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)abbrev package EXPRODE ExpressionSpaceODESolver
++ Taylor series solutions of ODE's
++ Author: Manuel Bronstein
++ Date Created: 5 Mar 1990
++ Date Last Updated: 30 September 1993
++ Description: Taylor series solutions of explicit ODE's;
++ Keywords: differential equation, ODE, Taylor series
ExpressionSpaceODESolver(R, F): Exports == Implementation where
R: Join(IntegralDomain, ConvertibleTo InputForm)
F: FunctionSpace R
K ==> Kernel F
P ==> SparseMultivariatePolynomial(R, K)
OP ==> BasicOperator
SY ==> Symbol
UTS ==> UnivariateTaylorSeries(F, x, center)
MKF ==> MakeUnaryCompiledFunction(F, UTS, UTS)
MKL ==> MakeUnaryCompiledFunction(F, List UTS, UTS)
A1 ==> AnyFunctions1(UTS)
AL1 ==> AnyFunctions1(List UTS)
EQ ==> Equation F
ODE ==> UnivariateTaylorSeriesODESolver(F, UTS)
Exports ==> with
seriesSolve: (EQ, OP, EQ, EQ) -> Any
++ seriesSolve(eq,y,x=a, y a = b) returns a Taylor series solution
++ of eq around x = a with initial condition \spad{y(a) = b}.
++ Note: eq must be of the form
++ \spad{f(x, y x) y'(x) + g(x, y x) = h(x, y x)}.
seriesSolve: (EQ, OP, EQ, List F) -> Any
++ seriesSolve(eq,y,x=a,[b0,...,b(n-1)]) returns a Taylor series
++ solution of eq around \spad{x = a} with initial conditions
++ \spad{y(a) = b0}, \spad{y'(a) = b1},
++ \spad{y''(a) = b2}, ...,\spad{y(n-1)(a) = b(n-1)}
++ eq must be of the form
++ \spad{f(x, y x, y'(x),..., y(n-1)(x)) y(n)(x) +
++ g(x,y x,y'(x),...,y(n-1)(x)) = h(x,y x, y'(x),..., y(n-1)(x))}.
seriesSolve: (List EQ, List OP, EQ, List EQ) -> Any
++ seriesSolve([eq1,...,eqn],[y1,...,yn],x = a,[y1 a = b1,...,yn a = bn])
++ returns a taylor series solution of \spad{[eq1,...,eqn]} around
++ \spad{x = a} with initial conditions \spad{yi(a) = bi}.
++ Note: eqi must be of the form
++ \spad{fi(x, y1 x, y2 x,..., yn x) y1'(x) +
++ gi(x, y1 x, y2 x,..., yn x) = h(x, y1 x, y2 x,..., yn x)}.
seriesSolve: (List EQ, List OP, EQ, List F) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
++ is equivalent to
++ \spad{seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,
++ [y1 a = b1,..., yn a = bn])}.
seriesSolve: (List F, List OP, EQ, List F) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn], x=a, [b1,...,bn])
++ is equivalent to
++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x=a, [b1,...,bn])}.
seriesSolve: (List F, List OP, EQ, List EQ) -> Any
++ seriesSolve([eq1,...,eqn], [y1,...,yn], x = a,[y1 a = b1,..., yn a = bn])
++ is equivalent to
++ \spad{seriesSolve([eq1=0,...,eqn=0], [y1,...,yn], x = a,
++ [y1 a = b1,..., yn a = bn])}.
seriesSolve: (EQ, OP, EQ, F) -> Any
++ seriesSolve(eq,y, x=a, b) is equivalent to
++ \spad{seriesSolve(eq, y, x=a, y a = b)}.
seriesSolve: (F, OP, EQ, F) -> Any
++ seriesSolve(eq, y, x = a, b) is equivalent to
++ \spad{seriesSolve(eq = 0, y, x = a, y a = b)}.
seriesSolve: (F, OP, EQ, EQ) -> Any
++ seriesSolve(eq, y, x = a, y a = b) is equivalent to
++ \spad{seriesSolve(eq=0, y, x=a, y a = b)}.
seriesSolve: (F, OP, EQ, List F) -> Any
++ seriesSolve(eq, y, x = a, [b0,...,bn]) is equivalent to
++ \spad{seriesSolve(eq = 0, y, x = a, [b0,...,b(n-1)])}.
Implementation ==> add
checkCompat: (OP, EQ, EQ) -> F
checkOrder1: (F, OP, K, SY, F) -> F
checkOrderN: (F, OP, K, SY, F, NonNegativeInteger) -> F
checkSystem: (F, List K, List F) -> F
div2exquo : F -> F
smp2exquo : P -> F
k2exquo : K -> F
diffRhs : (F, F) -> F
diffRhsK : (K, F) -> F
findCompat : (F, List EQ) -> F
findEq : (K, SY, List F) -> F
localInteger: F -> F
opelt := operator('elt)$OP
--opex := operator('exquo)$OP
opex := operator('fixedPointExquo)$OP
opint := operator('integer)$OP
Rint? := R has IntegerNumberSystem
localInteger n == (Rint? => n; opint n)
diffRhs(f, g) == diffRhsK(retract(f)@K, g)
k2exquo k ==
is?(op := operator k,'%diff) =>
error "Improper differential equation"
kernel(op, [div2exquo f for f in argument k]$List(F))
smp2exquo p ==
map(k2exquo,#1::F,p)$PolynomialCategoryLifting(IndexedExponents K,
K, R, P, F)
div2exquo f ==
one?(d := denom f) => f
opex(smp2exquo numer f, smp2exquo d)
-- if g is of the form a * k + b, then return -b/a
diffRhsK(k, g) ==
h := univariate(g, k)
(degree(numer h) <= 1) and ground? denom h =>
- coefficient(numer h, 0) / coefficient(numer h, 1)
error "Improper differential equation"
checkCompat(y, eqx, eqy) ==
lhs(eqy) =$F y(rhs eqx) => rhs eqy
error "Improper initial value"
findCompat(yx, l) ==
for eq in l repeat
yx =$F lhs eq => return rhs eq
error "Improper initial value"
findEq(k, x, sys) ==
k := retract(differentiate(k::F, x))@K
for eq in sys repeat
member?(k, kernels eq) => return eq
error "Improper differential equation"
checkOrder1(diffeq, y, yx, x, sy) ==
div2exquo subst(diffRhs(differentiate(yx::F,x),diffeq),[yx],[sy])
checkOrderN(diffeq, y, yx, x, sy, n) ==
zero? n => error "No initial value(s) given"
m := (minIndex(l := [retract(f := yx::F)@K]$List(K)))::F
lv := [opelt(sy, localInteger m)]$List(F)
for i in 2..n repeat
l := concat(retract(f := differentiate(f, x))@K, l)
lv := concat(opelt(sy, localInteger(m := m + 1)), lv)
div2exquo subst(diffRhs(differentiate(f, x), diffeq), l, lv)
checkSystem(diffeq, yx, lv) ==
for k in kernels diffeq repeat
is?(k, '%diff) =>
return div2exquo subst(diffRhsK(k, diffeq), yx, lv)
0
seriesSolve(l:List EQ, y:List OP, eqx:EQ, eqy:List EQ) ==
seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, eqy)
seriesSolve(l:List EQ, y:List OP, eqx:EQ, y0:List F) ==
seriesSolve([lhs deq - rhs deq for deq in l]$List(F), y, eqx, y0)
seriesSolve(l:List F, ly:List OP, eqx:EQ, eqy:List EQ) ==
seriesSolve(l, ly, eqx,
[findCompat(y rhs eqx, eqy) for y in ly]$List(F))
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, eqy:EQ) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, eqy)
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:F) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)
seriesSolve(diffeq:EQ, y:OP, eqx:EQ, y0:List F) ==
seriesSolve(lhs diffeq - rhs diffeq, y, eqx, y0)
seriesSolve(diffeq:F, y:OP, eqx:EQ, eqy:EQ) ==
seriesSolve(diffeq, y, eqx, checkCompat(y, eqx, eqy))
seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:F) ==
x := symbolIfCan(retract(lhs eqx)@K)::SY
sy := name y
yx := retract(y lhs eqx)@K
f := checkOrder1(diffeq, y, yx, x, sy::F)
center := rhs eqx
coerce(ode1(compiledFunction(f, sy)$MKF, y0)$ODE)$A1
seriesSolve(diffeq:F, y:OP, eqx:EQ, y0:List F) ==
x := symbolIfCan(retract(lhs eqx)@K)::SY
sy := new()$SY
yx := retract(y lhs eqx)@K
f := checkOrderN(diffeq, y, yx, x, sy::F, #y0)
center := rhs eqx
coerce(ode(compiledFunction(f, sy)$MKL, y0)$ODE)$A1
seriesSolve(sys:List F, ly:List OP, eqx:EQ, l0:List F) ==
x := symbolIfCan(kx := retract(lhs eqx)@K)::SY
fsy := (sy := new()$SY)::F
m := (minIndex(l0) - 1)::F
yx := concat(kx, [retract(y lhs eqx)@K for y in ly]$List(K))
lelt := [opelt(fsy, localInteger(m := m+1)) for k in yx]$List(F)
sys := [findEq(k, x, sys) for k in rest yx]
l := [checkSystem(eq, yx, lelt) for eq in sys]$List(F)
center := rhs eqx
coerce(mpsode(l0,[compiledFunction(f,sy)$MKL for f in l])$ODE)$AL1
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