PtjldZddlmZddlmZmZmZmZddlm Z ddl m Z m Z m Z ddlmZmZddlmZmZmZmZddlmZmZdd lmZdd lmZdd lmZmZdd l m!Z!m"Z"m#Z#m$Z$dd l%m&Z&ddl'm(Z(ddl)m*Z*ddl+m,Z,m-Z-ddl.m/Z/dZ0dZ1dZ2d$dZ3d%dZ4d%dZ5d%dZ6d%dZ7d%dZ8d%dZ9d%d Z:d%d!Z;d%d"Z#t$rYKwxYw|rfd|DSdS)Nxietaryhr!)hintfuncordermatch)r2c,g|]}t|S)_lie_group_remove).0coords Y/var/www/html/CA-Chatbot/venv/lib/python3.11/site-packages/sympy/solvers/ode/lie_group.py z0_ode_lie_group_try_heuristic..Ps!AAA%e,,AAArsC1T)dict)rrAcoeffcBg|]}t|Sr6)r )r8solfxs r:r;z0_ode_lie_group_try_heuristic..os)>>>#11Q44 >>>r<c jg|]/}t|d0S)r)r subs)r8rDrErFr/s r:r;z0_ode_lie_group_try_heuristic..zs9<<newcoordr?rcoordscoordrDxsubysubnumdenomdiffeqsepdeqsdeqrErFr/s @@@r:_ode_lie_group_try_heuristicrc6sj $B 5//C A ! A c A c AG   )$auUUUCC   44 2<2<1d $**433Q&,,T155 F5L ! !Q & & qAww||Au$qqAww||Av'== DqqAww ' ' ' + D1H11Q77 + + + /AA1a&AAA #JJ #JJ D\\!" <VQZ0!QTBBBCbyy a&Cq6Dq6D6;;q>>FKKNN1,<<==CV[[^^fkk!nnQ.>>??E >C?$S!}} ?>>>>>>>>>>/,,,s+++++, ? <11Q44v{A!6!6q!9::;;;; <11Q44v{A!6!6q!9::;;;; <7#    H >=<<<<<N?c&t}i}|j}|jd}||}t d} t d} |d} |d} |dd} | r,t ||d||dz  }n%td} ||| }| | | |||t| | |||t| ig}tt|||||dddrd gt|z}|| d }d}|D])}tt||||||}|r|cS*|S) Nrr-r.r/der!)r2r3 user_defined)r0r/)lie_heuristicsr2rIrLr poprrrHrr"r listrc)r>r2r3r4 heuristicsrRrErFdfr-r.xisetasr/r0rDrQs r:_ode_lie_groupro~sJ C A ! A 1B $B 5//C +C >> from sympy import Function, Eq, pprint >>> from sympy.abc import x, y >>> xi, eta, h = map(Function, ['xi', 'eta', 'h']) >>> h = h(x, y) # dy/dx = h >>> eta = eta(x, y) >>> xi = xi(x, y) >>> genform = Eq(eta.diff(x) + (eta.diff(y) - xi.diff(x))*h ... - (xi.diff(y))*h**2 - xi*(h.diff(x)) - eta*(h.diff(y)), 0) >>> pprint(genform) /d d \ d 2 d d d |--(eta(x, y)) - --(xi(x, y))|*h(x, y) - eta(x, y)*--(h(x, y)) - h (x, y)*--(xi(x, y)) - xi(x, y)*--(h(x, y)) + --(eta(x, y)) = 0 \dy dx / dy dy dx dx Solving the above mentioned PDE is not trivial, and can be solved only by making intelligent assumptions for `\xi` and `\eta` (heuristics). Once an infinitesimal is found, the attempt to find more heuristics stops. This is done to optimise the speed of solving the differential equation. If a list of all the infinitesimals is needed, ``hint`` should be flagged as ``all``, which gives the complete list of infinitesimals. If the infinitesimals for a particular heuristic needs to be found, it can be passed as a flag to ``hint``. Examples ======== >>> from sympy import Function >>> from sympy.solvers.ode.lie_group import infinitesimals >>> from sympy.abc import x >>> f = Function('f') >>> eq = f(x).diff(x) - x**2*f(x) >>> infinitesimals(eq) [{eta(x, f(x)): exp(x**3/3), xi(x, f(x)): 0}] References ========== - Solving differential equations by Symmetry Groups, John Starrett, pp. 1 - pp. 14 r!z(ODE's have only one independent variablerz?Infinitesimals for only first order ODE's have been implementedaexcludebr0r/z9Infinitesimals for the first order ODE could not be foundu)r0r2hxhyr/hinvalllie_heuristic_T)compcg|]}|v| Sr6r6)r8rRxietas r:r;z"infinitesimals..s#%Q%Q%Qc5@P@Pc@P@P@Pr<z3Infinitesimals could not be found for the given ODErpFzHeuristic not recognized: z;Infinitesimals could not be found using the given heuristic) isinstancer lhsrMrrIlenrKr rNrLrrr r4rrrrHrhglobalsextend)rPr2r3r1r4 variablesrFrlrrrur0r/rDrvrwrxryrQfunctioninflistr~s @r:rJrJs|"h Vbf_ #r??D I 9~~CDDD aL (b$''E A::%':;; ;1BSRD)))ASRD)))A $#J#Jr B//55adQh?? #!%(58"3444AA##Bmm  F /BBB13ABBBB #JJFF4OOc ABBqSJJAA/0066q!<> I  I c4g}|d}|d}|d}|d}|jd}td||}td||} tt||||dzz d ||g } | r| d r| |} t | d | |zz |z} | j} || vrdtt| |} | tj || | z ||i} |s| gS|r| |vr| | td }tt||||dzz d ||g } | r| d r| |} t | d | |zz |z} | j} || vrtt| |} | | z}| ||f||fg ||}| | |||tj i} |s| gS|r| |vr| | |r|SdS)a The second heuristic uses the following two assumptions on `\xi` and `\eta` .. math:: \eta = 0, \xi = f(x)*g(y) .. math:: \eta = f(x)*g(y), \xi = 0 The first assumption of this heuristic holds good if `\frac{1}{h^{2}}\frac{\partial^2}{\partial x \partial y}\log(h)` is separable in `x` and `y`, then the separated factors containing `x` is `f(x)`, and `g(y)` is obtained by .. math:: e^{\int f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)\,dy} provided `f\frac{\partial}{\partial x}\left(\frac{1}{f*h}\right)` is a function of `y` only. The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption satisfies. After obtaining `f(x)` and `g(y)`, the coordinates are again interchanged, to get `\eta` as `f(x)*g(y)` References ========== - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order ODE Patterns, pp. 7 - pp. 8 r/r0ryr2rr-r.rTrArrBr!u1N)rIr rrrLrrrrrrrHrOr)r4r|r~r/r0ryr2rFr-r.rRrgygysymsretavals r:lie_heuristic_abaco1_productrs> E c A c A =D =D ! A $4 B (5//!T " "C QQ--a00!Q$6TAq6 R R RC "s7| " V b1bd8//!,,- . . F??Yr1%%&&BRULLD$9$9:C u  "5(( S!!! tB T**0033T1W<4RSUVQW X X XC "s7| " V b1bg;,,Q//0 1 1 F??Yr1%%&&BUFkkAr7QF"344::2qAAF At,,b!&9C u  "5(( S!!!  r<c  |d}|d}|d}|d}|jd|d td|}td|}|rWtd \}} } } } } || | z |zz| |d zzz | |zz | |zz }t |\}}t | }td  }td  }|| |z |zz| |d zzz ||zz ||zz }td }td}t|dzD] rT|t fdtdzDz }|t fdtdzDz }| ||||i \}}t|}| r*|jr#t | }|r|j|j hz }t)|g|R}t-|t.r|d}t1|r| |}| |}t2|d}|| | ||| | |i}|gcS dSdS)a The third heuristic assumes the infinitesimals `\xi` and `\eta` to be bi-variate polynomials in `x` and `y`. The assumption made here for the logic below is that `h` is a rational function in `x` and `y` though that may not be necessary for the infinitesimals to be bivariate polynomials. The coefficients of the infinitesimals are found out by substituting them in the PDE and grouping similar terms that are polynomials and since they form a linear system, solve and check for non trivial solutions. The degree of the assumed bivariates are increased till a certain maximum value. References ========== - Lie Groups and Differential Equations pp. 327 - pp. 329 r0rwrxr2rr/r-r.zetax etay etad xix xiy xidrdetadxixi0eta0r!c g|]F}tdt|zdzt|z z|zz|z zzGS)xi__rstrr8powerirFr/s r:r;z+lie_heuristic_bivariate.. si///53u::-3c!e)nnDEEahNqSTW\S\~]///r<c g|]F}tdt|zdzt|z z|zz|z zzGS)eta_rrrs r:r;z+lie_heuristic_bivariate.. sk///6CJJ.4s1u9~~EFFq%xOPQTUX]T]P^^///r<N)rIr is_rational_functionrras_numer_denomr total_degreerLrrangerrHdoitr is_polynomialis_Addas_dictrunionrvaluesrrjanyrAfromkeys)!r4r|r0rwrxr2r-r.etaxetayetadxixxiyxidipder]r^degrrxieqetaeqpdenpolyysymsetsoldictxiredetareddict_rRrrFr/s! @@@r:lie_heuristic_bivariaters& c A tB tB =D ! A c A $4 B (5//!T " "C*!+22N*O*O'dD#sCtcz1n$s1a4x/#b&847BD\\0022 U31oo**,,x1%%huooa## !  ! sxx{{ :A==!aQRd@RR"fBwe}}vsQw ! !A 0//////!&q1u///00//////!&q1u///00 99c4u%=>>CCEEUUWWKD%$<.Rsi+++vE *S03q5y>>ABB1e8KAPQTYPYNZ+++r<N)rIr rrrrrrrLrrrrHrr rrrrrrrrjrrArr)r4r|r0rxr2r-r.schischixschiycpder]r^rr+chixchiychieqcnumcdencpolysolsymsrrxicetacrRrrFr/s @@@r:lie_heuristic_chir&s0 c A tB =D ! A c A $4 B (5//!T " "C%%$%9::eUqwD(D\\0022 U31oo**,,huooa##xx{{xx{{af}r#v%u q#'"" % %A S++++++"1q5\\+++, ,E  3- 8 8 = = ? ?@@OOQQJD$$<>++,, % % 7 3 3 $ gq 9 9 % 5 1 1%(qMM T"DIIa$6$6SXXa=N=N.s+&S&S&SsSWWQPQ]]&Ss&S&S&Sr<N)rIr rLrrrrrrHrris_MulrrrrOrN)r4r|r~r0r2ryr-r.odefacrr`rrfddrcheckrDrrRrFr/s @@r:lie_heuristic_function_sumrmsL E c A =D =D ! A c A $4 B (5//!T " "Cd)))!F(A../AfH??1--D1a&III # *3w<# *CFJJqMM# *c!fjjmm# *c A! *yQ+++3q6#g,./c&j2o.. A!455* %WWQ]] d#E1oo%"%a&C!#CB"$Q$BB$~~WWQ]]WWQ]])"r'22F}U!$&S&S&S&S&Sfk&S&S&S!T~~"FKK4$8$8"qvF!6;;q$#7#7qvF* #u  S)))?'    B  LLL Q))sJ J+*J+c |d}|d}|d}|d}|d}|jd}|d}td||} td ||} t||||d z } | |} | |} | |s| |st d |g }t d |g }t d||g }|||t||z zz} tt||||z  |||z }|||z}| || |igS#t$rYnwxYwt| | z }||st||z||z |z ||zz }||s; tt||}| |z}| || |igS#t$rYnwxYwt||||d z } | |} | |} | |s| |st d |g }t d |g }t d||g }|||t||z zz} tt||||z  |||z }|||z}| | ||| | ||igS#t$rYdSwxYwt| | z }||st|||z||z ||z ||zz }||sf tt||}| |z}| | ||| | ||igS#t$rYdSwxYwdSdS)aW This heuristic uses the following two assumptions on `\xi` and `\eta` .. math:: \eta = g(x), \xi = f(x) .. math:: \eta = f(y), \xi = g(y) For the first assumption, 1. First `\frac{\frac{\partial h}{\partial y}}{\frac{\partial^{2} h}{ \partial yy}}` is calculated. Let us say this value is A 2. If this is constant, then `h` is matched to the form `A(x) + B(x)e^{ \frac{y}{C}}` then, `\frac{e^{\int \frac{A(x)}{C} \,dx}}{B(x)}` gives `f(x)` and `A(x)*f(x)` gives `g(x)` 3. Otherwise `\frac{\frac{\partial A}{\partial X}}{\frac{\partial A}{ \partial Y}} = \gamma` is calculated. If a] `\gamma` is a function of `x` alone b] `\frac{\gamma\frac{\partial h}{\partial y} - \gamma'(x) - \frac{ \partial h}{\partial x}}{h + \gamma} = G` is a function of `x` alone. then, `e^{\int G \,dx}` gives `f(x)` and `-\gamma*f(x)` gives `g(x)` The second assumption holds good if `\frac{dy}{dx} = h(x, y)` is rewritten as `\frac{dy}{dx} = \frac{1}{h(y, x)}` and the same properties of the first assumption satisfies. After obtaining `f(x)` and `g(x)`, the coordinates are again interchanged, to get `\xi` as `f(x^*)` and `\eta` as `g(y^*)` References ========== - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order ODE Patterns, pp. 10 - pp. 12 r0rwrxr2ryrr/r-r.rArsBCN) rIr rrLrrr4rrrNrH)r4r|r0rwrxr2ryrFr/r-r.rfactorxfactoryrrrtaugxgammatauints r:lie_heuristic_abaco2_similarrsL c A tB tB =D =D ! A c A $4 B (5//!T " "C AFF1IIaffQll* + +Fkk!nnGkk!nnG ::a==0A0 qc " " " qc " " " q!f % % %Ac!A#hhJ'' (yq%(!2333Q77a@Cq#Bc2&' ' #    D ww''yy|| 0U2X 1 5:QYGHHF::a== 00i2233CeBc2.// +D DIIaLL1a0 1 1Fkk!nnGkk!nnG ::a==LAL qc " " " qc " " " q!f % % %Ac!A#hhJ'' Dyq%(!2333Q77a@Cq#B#((1d++RD1A1ABC C #    DD ww''yy|| LU499Q<</%**Q--?$))A,,Nu F::a== LLi2233CeB #((1d"3"3RD9I9IJKK +DD L L L LsH 6F(( F54F5&I II6O O O%R77 SSc|d}|d}|d}|d}|jd}|d}td||}td||} g} |tD]_} | \} } | |r1| |r| js| | `|tD]&}|j }||vr||vr| |'| D]X}t| || |z }t|d ||g }|r|d r||}d |||d zz }| ||z| ||zz ||zz ||zz }t|s||| |||igcSd |z }d |z }| || ||dzzz ||zz ||zz }tt|s||||| |igcSbd |z }| || ||zz|z ||zz }t|s'|t j| |||igcS| }| | |z| ||dzzz ||zz |z }tt|s'||||| t jigcSZdS)a This heuristic assumes the presence of unknown functions or known functions with non-integer powers. 1. A list of all functions and non-integer powers containing x and y 2. Loop over each element `f` in the list, find `\frac{\frac{\partial f}{\partial x}}{ \frac{\partial f}{\partial x}} = R` If it is separable in `x` and `y`, let `X` be the factors containing `x`. Then a] Check if `\xi = X` and `\eta = -\frac{X}{R}` satisfy the PDE. If yes, then return `\xi` and `\eta` b] Check if `\xi = \frac{-R}{X}` and `\eta = -\frac{1}{X}` satisfy the PDE. If yes, then return `\xi` and `\eta` If not, then check if a] :math:`\xi = -R,\eta = 1` b] :math:`\xi = 1, \eta = -\frac{1}{R}` are solutions. References ========== - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order ODE Patterns, pp. 10 - pp. 12 r0rwrxr2rr/r-r.TrrBr@r!rN)rIr atomsr as_base_expr is_IntegerrOr rrrLrrrHr rOne)r4r|r0rwrxr2rFr/r-r.funclistatombaserrsymsrEfracr`xitry1etatry1pde1xitry2etatry2pde2etatrypdexitrys r:#lie_heuristic_abaco2_unique_unknownr3s> c A tB tB =D ! A c A $4 B (5//!T " "CH &&$$&& c 88A;; &488A;; &> &%%%GGL))&&$ 99d OOH % % % ??affQiiq )**4dQF;;;  ?3w< ?VF#a&W-.G<<??1$v{{1~~a'77&)CgbjPDD>> BVS',,q$*?*?@AAAAwYFhG<<??fkk!nnad%::VBYFQSSDF4LL)) BV[[D113@AAAA BWF++a..6;;q>>!#33b86"9DCC== @QUCQ)=)=>????EE::a==."EJJqMM!Q$$66rA2ECF3KK(( ?UZZ400#qu=>>>> ?-??r<cT|d}|d}|d}|jd}|d}td||}td||}||} |||dzz} |||dzz } | r| r| sd S| |} | |} | |}| |}| |}td|z|| zz| |zz ||zd| zz| zz| zd | z| zz }|sZtd | dzz|dzd| zz| zd|zz | zz}|r"td|zd| z|dzz | zz| zd | dzzz }|st|d | zd |z| zz| d zz||| z| zzz z||d z| d zzz }|st d | d zz| || zz z||| z| z zzt d| z|zz }||vrd t t||}d| d zz|z|z }||vr||||||igSd S#t$rYd SwxYwd Sd Sd Sd Std|zd| z|dzz | zz| zd | dzzz }|r@td | d zz|z|dzz |d|z|dzd| zz| zz | zd | dzzz zz}|st| |z ||z||||zz | zd | z|zz| |zd | zz |zz|zz}|st | |z| z |z| | |zz|zz t d| z|zz }||vr_ t t||}| |z|z }||vr||||||igSd S#t$rYd SwxYwd Sd Sd Sd S)aQ This heuristic finds if infinitesimals of the form `\eta = f(x)`, `\xi = g(y)` without making any assumptions on `h`. The complete sequence of steps is given in the paper mentioned below. References ========== - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order ODE Patterns, pp. 10 - pp. 12 rwrxr2rr/r-r.rN) rIr rLrrrrrrHrN)r4r|rwrxr2rFr/r-r.rrrAxAyAxyAxxAyyDE1E2E3rxivals r:#lie_heuristic_abaco2_unique_generalrs_ tB tB =D ! A c A $4 B (5//!T " "C  A  RUA  RUA ! B B ''!**C ''!**C ''!**C3BB&"R%!A#+q899!;ad2gEA &P aAg"a%!A#+q1S5!8! ;; < <  P1S5AaC"a%K?2A5"a%?@@B PB2aA-BqD2I>?"''!**Q,qRStBSSUU P#Qq!tVR"Q$Y%7"bdRi.%H1Q44PQ6RT9$UVVFP%(61)=)=%>%>F%'q!tGFN2$5E ~~)+UCQ9M9M(N'O O .~ 3!!! DD! P P P P P P&qu!b!e Q.1Ab!eG; < <  P!Q$q1a4"quA! Q'>&AAb!eG&K"LLNNB PqS6"''!**$bd):A(="Q(FrTAbD["$)%&(())** P#adRi^rAbDy!m%Cadd1fQh$OPPFP%(61)=)=%>%>F&(CJqLE ~~)+UCQ9M9M(N'O O .~ 3!!! DD! P P P P P P&s$ J J'&J' P PPcD|d}|d}|d}|d}|jd|dtd|}td|}i}td t fd t d D} | \} } } } }}| || z |zz| z| zz| z|zz | z|zz|z|zz | |d zzz }|\}}t t|}|jr||j}|D]r}|j r;tfd|jD}||z }||vr|||<3||xx|z cc<D||vrtj ||<X||xxtj z cc<s| }t|| }|rt|t r|d}| }t#d|Drt%t'| dgd z}| z| |zz| z}| z||zz|z}||}||}||}||}||||igSdSdS)a This heuristic assumes 1. `\xi = ax + by + c` and 2. `\eta = fx + gy + h` After substituting the following assumptions in the determining PDE, it reduces to .. math:: f + (g - a)h - bh^{2} - (ax + by + c)\frac{\partial h}{\partial x} - (fx + gy + c)\frac{\partial h}{\partial y} Solving the reduced PDE obtained, using the method of characteristics, becomes impractical. The method followed is grouping similar terms and solving the system of linear equations obtained. The difference between the bivariate heuristic is that `h` need not be a rational function in this case. References ========== - E.S. Cheb-Terrab, A.D. Roche, Symmetries and First Order ODE Patterns, pp. 10 - pp. 12 r0rwrxr2rr/r-r.c)clsc.g|]}tSr6)next)r8rrs r:r;z(lie_heuristic_linear..s999tG}}999r<rc@g|]}||Sr6r)r8mrFr/s r:r;z(lie_heuristic_linear..s*DDD!a DADDDr<c3K|]}|VdS)Nr6)r8ts r: z'lie_heuristic_linear.. s"!!Qq!!!!!!r<r!N)rIr rrrrrr rrrrrrrrrjrrAziprH)r4r|r0rwrxr2r-r. coeffdictsymlistC0r?C2C3C4C5rr^termstermremxypartsollistrsubvalonedictrrrrFr/s @@@r:lie_heuristic_linearr+s0 c A tB tB =D ! A c A $4 B (5//!T " "CIs...G9999fWa&8&8999G$BBB R{ bdRTkB.2 2bdRTkB6F5J JRPQSTPTW TC##%%JC &++  C z- - -D{ -DDDDDtyDDDEc**(+If%%f%%%,%%%%y((&'eIdOOdOOOqu,OOOO  GGW%%G . gt $ $ !ajG!! !!&!!! ! ! .3wA..//GqD2d7NR'ETBtG^b(FJJw''E[[))FJJw''E[[))FV,- - . . . .r<ct|tr |jdS|jr9|t}|r|D]}||d}|S|jr:|\}}t|}t|}||zS|j rNg}|j}|D]9}t|ts"| t|:t|S|S)a This function is strictly meant for internal use by the Lie group ODE solving method. It replaces arbitrary functions returned by pdsolve as follows: 1] If coords is an arbitrary function, then its argument is returned. 2] An arbitrary function in an Add object is replaced by zero. 3] An arbitrary function in a Mul object is replaced by one. 4] If there is no arbitrary function coords is returned unchanged. Examples ======== >>> from sympy.solvers.ode.lie_group import _lie_group_remove >>> from sympy import Function >>> from sympy.abc import x, y >>> F = Function("F") >>> eq = x**2*y >>> _lie_group_remove(eq) x**2*y >>> eq = F(x**2*y) >>> _lie_group_remove(eq) x**2*y >>> eq = x*y**2 + F(x**3) >>> _lie_group_remove(eq) x*y**2 >>> eq = (F(x**3) + y)*x**4 >>> _lie_group_remove(eq) x**4*y r) rr rIrrrHis_Powrr7rrOr)coordssubfuncr2rexprmulargs coordargsrs r:r7r7s>&,''{1~ ,,|,,  . . .T1--  '')) d && &&Tz K  7 7Cfl33 7055666G} Mr<)NNrpN)F)>__doc__ itertoolsr sympy.corerrrrsympy.core.exprtoolsrsympy.core.functionr r r sympy.core.relationalr r sympy.core.symbolrrrrsympy.functionsrrsympy.integrals.integralsr sympy.polysrsympy.polys.polytoolsrrsympy.simplifyrrrr sympy.solversrsympy.solvers.pdersympy.utilitiesrsympy.solvers.deutilsrr oder"rhrcrorJrrrrrrrrr+r7r6r<r:rDs&''''''''''''------>>>>>>>>>>........::::::::::::$$$$$$$$//////--------%%%%%%,,,,,,88888888 EEEP!!!HI-I-I-I-XeeeeNGGGGRF!F!F!F!PE%E%E%E%NXXXXtiLiLiLiLXK?K?K?K?\HPHPHPHPVF.F.F.F.R33333r<