F&6$-%#V1GF*F-#!\"\"\"\"#/-F&6$-%#V2GF*F+\"\"!/-F&6$F1F+F;/-F&6$ F(F,,$*&-F&6$F(F-\"\"\"F+F4F4/-F&6$F9F,,$F/#FFF5/-F&6$F1F,F;/-F&6$F(-% \"$G6$F-F5F;/-F&6$F1FR,$FC!\"#/-F&6$F9F-,&*&F+FFF/FFFKF(FF" }{TEXT 294 1 " " }{TEXT 294 1 " " }{TEXT 294 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 94 "We can see that there are three independent ordinary dif ferential polynomials with respect to " }{TEXT 292 1 "t" }{TEXT 291 81 " (the last three of the differential characteristic set). All othe r ordinary (in " }{TEXT 292 1 "t" }{TEXT 291 57 ") differential polyno mials vanishing on the solutions of " }{TEXT 292 11 "BurgersSymm" } {TEXT 291 90 " can be written as linear combination of these three one s together with their derivatives." }}{PARA 236 "" 0 "" }}{PARA 236 "" 0 "" }}{SECT 1 {PARA 234 "" 0 "weierstrassp" {TEXT 286 42 "Analysis o f singular solutions: an example" }}{PARA 236 "" 0 "" {TEXT 291 48 "Al l singular zeros are present in the output of " }{HYPERLNK 290 "Rosenf eld_Groebner" 2 "diffalg[Rosenfeld_Groebner]" "" }{TEXT 291 65 ". Only the essential singular zeros are present in the output of " } {HYPERLNK 290 "essential_components" 2 "diffalg[essential_components]" "" }{TEXT 291 1 "." }}{PARA 236 "" 0 "" {TEXT 291 39 "For first order differential equations:" }}{PARA 236 "" 0 "" {TEXT 291 70 "- the esse ntial singular zeros are envelopes of the non singular zeros" }}{PARA 236 "" 0 "" {TEXT 291 82 "- the other singular zeros are analytically \+ embedded among the non singular zeros." }}{PARA 236 "" 0 "" {TEXT 291 63 "We can illustrate this analysis on the differential polynomial " } {TEXT 292 2 "Wp" }{TEXT 291 6 " (cf. " }{HYPERLNK 290 "WeierstrassP" 2 "WeierstrassP" "" }{TEXT 291 2 "):" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 42 "Wp := diff(w(z),z)^2-4*w(z)^3+g2*w(z) +g3;" }} {PARA 239 "" 1 "" {XPPMATH 20 "6#>%#WpG,**$-%%diffG6$-%\"wG6#%\"zGF-\" \"#\"\"\"*$F*\"\"$!\"%*&%#g2GF/F*F/F/%#g3GF/" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 235 "" 0 "" {TEXT 287 14 "Generic \+ case (" }{TEXT 288 15 "g2^3 <> 27*g3^2" }{TEXT 287 3 ") :" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 14 "with(diffalg):" }{MPLTEXT 1 293 55 "\nK := field_extension(transcendental_elements=[g2,g3]):" } {MPLTEXT 1 293 88 "\nR := differential_ring(ranking=[w],derivations=[z ],notation=diff,field_of_constants=K):" }{MPLTEXT 1 293 39 "\nequation s(Rosenfeld_Groebner([Wp],R));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7$7 #,**$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F*\"\"$!\"%*&%#g2GF/F*F/F/% #g3GF/7#,(F0\"\"%F3!\"\"F5F9" }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 38 "equat ions(essential_components(Wp,R));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6# 7$7#,**$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F*\"\"$!\"%*&%#g2GF/F*F/ F/%#g3GF/7#,(F0\"\"%F3!\"\"F5F9" }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 32 "There are three sin gular zeros: " }{TEXT 292 7 "w(z)=ri" }{TEXT 291 8 ", where " }{TEXT 292 2 "ri" }{TEXT 291 26 " is one of the root of of " }{TEXT 292 13 "4 *r^3-g2*r-g3" }{TEXT 291 104 ". There are all three essential singular zeros. As such they are envelopes of the non-singular zeros of " } {TEXT 292 2 "Wp" }{TEXT 291 54 ". This can be observed by plotting the real zeros for " }{TEXT 292 10 "g2=4,g3=-1" }{TEXT 291 1 ":" }} {EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 76 "singsol := \{plot(\{fsol ve(-4*r^3+4*r-1,r)\},z=-3..3, color=red, thickness=2)\}:" }{MPLTEXT 1 293 89 "\ngalsol1 :=\{seq(plots[odeplot](dsolve(\{diff(w(z),z,z)-6*w(z )^2+2,w(ic)=-1,D(w)(ic)=-1.0\}," }{MPLTEXT 1 293 53 "\n w(z),type= numeric),-3..3,color=blue),ic=0..4)\}:" }{MPLTEXT 1 293 90 "\ngalsol2 \+ := \{seq(plots[odeplot](dsolve(\{diff(w(z),z,z)-6*w(z)^2+2,w(ic)=2,D( w)(ic)=-5.0\}," }{MPLTEXT 1 293 65 "\n w(z),type=numeric),ic-0.15. .ic+1.7,color=green),ic=-3..1)\}:" }{MPLTEXT 1 293 93 "\nplots[display ](singsol union galsol1 union galsol2,title=\"g2=4,g3=-1\",view=[-3..3 ,-3/2..3]):" }}}{PARA 235 "" 0 "" {TEXT 288 14 "g2^3 = 27*g3^2" }} {PARA 236 "" 0 "" {TEXT 291 34 "We parameterise the equation with " } {TEXT 292 1 "g" }{TEXT 291 1 ":" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 32 "Wps := subs(g2=3*g^2,g3=g^3,Wp);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%$WpsG,**$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F* \"\"$!\"%*&%\"gGF.F*F/F1*$F4F1F/" }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 51 "Ks := field_extension(transcendental_elements=[g]):" }{MPLTEXT 1 293 90 "\n Rs := differential_ring(ranking=[w],derivations=[z],notation=diff,fiel d_of_constants=Ks):" }{MPLTEXT 1 293 41 "\nequations(Rosenfeld_Groebne r([Wps],Rs));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7$7#,**$-%%diffG6$-% \"wG6#%\"zGF-\"\"#\"\"\"*$F*\"\"$!\"%*&%\"gGF.F*F/F1*$F4F1F/7#,(*$F4F. !\"\"*&F4F/F*F/F9*$F*F.F." }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 40 "equations(e ssential_components(Wps,Rs));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7$7# ,**$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F*\"\"$!\"%*&%\"gGF.F*F/F1*$ F4F1F/7#,&F4!\"\"F*F/" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 30 "There are two singular zeros: " }{TEXT 292 6 "w(z)=g" }{TEXT 291 5 " and " }{TEXT 292 9 "w(z)=-g/2" } {TEXT 291 2 ". " }{TEXT 292 6 "w(z)=g" }{TEXT 291 34 " is an essential singular zero of " }{TEXT 292 3 "Wps" }{TEXT 291 59 " and therefore i s an envelope of the non singular zeros of " }{TEXT 292 3 "Wps" } {TEXT 291 37 ". On the contrary, the singular zero " }{TEXT 292 10 "w( z) =-g/2" }{TEXT 291 4 " of " }{TEXT 292 3 "Wps" }{TEXT 291 80 " is no t essential and therefore is a limiting case of the non singular zeros of " }{TEXT 292 3 "Wps" }{TEXT 291 1 "." }}{PARA 236 "" 0 "" {TEXT 291 33 "We can observe the situation for " }{TEXT 292 4 "g=-1" }{TEXT 291 1 ":" }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 46 "singsol:=plo t(\{-1,1/2\},color=red,thickness=2):" }{MPLTEXT 1 293 84 "\ngalsol1:=p lot(\{seq(-1+3/2*tanh(1/2*(C-z)*sqrt(6))^2,C=-1..1)\},z=-3..3,color=gr een):" }{MPLTEXT 1 293 83 "\ngalsol2:=plot(\{seq(-1+3/2*coth(1/2*(C-z) *sqrt(6))^2,C=-1..1)\},z=-3..3,color=blue):" }{MPLTEXT 1 293 83 "\nplo ts[display](\{singsol,galsol1,galsol2\},view=[-3..3,-3/2..3],title=\"g 2=3,g3=-1\"):" }}}{PARA 235 "" 0 "" {TEXT 288 10 "g2=0, g3=0" }}{PARA 236 "" 0 "" {TEXT 291 20 "The equation is now:" }}{EXCHG {PARA 237 "> \+ " 0 "" {MPLTEXT 1 293 24 "Wp0:=subs(g2=0,g3=0,Wp);" }}{PARA 239 "" 1 " " {XPPMATH 20 "6#>%$Wp0G,&*$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F*\" \"$!\"%" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 67 "R0 := differential_ring(ranking =[w],derivations=[z],notation=diff):" }{MPLTEXT 1 293 41 "\nequations( Rosenfeld_Groebner([Wp0],R0));" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#7$7 #,&*$-%%diffG6$-%\"wG6#%\"zGF-\"\"#\"\"\"*$F*\"\"$!\"%7#F*" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 40 "equations(essential_components(Wp0,R0));" }} {PARA 239 "" 1 "" {XPPMATH 20 "6#7#7#,&*$-%%diffG6$-%\"wG6#%\"zGF-\"\" #\"\"\"*$F*\"\"$!\"%" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{PARA 236 "" 0 "" {TEXT 291 33 "There is only one singular zero, " }{TEXT 292 6 "w(z)=0" }{TEXT 291 99 ", and it is not essential. We \+ can see its limiting property with respect to the non singular zeros." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 49 "singsol:=plot(0,s=-4. .4, color=red, thickness=2):" }{MPLTEXT 1 293 51 "\ngalsol:= plot(\{se q(1/((z-C)^2),C=-2..1)\},z=-4..4):" }{MPLTEXT 1 293 55 "\nplots[displa y](\{singsol,galsol\},view=[-4..4,-1/2..2]):" }}}{PARA 235 "" 0 "" {TEXT 287 29 "More generally, the commands " }{HYPERLNK 290 "essential _components" 2 "diffalg[essential_components]" "" }{TEXT 287 6 ", and \+ " }{HYPERLNK 290 "preparation_polynomial" 2 "diffalg[preparation_polyn omial]" "" }{TEXT 287 112 ", allows one to interpret some qualitative \+ feature of singular zeros. See their help pages for further examples." }}{PARA 235 "" 0 "" {TEXT 287 12 "Reference : " }{HYPERLNK 290 "[H99] " 2 "diffalg" "biblio" }{TEXT 287 1 " " }}}{SECT 1 {PARA 241 "" 0 "" {TEXT 298 47 "Derivations with non trivial commutation rules " }{TEXT 299 5 "[NEW]" }}{PARA 237 "" 0 "" {TEXT 300 203 "Change of derivations can provide computational advantages. Also some problems are more n aturally expressed in terms of derivations other than the partial der ivtions w.r.t. the independant variables. " }{TEXT 300 140 "\nThe chan ge of derivations does not need to come from a change of variables and the appropriate derivations for a problem need not commute. " }{TEXT 300 156 "\nGenerally the commutator of two derivations is a linear com bination of the derivations, the coefficients of the combination being differential polynomials." }{TEXT 300 15 "\nReference : " } {HYPERLNK 290 "[H05]" 2 "diffalg" "biblio" }{TEXT 300 3 " . " }}{SECT 1 {PARA 242 "" 0 "" {TEXT 301 62 "An equivalence problem : linearisabi lity of a second order ODE" }{TEXT 301 1 "\n" }}{PARA 243 "" 0 "" {TEXT 302 72 "For description of the problem and of the approach for r esolution see " }{TEXT 302 112 "[S. Neut, 2003 : Implantation et nou velles application de la methode d'equivalence de Cartan, PhD thesis, \+ Lille]" }{TEXT 303 36 "\nWe consider a second order ODE : " } {XPPEDIT 18 0 "Diff(y,x,x)= f(x,y,Diff(y,x))" "6#/-%%DiffG6%%\"yG%\"xG F(-%\"fG6%F(F'-F%6$F'F(" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 20 " that we rewrite as " }{XPPEDIT 18 0 "y2 = f(x,y0 ,y1)" "6#/%#y2G-%\"fG6%%\"xG%#y0G%#y1G" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 2 " ." }{TEXT 304 24 "\nWe search cond ition on " }{TEXT 305 1 "f" }{TEXT 304 39 " for this equation to be eq uivalent to " }{XPPEDIT 2 0 "y2=0" "6#/I#y2G6\"\"\"!" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 25 " by a change of variable " }{XPPEDIT 2 0 "x= xi(x,y), y=eta0(x,y)" "6#6$/I\"xG6\"-I#xiGF&6$F%I\"yGF&/F*-I%e ta0GF&F)" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 1 "." }{TEXT 303 1 "\n" }{TEXT 303 144 "\nThe name y0 and y1 correspond to partial \+ derivation with respect to those variables. The derivation defined by \+ the index x is the derivation " }{XPPEDIT 18 0 "proc (p) options ope rator, arrow; Diff(p,x)+y1*Diff(p,y[0])+f*Diff(p,y1) end proc;" "6#f*6 #%\"pG7\"6$%)operatorG%&arrowG6\",(-%%DiffG6$F%%\"xG\"\"\"*&%#y1GF0-F- 6$F%&%\"yG6#\"\"!F0F0*&%\"fGF0-F-6$F%F2F0F0F*F*F*" }{TEXT 304 1 " " } {TEXT 304 1 " " }{TEXT 304 2 " ." }}{PARA 243 "" 0 "" {TEXT 303 35 "We consider a second order ODE : " }{XPPEDIT 18 0 "Diff(y,x,x)= f(x,y, Diff(y,x))" "6#/-%%DiffG6%%\"yG%\"xGF(-%\"fG6%F(F'-F%6$F'F(" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 303 20 " that we re write as " }{XPPEDIT 18 0 "y2 = f(x,y0,y1)" "6#/%#y2G-%\"fG6%%\"xG%#y0 G%#y1G" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 303 2 " ." }{TEXT 303 24 "\nWe search condition on " }{TEXT 306 1 "f" } {TEXT 303 39 " for this equation to be equivalent to " }{XPPEDIT 2 0 " y2=0" "6#/I#y2G6\"\"\"!" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 303 25 " by a change of variable " }{XPPEDIT 2 0 "x= xi(x,y), y=eta0(x,y)" "6#6$/I\"xG6\"-I#xiGF&6$F%I\"yGF&/F*-I%eta0GF&F)" }{TEXT 304 1 " " } {TEXT 304 1 " " }{TEXT 303 1 "." }{TEXT 303 71 "\nFor description of t he problem and of the approach for resolution see " }}{PARA 243 "" 0 " " {TEXT 303 115 " [S. Neut, 2003 : Implantation et nouvelles applica tion de la methode d'equivalence de Cartan, PhD thesis, Lille]" } {TEXT 303 1 "\n" }{TEXT 303 144 "\nThe name y0 and y1 correspond to pa rtial derivation with respect to those variables. The derivation defin ed by the index x is the derivation " }{XPPEDIT 18 0 "proc (p) optio ns operator, arrow; Diff(p,x)+y1*Diff(p,y[0])+f*Diff(p,y1) end proc;" "6#f*6#%\"pG7\"6$%)operatorG%&arrowG6\",(-%%DiffG6$F%%\"xG\"\"\"*&%#y1 GF0-F-6$F%&%\"yG6#\"\"!F0F0*&%\"fGF0-F-6$F%F2F0F0F*F*F*" }{TEXT 304 1 " " }{TEXT 304 1 " " }{TEXT 304 2 " ." }}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 80 "R := differential_ring(ranking=[eta2,eta1,[eta0,xi] ,[f]],derivations=[x,y0,y1], " }{MPLTEXT 1 293 76 "\n \+ commutations=[[y1,x]=[0,1,f[y1]],[y0,x]=[0,0,f[y0]]]):" }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 63 "eta0[x, y1] = differentiate(dif ferentiate( eta0, y1, R), x, R);" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/ &%%eta0G6$%\"xG%#y1GF$" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 38 "eta0[y1, x] = r educe(eta0[y1, x], R) ;" }}{PARA 239 "" 1 "" {XPPMATH 20 "6#/&%%eta0G6 $%#y1G%\"xG,(&F%6$F(F'\"\"\"&F%6#%#y0GF,*&&%\"fG6#F'F,&F%F3F,F," } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 70 "S := [eta2, seq( xi[x]*eta||(i+1)-eta||i[ x],i=0..1), eta0[y1],xi[y1]];" }{MPLTEXT 1 293 53 "\nH := [xi[x], xi[ x]*eta0[y0]-xi[y0]*eta0[x], f[y1]];" }}{PARA 239 "" 1 "" {XPPMATH 20 " 6#>%\"SG7'%%eta2G,&*&&%#xiG6#%\"xG\"\"\"%%eta1GF-F-&%%eta0GF+!\"\",&*& F)F-F&F-F-&F.F+F1&F06#%#y1G&F*F6" }{TEXT 296 1 " " }{TEXT 296 1 " " } {TEXT 296 1 " " }}{PARA 239 "" 1 "" {XPPMATH 20 "6#>%\"HG7%&%#xiG6#%\" xG,&*&F&\"\"\"&%%eta0G6#%#y0GF,F,*&&F'F/F,&F.F(F,!\"\"&%\"fG6#%#y1G" } {TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 34 "CR :=Rosenfeld_Groebner(S , H, R);" }} {PARA 239 "" 1 "" {XPPMATH 20 "6#>%#CRG7$%0characterisableGF&" }{TEXT 296 1 " " }{TEXT 296 1 " " }{TEXT 296 1 " " }}}{EXCHG {PARA 237 "> " 0 "" {MPLTEXT 1 293 35 "for c in CR do rewrite_rules(c) od;" }}{PARA 238 "" 1 "" {XPPMATH 20 "6#70/&%%eta2G6\"\"\"!/&%%eta1GF'*&&%%eta0G6#% \"xG\"\"\"&%#xiGF/!\"\"/&F.6$F0F0,$*&*&F-F1,&*&&%\"fG6#%#y1GF1&F.6#%#y 0GF1F1&F.6$F0FCF1F1F1FAF4\"\"#/&F3F7,$*&*&F2F1F;F1F1FAF4FF/&F.6$FCFC,$ *&FAF1&F>6$F@F@F1F4/&F3FBF(/&F.F?F(/&F3F?F(/&F>6%FCFCFC,&*&FQF1&F>FNF1 F1*$)&F>6$FCF@FFF1F4/&F>6%F0FCF@,(FhnFF*(FFF1FQF1&F>FBF1F4*(FFF1F=F1F[ oF1F1/&F>6%FCFCF@F(/&F>6%F0F@F@F[o/&F>6%FCF@F@F(/&F>6%F@F@F@F(" } {TEXT 294 1 " " }{TEXT 294 1 " " }{TEXT 294 1 " " }}{PARA 238 "" 1 "" {XPPMATH 20 "6#71/&%%eta2G6\"\"\"!/&%%eta1GF'*&&%%eta0G6#%\"xG\"\"\"&% #xiGF/!\"\"/&F36%F0%#y0GF8,$*&,0*(F2F1)&F36#F8\"\"#F1)&%\"fG6$%#y1GFEF @F1!\"$*(\"\")F1)F>\"\"$F1&FC6$F8FEF1F4*,\"\"'F1F2F1&F36$F8F8F1FBF1F>F 1F4*(FJF1F2F1)FOF@F1F4*(F@F1FIF1&FC6%F0FEFEF1F1**FNF1F=F1FBF1&F36$F0F8 F1F1**\"#7F1FOF1FWF1F>F1F1F1*$F=F1F4#F1FN/&F36%F8F8F8,$*&,&*&F=F1FAF1F 4*&FJF1FRF1F1F1F>F4#F1F@/&F.6$F0F0,$*&*&F-F1,**&F2F1FOF1F1*(F2F1FBF1F> F1F1*(F@F1F=F1&FC6#FEF1F4*(F@F1F>F1FWF1F4F1F1FenF4F4/&F3Fbo,$*&*&F2F1F foF1F1FenF4F4/&F.FX,$*&,,*(F-F1FBF1F=F1F1*(F>F1F-F1FOF1F1**F@F1F>F1&F. F?F1FWF1F1**F>F1F2F1FjpF1FBF1F4*(F2F1FOF1FjpF1F4F1FenF4F_o/&F.FP*&*&FO F1FjpF1F1F>F4/&F.F[pF(/&F3F[pF(/&FC6&F0F0FEFE,,&FC6%F0F8FE\"\"%*(FHF1F joF1FKF1F4*&FNF1&FCFPF1F4*(F@F1FjoF1FTF1F1*(FNF1FBF1&FCF?F1F1/&FCFin,0 *(FBF1FjoF1FKF1!\"%*&#F[rFJF1*$)FKF@F1F1F4*&#F1FJF1*&FKF1FTF1F1F4*(F@F 1FBF1FiqF1F1*&#F@FJF1)FTF@F1F1*(F[rF1FAF1FarF1F1*(FJF1FBF1F^rF1F4/&FC6 %F8F8FE*&FBF1,&FKF4FTF1F1/&FC6%F8FEFEF(/&FC6%FEFEFEF(" }{TEXT 294 1 " \+ " }{TEXT 294 1 " " }{TEXT 294 1 " " }}}}}{SECT 0 {PARA 233 "" 0 "" {TEXT 307 12 "Bibliography" }}{PARA 244 "" 0 "biblio" {TEXT 308 10 "\n [BLOP95] " }{TEXT 309 6 " " }{TEXT 310 173 " Boulier, F., Lazar d, D., Ollivier, F., Petitot, M. Computing Representations for Radica ls of Finitely Generated Differential Ideals. Proceedings of ISSAC'95 , ACM Press." }{TEXT 308 12 "\n[BLOP97] " }{TEXT 309 6 " " } {TEXT 310 169 "Boulier, F., Lazard, D., Ollivier, F., Petitot, M. Comp uting Representations for Radicals of Finitely Generated Differential \+ Ideals. Technical Report IT-306, LIFL. 1997." }{TEXT 308 6 "\n[H99]" } {TEXT 309 9 " " }{TEXT 310 144 "Hubert, E. Essential Component s of Algebraic Differential Equations. Journal of Symbolic Computation s (1999) volume 28 number 4-5 pages 657-680." }{TEXT 308 6 "\n[H00]" } {TEXT 309 7 " " }{TEXT 310 156 " Hubert, E. Factorization Free D ecomposition Algorithms in Differential Algebra. Journal of Symbolic C omputations (2000) volume 29 number 4-5 pages 641-662." }{TEXT 308 10 "\n[BLMM01] " }{TEXT 309 7 " " }{TEXT 310 88 " Boulier, F., Lema ire, F. and Moreno-Maza, M. PARDI! Proceedings of ISSAC'01, ACM Press. " }{TEXT 308 8 "\n[H03p] " }{TEXT 309 7 " " }{TEXT 310 243 " Hub ert, E. Notes on triangular sets and triangulation-decomposition algor ithms I: Polynomial systems. Chapter of Symbolic and Numerical Scienti fic Computations Edited by U. Langer and F. Winkler. LNCS, volume 2630 , Springer-Verlag Heidelberg." }{TEXT 308 8 "\n[H03d] " }{TEXT 309 6 " " }{TEXT 310 247 " Hubert, E. Notes on triangular sets and tria ngulation-decomposition algorithms II: Differential Systems. Chapter o f Symbolic and Numerical Scientific Computations Edited by U. Langer a nd F. Winkler. LNCS, volume 2630, Springer-Verlag Heidelberg." }{TEXT 308 6 "\n[H04]" }{TEXT 309 8 " " }{TEXT 310 161 " Hubert, E. Im provements to a triangulation-decomposition algorithm for ordinary dif ferential systems in higher degree cases, Proceedings of ISSAC'04, ACM Press." }{TEXT 308 7 "\n[H05] " }{TEXT 309 8 " " }{TEXT 310 130 "Hubert, E. Differential Algebra for Derivations with Nontrivial C ommutation Rules, Journal of Pure and Applied Algebra, to appear." }}} {SECT 1 {PARA 234 "" 0 "" {TEXT 286 15 "Functionalities" }}{PARA 245 " " 0 "" {TEXT 311 37 "Set up functions, notational aspects:" }}{PARA 236 "" 0 "" {HYPERLNK 290 "differential_ring" 2 "diffalg[differential_ ring]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "field_extension" 2 "diffal g[field_extension]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "print_ranking " 2 "diffalg[print_ranking]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "deno te" 2 "diffalg[denote]" "" }{TEXT 291 1 "." }}{PARA 245 "" 0 "" {TEXT 311 26 "Differential elimination: " }}{PARA 236 "" 0 "" {HYPERLNK 290 "Rosenfeld_Groebner" 2 "diffalg[Rosenfeld_Groebner]" "" }{TEXT 291 2 " , " }{HYPERLNK 290 "equations" 2 "diffalg[equations]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "inequations" 2 "diffalg[equations]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "rewrite_rules" 2 "diffalg[equations]" "" } {TEXT 291 3 ", " }{HYPERLNK 290 "reduce" 2 "diffalg[reduce]" "" } {TEXT 291 3 ", " }{HYPERLNK 290 "is_prime" 2 "diffalg[is_prime]" "" } {TEXT 291 1 "." }{TEXT 312 3 "\n( " }{XPPEDIT 277 0 "is_orthonomic" "6 #%.is_orthonomicG" }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 312 18 " obsoleted out by " }{XPPEDIT 276 0 "is_prime" "6#%)is_ primeG" }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 312 2 "; " }{XPPEDIT 275 0 "belongs_to, differential_sprem " "6$%+belongs_ toG%3differential_spremG" }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 312 17 "obsoleted out by " }{XPPEDIT 272 0 "reduce" " 6#%'reduceG" }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " } {TEXT 312 1 ")" }}{PARA 245 "" 0 "" {TEXT 311 23 "Power series solutio ns:" }}{PARA 236 "" 0 "" {HYPERLNK 290 "power_series_solutions" 2 "dif falg[power_series_solutions]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "ini tial_conditions" 2 "diffalg[initial_conditions]" "" }{TEXT 291 2 " ." }}{PARA 245 "" 0 "" {TEXT 311 30 "Analysis of singular solutions" }} {PARA 236 "" 0 "" {HYPERLNK 290 "essential_components" 2 "diffalg[esse ntial_components]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "preparation_po lynomial" 2 "diffalg[preparation_polynomial]" "" }{TEXT 291 1 "." }} {PARA 245 "" 0 "" {TEXT 311 48 "Function to manipulate differential po lynomials:" }}{PARA 236 "" 0 "" {HYPERLNK 290 "initial" 2 "diffalg[lea der]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "leader" 2 "diffalg[leader]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "separant" 2 "diffalg[leader]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "rank" 2 "diffalg[leader]" "" } {TEXT 291 2 ", " }{HYPERLNK 290 "differentiate" 2 "diffalg[differentia te]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "derivatives" 2 "diffalg[deri vatives]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "delta_polynomial" 2 "di ffalg[delta_polynomial]" "" }{TEXT 291 2 ", " }{HYPERLNK 290 "greater" 2 "diffalg[greater]" "" }{TEXT 291 1 "." }{TEXT 312 3 "\n( " } {XPPEDIT 269 0 "reduced, reduced_form" "6$%(reducedG%-reduced_formG" } {TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 312 18 " obso leted out by " }{XPPEDIT 259 0 "reduce" "6#%'reduceG" }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 313 1 " " }{TEXT 312 1 ")" }}{PARA 235 "" 0 " " {TEXT 287 9 "To use a " }{TEXT 288 7 "diffalg" }{TEXT 287 61 " funct ion, either load that function alone using the command " }{TEXT 288 23 "with(diffalg, function)" }{TEXT 287 50 ", or load all diffalg func tions using the command " }{TEXT 288 13 "with(diffalg)" }{TEXT 287 57 ". Alternatively, invoke the function using the long form " }{TEXT 288 17 "diffalg[function]" }{TEXT 287 146 ". This long form notation i s necessary whenever there is a conflict between a package function na me and another function used in the same session." }}{PARA 235 "" 0 "" {TEXT 287 50 "For more information on a particular function see " } {TEXT 288 17 "diffalg[function]" }{TEXT 287 2 ". " }}}{SECT 0 {PARA 234 "" 0 "seealso" {TEXT 286 10 "See Also: " }}{PARA 237 "" 0 "" {HYPERLNK 290 "diffalg[differential_algebra]" 2 "diffalg[differential_ algebra]" "" }{TEXT 300 2 ". " }}}{PARA 246 "" 0 "" }{PARA 247 "" 0 "" }{PARA 248 "" 0 "" }}{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }