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)abbrev package UTSODE UnivariateTaylorSeriesODESolver
++ Taylor series solutions of explicit ODE's.
++ Author: Stephen Watt (revised by Clifton J. Williamson)
++ Date Created: February 1988
++ Date Last Updated: 30 September 1993
++ Keywords: differential equation, ODE, Taylor series
++ Examples:
++ References:
UnivariateTaylorSeriesODESolver(Coef,UTS):_
Exports == Implementation where
++ This package provides Taylor series solutions to regular
++ linear or non-linear ordinary differential equations of
++ arbitrary order.
Coef : Algebra Fraction Integer
UTS : UnivariateTaylorSeriesCategory Coef
L ==> List
L2 ==> ListFunctions2
FN ==> (L UTS) -> UTS
ST ==> Stream Coef
YS ==> Y$ParadoxicalCombinatorsForStreams(Coef)
STT ==> StreamTaylorSeriesOperations(Coef)
Exports ==> with
stFunc1: (UTS -> UTS) -> (ST -> ST)
++ stFunc1(f) is a local function exported due to compiler problem.
++ This function is of no interest to the top-level user.
stFunc2: ((UTS,UTS) -> UTS) -> ((ST,ST) -> ST)
++ stFunc2(f) is a local function exported due to compiler problem.
++ This function is of no interest to the top-level user.
stFuncN: FN -> ((L ST) -> ST)
++ stFuncN(f) is a local function xported due to compiler problem.
++ This function is of no interest to the top-level user.
fixedPointExquo: (UTS,UTS) -> UTS
++ fixedPointExquo(f,g) computes the exact quotient of \spad{f} and
++ \spad{g} using a fixed point computation.
ode1: ((UTS -> UTS),Coef) -> UTS
++ ode1(f,c) is the solution to \spad{y' = f(y)}
++ such that \spad{y(a) = c}.
ode2: ((UTS, UTS) -> UTS,Coef,Coef) -> UTS
++ ode2(f,c0,c1) is the solution to \spad{y'' = f(y,y')} such that
++ \spad{y(a) = c0} and \spad{y'(a) = c1}.
ode: (FN,List Coef) -> UTS
++ ode(f,cl) is the solution to \spad{y<n>=f(y,y',..,y<n-1>)} such that
++ \spad{y<i>(a) = cl.i} for i in 1..n.
mpsode:(L Coef,L FN) -> L UTS
++ mpsode(r,f) solves the system of differential equations
++ \spad{dy[i]/dx =f[i] [x,y[1],y[2],...,y[n]]},
++ \spad{y[i](a) = r[i]} for i in 1..n.
Implementation ==> add
stFunc1 f == coefficients f series(#1)
stFunc2 f == coefficients f(series(#1),series(#2))
stFuncN f == coefficients f map(series,#1)$ListFunctions2(ST,UTS)
import StreamTaylorSeriesOperations(Coef)
divloopre:(Coef,ST,Coef,ST,ST) -> ST
divloopre(hx,tx,hy,ty,c) == delay(concat(hx*hy,hy*(tx-(ty*c))))
divloop: (Coef,ST,Coef,ST) -> ST
divloop(hx,tx,hy,ty) == YS(divloopre(hx,tx,hy,ty,#1))
sdiv:(ST,ST) -> ST
sdiv(x,y) == delay
empty? x => empty()
empty? y => error "stream division by zero"
hx := frst x; tx := rst x
hy := frst y; ty := rst y
zero? hy =>
zero? hx => sdiv(tx,ty)
error "stream division by zero"
rhy := recip hy
rhy case "failed" => error "stream division:no reciprocal"
divloop(hx,tx,rhy::Coef,ty)
fixedPointExquo(f,g) == series sdiv(coefficients f,coefficients g)
-- first order
ode1re: (ST -> ST,Coef,ST) -> ST
ode1re(f,c,y) == lazyIntegrate(c,f y)$STT
iOde1: ((ST -> ST),Coef) -> ST
iOde1(f,c) == YS ode1re(f,c,#1)
ode1(f,c) == series iOde1(stFunc1 f,c)
-- second order
ode2re: ((ST,ST)-> ST,Coef,Coef,ST) -> ST
ode2re(f,c0,c1,y)==
yi := lazyIntegrate(c1,f(y,deriv(y)$STT))$STT
lazyIntegrate(c0,yi)$STT
iOde2: ((ST,ST) -> ST,Coef,Coef) -> ST
iOde2(f,c0,c1) == YS ode2re(f,c0,c1,#1)
ode2(f,c0,c1) == series iOde2(stFunc2 f,c0,c1)
-- nth order
odeNre: (List ST -> ST,List Coef,List ST) -> List ST
odeNre(f,cl,yl) ==
-- yl is [y, y', ..., y<n>]
-- integrate [y',..,y<n>] to get [y,..,y<n-1>]
yil := [lazyIntegrate(c,y)$STT for c in cl for y in rest yl]
-- use y<n> = f(y,..,y<n-1>)
concat(yil,[f yil])
iOde: ((L ST) -> ST,List Coef) -> ST
iOde(f,cl) == first YS(odeNre(f,cl,#1),#cl + 1)
ode(f,cl) == series iOde(stFuncN f,cl)
simulre:(L Coef,L ((L ST) -> ST),L ST) -> L ST
simulre(cst,lsf,c) ==
[lazyIntegrate(csti,lsfi concat(monom(1,1)$STT,c))_
for csti in cst for lsfi in lsf]
iMpsode:(L Coef,L ((L ST) -> ST)) -> L ST
iMpsode(cs,lsts) == YS(simulre(cs,lsts,#1),# cs)
mpsode(cs,lsts) ==
-- stSol := iMpsode(cs,map(stFuncN,lsts)$L2(FN,(L ST) -> ST))
stSol := iMpsode(cs,[stFuncN(lst) for lst in lsts])
map(series,stSol)$L2(ST,UTS)
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