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1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 257 1 {CSTYLE "" -1 -1 "Time s" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "List Item" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Heading 3" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List Item" -1 260 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 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "List Item" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 3 3 1 0 1 0 2 2 14 5 }{PSTYLE "Normal " -1 262 1 {CSTYLE "" -1 -1 "Courier" 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 "Normal" -1 263 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 "Normal" -1 264 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "List It em" -1 265 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }3 1 0 0 3 3 1 0 1 0 2 2 14 5 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 52 "Computing Mathieu function solutions for line ar ODEs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 256 17 "E.S. Cheb-Terrab\n" }{TEXT 257 39 "MITACS - CECM, \+ Simon Fraser University." }}{PARA 256 "" 0 "" {TEXT 258 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 47 "This talk pre sents an algorithm for computing " }{XPPEDIT 18 0 "Mathieu(a,q,x)" "6# -%(MathieuG6%%\"aG%\"qG%\"xG" }{TEXT -1 110 " function solutions for l inear ODEs, by solving an \"equivalence\" (when it exists) between any given linear ODE" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "`y''`+c[1](x)*`y '`+c[0](x)*y = 0;" "6#/,(%$y''G\"\"\"*&-&%\"cG6#F&6#%\"xGF&%#y'GF&F&*& -&F*6#\"\"!6#F-F&%\"yGF&F&F3" }{TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 19 "and Mathieu's ODE, " }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "`y''`+( a-2*q*cos(2*x))*y = 0;" "6#/,&%$y''G\"\"\"*&,&%\"aGF&*(\"\"#F&%\"qGF&- %$cosG6#*&F+F&%\"xGF&F&!\"\"F&%\"yGF&F&\"\"!" }{TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 21 "under transformations" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "proc (x) options operator, arrow; (A*x^k+B)/(C*x^k+D) e nd proc;" "6#f*6#%\"xG7\"6$%)operatorG%&arrowG6\"*&,&*&%\"AG\"\"\")F%% \"kGF/F/%\"BGF/F/,&*&%\"CGF/)F%F1F/F/%\"DGF/!\"\"F*F*F*" }{TEXT -1 8 " , " }{XPPEDIT 18 0 "y -> P(x)*y" "6#f*6#%\"yG7\"6$%)operatorG%&a rrowG6\"*&-%\"PG6#%\"xG\"\"\"F%F0F*F*F*" }{TEXT -1 1 "," }}{PARA 14 " " 0 "" {TEXT -1 0 "" }}{PARA 14 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "\{A,B,C,D, k\}" "6#<'%\"AG%\"BG%\"CG%\"DG%\"kG" }{TEXT -1 31 " a re constants with respect to " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "P(x)" "6#-%\"PG6#%\"xG" }{TEXT -1 107 " is \+ any not zero function . The algorithm includes computing the values of the Mathieu function parameters " }{XPPEDIT 18 0 "\{a,q\}" "6#<$%\"aG %\"qG" }{TEXT -1 39 " such that the equivalence is possible." }}{PARA 258 "" 0 "" {TEXT -1 62 "_____________________________________________ _________________" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 26 "Mathieu fun ctions - review" }}{PARA 15 "" 0 "" {TEXT -1 22 "The Mathieu functions " }{XPPEDIT 18 0 "MathieuC(a,q,x);" "6#-%)MathieuCG6%%\"aG%\"qG%\"xG " }{TEXT -1 7 " and " }{XPPEDIT 18 0 "MathieuS(a,q,x);" "6#-%)Mathie uSG6%%\"aG%\"qG%\"xG" }{TEXT -1 18 " are solutions of " }{TEXT 261 19 "Mathieu's equation\n" }{TEXT -1 0 "" }}{PARA 264 "" 0 "" {XPPEDIT 18 0 "`y''`+(a-2*q*cos(2*x))*y = 0;" "6#/,&%$y''G\"\"\"*&,&%\"aGF&*(\"\"# F&%\"qGF&-%$cosG6#*&F+F&%\"xGF&F&!\"\"F&%\"yGF&F&\"\"!" }{TEXT -1 0 " " }}{PARA 262 "" 0 "" {TEXT -1 0 "" }}{PARA 260 "" 0 "" {TEXT -1 7 "wh ere " }{XPPEDIT 18 0 "a" "6#%\"aG" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "q" "6#%\"qG" }{TEXT -1 54 " are arbitrary parameters. MathieuC \+ and MathieuS are " }{TEXT 262 4 "even" }{TEXT -1 5 " and " }{TEXT 263 3 "odd" }{TEXT -1 24 " functions respectively." }}{PARA 14 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 18 "Mathieu functions " } {TEXT 259 19 "cannot be expressed" }{TEXT -1 66 " in terms of elementa ry or hypergeometric functions for arbitrary " }{XPPEDIT 18 0 "\{a,q\} " "6#<$%\"aG%\"qG" }{TEXT -1 2 ". " }}{PARA 14 "" 0 "" {TEXT -1 0 "" } }{PARA 15 "" 0 "" {TEXT -1 11 "Given some " }{TEXT 267 5 "(a,q)" } {TEXT -1 42 ", three different situations can happen:\n\n" }{TEXT 264 2 "1." }{TEXT -1 9 " Neither " }{XPPEDIT 18 0 "MathieuC(a,q,x);" "6#-% )MathieuCG6%%\"aG%\"qG%\"xG" }{TEXT 268 4 " nor" }{TEXT -1 1 " " } {XPPEDIT 18 0 "MathieuS(a,q,x);" "6#-%)MathieuSG6%%\"aG%\"qG%\"xG" } {TEXT -1 16 " is periodic in " }{TEXT 298 1 "x" }{TEXT -1 39 " but a l inear combination of them is;\n\n" }{TEXT 265 2 "2." }{TEXT -1 50 " Ma thieu's equation admits a solution with period " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%# PiGF%" }{TEXT -1 18 " and only one of " }{XPPEDIT 18 0 "MathieuC(a,q, x);" "6#-%)MathieuCG6%%\"aG%\"qG%\"xG" }{TEXT 269 3 " or" }{TEXT -1 1 " " }{XPPEDIT 18 0 "MathieuS(a,q,x);" "6#-%)MathieuSG6%%\"aG%\"qG%\"xG " }{TEXT -1 38 " is periodic - the other one is not;\n\n" }{TEXT 266 2 "3." }{TEXT -1 49 " Mathieu's equation admits solutions with period \+ " }{XPPEDIT 18 0 "2*n*Pi;" "6#*(\"\"#\"\"\"%\"nGF%%#PiGF%" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "n >= 2" "6#1\"\"#%\"nG" }{TEXT -1 33 " is a positive integer, and both " }{XPPEDIT 18 0 "MathieuC(a,q,x);" "6#- %)MathieuCG6%%\"aG%\"qG%\"xG" }{TEXT 270 6 " and " }{TEXT -1 1 " " } {XPPEDIT 18 0 "MathieuS(a,q,x);" "6#-%)MathieuSG6%%\"aG%\"qG%\"xG" } {TEXT -1 14 " are periodic." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 15 "" 0 "" {TEXT -1 112 "These functions frequently appear in the solu tions to problems involving periodic potentials or elliptic shapes." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 7 "Example " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 230 "Any \+ oscillating system subject to a periodic external force equivalent to \+ having a variation with time of the parameters of the system. For inst ance: a pendulum where the suspension point moves periodically over th e vertical axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "These systems can be represented by the well know oscilla tor equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "`y''`+omega(t)*y = 0;" "6#/,&%$y''G\"\"\"*&-%&omegaG6#% \"tGF&%\"yGF&F&\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 68 "where the novelty consists of having the frequency of oscillations, " }{XPPEDIT 18 0 "omega;" "6#%&omegaG" } {TEXT -1 33 ", as a periodic function of time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 41 "An Equivalence a pproach for Mathieu's ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 263 "" 0 "" {TEXT 271 17 "Problem proposed:" }{TEXT 272 1 "\n" }} {PARA 261 "" 0 "" {TEXT -1 58 "Given a linear second order ODE, compu te its equivalence " }{TEXT 274 1 "(" }{TEXT -1 14 "when it exists" } {TEXT 275 1 ")" }{TEXT -1 28 " to the Mathieu ODE, under " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "x = (A*t^k+B)/(C*t^k+D),` \+ y`(x) = P(t)*u(t);" "6$/%\"xG*&,&*&%\"AG\"\"\")%\"tG%\"kGF)F)%\"BG F)F),&*&%\"CGF))F+F,F)F)%\"DGF)!\"\"/-%)~~~~~~~yG6#F$*&-%\"PG6#F+F)-% \"uG6#F+F)" }{TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{PARA 261 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 276 0 "\{t, u(t)\};" "6#<$% \"tG-%\"uG6#F$" }{TEXT -1 24 " are the new variables, " }{XPPEDIT 277 0 "\{A, B, C, D, k\};" "6#<'%\"AG%\"BG%\"CG%\"DG%\"kG" }{TEXT -1 32 " \+ are constants with respect to " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 279 0 "P(t);" "6#-%\"PG6#%\"tG" }{TEXT -1 27 " \+ is an arbitrary function. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 8 "Diagram:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 264 "" 0 "" {TEXT -1 65 " (Mathieu) >---------------Moebius --- -------------> (M_Mathieu)" }}{PARA 264 "" 0 "" {TEXT -1 265 " . \+ |\n . \+ |\n . TR = ? \+ x = t^k | \n . |\n . |" }}{PARA 264 "" 0 "" {TEXT -1 52 " \\/ \\/ " }}{PARA 264 "" 0 "" {TEXT -1 61 " Given ODE <------------ y(x) = P(t )u(t)-----------< (P_ODE)\n" }}{PARA 263 "" 0 "" {TEXT -1 0 "" }} {PARA 15 "" 0 "" {TEXT -1 91 "The knowledge of TR maping Mathieu's ODE into a given ODE suffices to solve the given ODE.\n" }}{PARA 15 "" 0 "" {TEXT -1 136 "Compute TR without solving auxiliary Differential Equ ations, nor computing resultants, nor running differential elimination processes. \n" }}{PARA 15 "" 0 "" {TEXT -1 71 "The steps taking \"Giv en ODE\" to \"M_Mathieu\" were already developed for " }{TEXT 283 7 "h yper3 " }{TEXT -1 41 "and are available as an external command " } {HYPERLNK 17 "DEtools[power_equivalent]" 2 "DEtools[power_equivalent] " "" }{TEXT -1 81 ". What remains is to classify Mathieu ODEs and comp ute the Mobius transformation." }}{PARA 258 "" 0 "" {TEXT -1 62 "_____ _________________________________________________________" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Preparing a " }{TEXT 282 6 "seed " }{TEXT -1 44 "for formulating the equivalence under Mobius" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Original form of Mathieu \+ ODE" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart; with(PDEtool s): with(ODEtools): declare(y(x),prime=x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(% " 0 "" {MPLTEXT 1 0 45 "odeM := diff(y(x),x,x )+(a-2*q*cos(2*x))*y(x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%odeMG,& %$y''G\"\"\"*&,&%\"aGF'*(\"\"#F'%\"qGF'-%$cosG6#,$*&F,F'%\"xGF'F'F'!\" \"F'%\"yGF'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Its solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "ansM := y(x) = _C1*MathieuC(a, q, x) + _C2*MathieuS(a, q, x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%ansMG/%\"yG,&*&%$_C1G\"\"\"-%)Mathi euCG6%%\"aG%\"qG%\"xGF*F**&%$_C2GF*-%)MathieuSGF-F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 107 "The equivalence pro blem is mounted over a rational form of Mathieu's ODE. Consider the c hange of variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x=arcc os(t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"xG-%'arccosG6#%\"tG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "rational_textbook := ch(%, o deM, expand);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%2rational_textbookG /%$y''G,&**%\"xG\"\"\",&F)F*F*!\"\"F,,&F)F*F*F*F,%#y'GF*F,**,(*&\"\"#F *%\"qGF*F,%\"aGF,*(\"\"%F*F3F*)F)F2F*F*F*F+F,F-F,%\"yGF*F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "The solution t o this rational form found in textbooks is obtained using the same cha nge of variables" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "ans_text book := y(x) = _C1*MathieuC(a, q, arccos(x)) + _C2*MathieuS(a, q, arcc os(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-ans_textbookG/%\"yG,&*&% $_C1G\"\"\"-%)MathieuCG6%%\"aG%\"qG-%'arccosG6#%\"xGF*F**&%$_C2GF*-%)M athieuSGF-F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 62 "_________________________________________________________ _____" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 48 " Yet to formulate the problem correctly we need:\n" }}{PARA 261 "" 0 " " {TEXT -1 1 "1" }{TEXT 32 0 "" }{TEXT -1 2 ". " }{TEXT 278 1 "A" } {TEXT -1 1 " " }{TEXT 280 11 "Normal form" }{TEXT -1 1 " " }{TEXT 284 45 "for this rational version of Mathieu equation" }{TEXT -1 1 "\n" }} {PARA 261 "" 0 "" {TEXT -1 3 "2. " }{TEXT 285 4 "This" }{TEXT -1 12 " \+ Normal form" }{TEXT 286 13 " should have " }{TEXT 281 17 "minimized de grees" }{TEXT 287 17 " in the invariant" }{TEXT -1 1 "\n" }}{PARA 263 "" 0 "" {TEXT -1 55 "Such a \"featured\" seed is generated using the e xisting " }{HYPERLNK 17 "DEtools[power_equivalent]" 2 "DEtools,power_e quivalent" "" }{TEXT -1 8 " command" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "o2,tr := power_equivalent( \+ rational_textbook, keep, y(x), u(t));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>6$%#o2G%#trG6$/%$y''G,$*,\"#;!\"\",*\"\"$\"\"\"*(F,F0%\"qGF0)%\"xG F/F0F0*&,(*&\"#CF0F2F0F-\"\"%F0*&F9F0%\"aGF0F-F0)F4\"\"#F0F0*&,(*&\"\" )F0F2F0F0F9F-*&F9F0F;F0F0F0F4F0F0F0F4!\"#,(*$F " 0 "" {MPLTEXT 1 0 78 "itr := convert(s ubs([y=u,u=y,x=t,t=x], solve(useIntat(tr),\{t,u(t)\})),radical):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "a2 := combine( isolate(ch(value(itr ),ans_textbook,expand),y(x)), symbolic);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#a2G/%\"yG*&,&*&%$_C1G\"\"\"-%)MathieuCG6%%\"aG%\"qG-%'arccosG 6#*$%\"xG#F+\"\"#F+F+*&%$_C2GF+-%)MathieuSGF.F+F+F+)*(F5F+,&F4F+F+!\" \"F+,&F4F+F+F+F+#F+\"\"%F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 62 "_____________________________________________ _________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Finally, a simpler normal form is obtained redefining the parameters " }{XPPEDIT 18 0 "\{a,q\}" "6#<$%\"aG%\"qG" }{TEXT -1 20 " (..can be improved)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "rel := zip(`=`, [a,q], subs(a=(kappa+4)/4,q=mu/8,kappa= -mu+2*kappa,kappa = kappa/2,mu=mu/2, [a,q] ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$rel G7$/%\"aG,(*&\"\")!\"\"%#muG\"\"\"F+*&\"\"%F+%&kappaGF-F-F-F-/%\"qG,$* &\"#;F+F,F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "seed := co llect(expand(subs(rel,o2)),y,u -> simplify(u,size));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%seedG/%$y''G,$*,\"#;!\"\",*\"\"$F**&%#muG\"\"\")% \"xGF,F/F**&,&F.F/%&kappaGF/F/)F1\"\"#F/F/*&F4F/F1F/F*F/F1!\"#,&F1F/F/ F*F8%\"yGF/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 32 "This is the solution to the seed" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "ans_seed := subs(rel,a2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)ans_seedG/%\"yG*&,&*&%$_C1G\"\"\"-%)MathieuCG6%,(*& \"\")!\"\"%#muGF+F2*&\"\"%F2%&kappaGF+F+F+F+,$*&\"#;F2F3F+F+-%'arccosG 6#*$%\"xG#F+\"\"#F+F+*&%$_C2GF+-%)MathieuSGF.F+F+F+)*(F>F+,&F=F+F+F2F+ ,&F=F+F+F+F+#F+F5F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "odet est( ans_seed, seed);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 0 "" }{TEXT 288 0 "" }{TEXT -1 34 "The singularities of Mat hieu's ODE" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 297 7 "Recall:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "seed;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$y''G, $*,\"#;!\"\",*\"\"$F(*&%#muG\"\"\")%\"xGF*F-F(*&,&F,F-%&kappaGF-F-)F/ \"\"#F-F-*&F2F-F/F-F(F-F/!\"#,&F/F-F-F(F6%\"yGF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "singularities( seed );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%(regularG<$\"\"!\"\"\"/%*irregularG<#%)infinityG" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 296 7 "Facts:\n " }}{PARA 261 "" 0 "" {TEXT -1 16 "* 2F1 ODEs have " }{TEXT 290 23 "3 \+ regular singularities" }{TEXT -1 76 ".\n\n* 1F1 ODEs are obtained from 2F1 through a \"confluence\" process and have " }{TEXT 291 25 "1 regu lar and 1 irregular" }{TEXT -1 90 " singularities.\n\n* 0F1 (Bessel) O DEs are a particular case of the 1F1 ODEs and also have " }{TEXT 299 25 "1 regular and 1 irregular" }{TEXT -1 16 " singularities. " }} {PARA 265 "" 0 "" {TEXT -1 15 "______________\n" }}{PARA 261 "" 0 "" {TEXT -1 22 "* Heun type ODEs have " }{TEXT 292 25 "4 regular singular ities.\n" }}{PARA 14 "" 0 "" {TEXT -1 2 "* " }{TEXT 289 108 "The Spher oidal wave function ODE is obtained from a Heun type ODE through a \"c onfluence\" process and it has " }{TEXT 293 25 "2 regular and 1 irregu lar" }{TEXT 294 16 " singularities.\n" }}{PARA 14 "" 0 "" {TEXT 295 89 "* Mathieu equation is a particular case of the Spheroidal wave fun ction equation and has " }{TEXT 300 25 "2 regular and 1 irregular" } {TEXT 302 16 " singularities.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 261 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 60 "Connection between the Spheroidal wave and Mathieu equati ons" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "Th e Spheroidal wave function ODE depends on three parameters " } {XPPEDIT 18 0 "\{b,c,q\}" "6#<%%\"bG%\"cG%\"qG" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "spheroidal_wave := diff(y(x) ,x,x) = 2*(b+1)*x/(1-x^2)*diff(y(x),x)+(4*q*x^2-c)/(1-x^2)*y(x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%0spheroidal_waveG/%$y''G,&*,\"\"#\" \"\",&%\"bGF*F*F*F*%\"xGF*,&F*F**$)F-F)F*!\"\"F1%#y'GF*F**(,&*(\"\"%F* %\"qGF*F0F*F*%\"cGF1F*F.F1%\"yGF*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "singularities(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /%(regularG<$!\"\"\"\"\"/%*irregularG<#%)infinityG" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "This is the particulari zation leading to Mathieu's equation which depends on only two paramet ers " }{XPPEDIT 18 0 "\{a,q\}" "6#<$%\"aG%\"qG" }{TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "b = -1/2, c = a+2*q;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$/%\"bG#!\"\"\"\"#/%\"cG,&%\"aG\"\"\"*& F'F,%\"qGF,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "eval( sphe roidal_wave, [%] );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$y''G,&*(%\"x G\"\"\",&F(F(*$)F'\"\"#F(!\"\"F-%#y'GF(F(*(,(*(\"\"%F(%\"qGF(F+F(F(%\" aGF-*&F,F(F3F(F-F(F)F-%\"yGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "collect( rational_textbook , y, u -> normal(u,expanded));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%$y''G,&*(,(*&\"\"#\"\"\"%\"qGF*F*%\" aGF**(\"\"%F*F+F*)%\"xGF)F*!\"\"F*,&*$F/F*F*F*F1F1%\"yGF*F**(F0F*%#y'G F*F2F1F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 301 9 "Summary: " }}{PARA 261 "" 0 "" {TEXT -1 220 "The sequence 0F1, 1F1, 2F1 is reproduced by the sequence Mathieu, Spheroidal, Heun by just adding one regular singularity to the problem. Mathieu functi ons are the \"Bessel\" functions of ODE problems with 4 singularities. " }}{PARA 258 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 62 "_ _____________________________________________________________" }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "Examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The traditional form of the equ ation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "odeM;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$y''G\"\" \"*&,&%\"aGF%*(\"\"#F%%\"qGF%-%$cosG6#,$*&F*F%%\"xGF%F%F%!\"\"F%%\"yGF %F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&%$_C1G\"\"\"-%)MathieuCG6%%\"aG%\" qG%\"xGF(F(*&%$_C2GF(-%)MathieuSGF+F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "Let's change \"cos into sin\" :)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "subs(cos=sin,odeM);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,&%$y''G\"\"\"*&,&%\"aGF%*(\"\"#F%%\"q GF%-%$sinG6#,$*&F*F%%\"xGF%F%F%!\"\"F%%\"yGF%F%" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "dsolve(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# /%\"yG,&*&%$_C1G\"\"\"-%)MathieuCG6%%\"aG%\"qG-%'arccosG6#,$*&#F(\"\"# F(*$,&*(\"\"%F(-%$sinG6#%\"xGF(-%$cosGF;F(F(F4F(F3F(F(F(F(*&%$_C2GF(-% )MathieuSGF+F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "odetest (%,%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 36 "The rational form found i n textbooks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "rational_text book;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$y''G,&**%\"xG\"\"\",&F'F(F (!\"\"F*,&F'F(F(F(F*%#y'GF(F***,(*&\"\"#F(%\"qGF(F*%\"aGF**(\"\"%F(F1F ()F'F0F(F(F(F)F*F+F*%\"yGF(F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "dsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*&%$_C1G\" \"\"-%)MathieuCG6%%\"aG%\"qG-%'arccosG6#%\"xGF(F(*&%$_C2GF(-%)MathieuS GF+F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The seed form" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "seed;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$y''G,$*,\"#;!\"\",*\"\"$F(*&%#muG \"\"\")%\"xGF*F-F(*&,&F,F-%&kappaGF-F-)F/\"\"#F-F-*&F2F-F/F-F(F-F/!\"# ,&F/F-F-F(F6%\"yGF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "ds olve(%);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"yG,&*(%$_C1G\"\"\")*&% \"xGF(,&F+F(F(!\"\"F(#F(\"\"%F(-%)MathieuCG6%,(*&\"\")F-%#muGF(F-*&F/F -%&kappaGF(F(F(F(,$*&\"#;F-F6F(F(-%'arccosG6#*$F+#F(\"\"#F(F(*(%$_C2GF (F)F(-%)MathieuSGF2F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 36 "An example from Kamke's book (2.268)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "2*x*(x-1)*diff(y(x),x,x)+(2*x-1)*di ff(y(x),x)+(a*x+b)*y(x) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(** \"\"#\"\"\"%\"xGF',&F(F'F'!\"\"F'%$y''GF'F'*&,&*&F&F'F(F'F'F'F*F'%#y'G F'F'*&,&*&%\"aGF'F(F'F'%\"bGF'F'%\"yGF'F'\"\"!" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 10 "dsolve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%\"yG,&*&%$_C1G\"\"\"-%)MathieuCG6%,&%\"aG!\"\"*&\"\"#F(%\"bGF(F.,$*& F0F.F-F(F(-%'arccosG6#*$%\"xG#F(F0F(F(*&%$_C2GF(-%)MathieuSGF+F(F(" }} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 147 "An exam ple full of symbols around - nothing special, just to test the ability to determine the parameters even in front of many symbolic parameters " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 630 "ode := NForm(diff(diff( y(x),x),x) = -2*G/(G*x+H)*diff(y(x),x)+1/16*(A*H-B*G)^2*(A*H+A*x*G-A*G *epsilon*x-G*epsilon*B)*(epsilon^2*mu*x^2*A^4-A^3*epsilon*mu*x*H-A^3*e psilon*mu*x^2*G+2*A^3*epsilon^2*mu*x*B-A^3*epsilon^2*mu*x^2*G+A^2*H*ep silon*mu*G*x-A^2*H*epsilon*mu*B-A^2*B*x*epsilon*mu*G+A^2*epsilon*mu*G^ 2*x^2+A^2*epsilon^2*mu*B^2-2*A^2*G*epsilon^2*mu*x*B-3*A*H^2*G-6*A*G^2* x*H+A*H*epsilon*mu*G*B-3*A*H*G^2*epsilon*x+A*B*x*epsilon*mu*G^2-3*A*G^ 3*x^2-A*G*epsilon^2*mu*B^2-3*A*G^3*x^2*epsilon+3*H^2*G^2-3*H*epsilon*G ^2*B+6*H*G^3*x+3*G^4*x^2-3*G^3*x*epsilon*B)/G/A/(G*x+H)^3/(A-G)/(A*x+B )^2/(epsilon*A*x+epsilon*B-G*x-H)^2*y(x),tidy);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$odeG/%$y''G,$*8\"#;!\"\",&*&%\"AG\"\"\"%\"HGF.F.*&% \"GGF.%\"BGF.F*\"\"#,4*&,0*()%(epsilonGF3F.%#muGF.)F-\"\"%F.F.**)F-\" \"$F.F8F.F:F.F1F.F***F?F.F-F.)F1F?F.F9F.F**(F?F.F-F.FAF.F***)F-F3F.F9F .F:F.)F1F3F.F.*&F?F.)F1FF.F9F.F:F.F1F.F*F.)%\"xGF3F.F.*&,6**F >F.F9F.F:F.F/F.F**,F-F.F2F.F9F.F:F.FEF.F.*(\"\"'F.F/F.FAF.F.*,FDF.F/F. F9F.F:F.F1F.F.*.F3F.FDF.F1F.F8F.F:F.F2F.F***FPF.F-F.FEF.F/F.F**,F?F.F- F.F/F.FEF.F9F.F**,F3F.F>F.F8F.F:F.F2F.F.*,FDF.F2F.F9F.F:F.F1F.F***F?F. FAF.F9F.F2F.F*F.FJF.F.*.F-F.F/F.F9F.F:F.F1F.F2F.F.**FDF.F8F.F:F.)F2F3F .F.*,FDF.F/F.F9F.F:F.F2F.F**,F?F.F/F.F9F.FEF.F2F.F**,F-F.F1F.F8F.F:F.F ZF.F***F?F.F-F.)F/F3F.F1F.F**(F?F.FinF.FEF.F.F.,(**F-F.F1F.,&F.F*F9F.F .FJF.F.F,F**(F1F.F9F.F2F.F.F.,(*&,&*&F9F.F-F.F.F1F*F.FJF.F.*&F9F.F2F.F .F/F*!\"#F1F*,&*&F1F.FJF.F.F/F.!\"$,&*&F-F.FJF.F.F2F.Fdo,&F-F.F1F*F*F- F*%\"yGF.F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "indets(ode,s ymbol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<)%\"AG%\"BG%\"xG%#muG%\"GG %(epsilonG%\"HG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "dsolve(o de);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%\"yG,&**%$_C1G\"\"\",&*&%\"G GF(%\"xGF(F(%\"HGF(#F(\"\"#)*&,&*&%\"AGF(F,F(F(%\"BGF(F(,**(%(epsilonG F(F4F(F,F(F(*&F8F(F5F(F(F*!\"\"F-F:F(#F(\"\"%F(-%)MathieuCG6%,$*,\"\") F:,.**\"\"$F(%#muGF(F+F()F4F/F(F:*(F4F(FFF()F+F/F(F(*(F/F(FFF()F4FEF(F (*&\"\"'F()F+FEF(F(*(FBF(FGF(F+F(F(*(FBF(F4F(FIF(F:F(,&F4F(F+F:F:F4F:F +F:F(,$*&\"#;F:FFF(F(-%'arccosG6#*$*(F8F(F2F(F)F:F.F(F(**%$_C2GF(F)F.F 0F(-%)MathieuSGF?F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "odetest(%,ode);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 62 "____ __________________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}}{MARK "3 21 32 0 0" 60 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }