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\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra nlode.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{package NODE1 NonLinearFirstOrderODESolver}
<<package NODE1 NonLinearFirstOrderODESolver>>=
)abbrev package NODE1 NonLinearFirstOrderODESolver
++ Author: Manuel Bronstein
++ Date Created: 2 September 1991
++ Date Last Updated: 14 October 1994
++ Description: NonLinearFirstOrderODESolver provides a function
++ for finding closed form first integrals of nonlinear ordinary
++ differential equations of order 1.
++ Keywords: differential equation, ODE
NonLinearFirstOrderODESolver(R, F): Exports == Implementation where
R: Join(EuclideanDomain, RetractableTo Integer,
LinearlyExplicitRingOver Integer, CharacteristicZero)
F: Join(AlgebraicallyClosedFunctionSpace R, TranscendentalFunctionCategory,
PrimitiveFunctionCategory)
N ==> NonNegativeInteger
Q ==> Fraction Integer
UQ ==> Union(Q, "failed")
OP ==> BasicOperator
SY ==> Symbol
K ==> Kernel F
U ==> Union(F, "failed")
P ==> SparseMultivariatePolynomial(R, K)
REC ==> Record(coef:Q, logand:F)
SOL ==> Record(particular: F,basis: List F)
BER ==> Record(coef1:F, coefn:F, exponent:N)
Exports ==> with
solve: (F, F, OP, SY) -> U
++ solve(M(x,y), N(x,y), y, x) returns \spad{F(x,y)} such that
++ \spad{F(x,y) = c} for a constant \spad{c} is a first integral
++ of the equation \spad{M(x,y) dx + N(x,y) dy = 0}, or
++ "failed" if no first-integral can be found.
Implementation ==> add
import ODEIntegration(R, F)
import ElementaryFunctionODESolver(R, F) -- recursive dependency!
checkBernoulli : (F, F, K) -> Union(BER, "failed")
solveBernoulli : (BER, OP, SY, F) -> Union(F, "failed")
checkRiccati : (F, F, K) -> Union(List F, "failed")
solveRiccati : (List F, OP, SY, F) -> Union(F, "failed")
partSolRiccati : (List F, OP, SY, F) -> Union(F, "failed")
integratingFactor: (F, F, SY, SY) -> U
unk := new()$SY
kunk:K := kernel unk
solve(m, n, y, x) ==
-- first replace the operator y(x) by a new symbol z in m(x,y) and n(x,y)
lk:List(K) := [retract(yx := y(x::F))@K]
lv:List(F) := [kunk::F]
mm := eval(m, lk, lv)
nn := eval(n, lk, lv)
-- put over a common denominator (to balance m and n)
d := lcm(denom mm, denom nn)::F
mm := d * mm
nn := d * nn
-- look for an integrating factor mu
(u := integratingFactor(mm, nn, unk, x)) case F =>
mu := u::F
mm := mm * mu
nn := nn * mu
eval(int(mm,x) + int(nn-int(differentiate(mm,unk),x), unk),[kunk],[yx])
-- check for Bernoulli equation
(w := checkBernoulli(m, n, k1 := first lk)) case BER =>
solveBernoulli(w::BER, y, x, yx)
-- check for Riccati equation
(v := checkRiccati(m, n, k1)) case List(F) =>
solveRiccati(v::List(F), y, x, yx)
"failed"
-- look for an integrating factor
integratingFactor(m, n, y, x) ==
-- check first for exactness
zero?(d := differentiate(m, y) - differentiate(n, x)) => 1
-- look for an integrating factor involving x only
not member?(y, variables(f := d / n)) => expint(f, x)
-- look for an integrating factor involving y only
not member?(x, variables(f := - d / m)) => expint(f, y)
-- room for more techniques later on (e.g. Prelle-Singer etc...)
"failed"
-- check whether the equation is of the form
-- dy/dx + p(x)y + q(x)y^N = 0 with N > 1
-- i.e. whether m/n is of the form p(x) y + q(x) y^N
-- returns [p, q, N] if the equation is in that form
checkBernoulli(m, n, ky) ==
r := denom(f := m / n)::F
(not freeOf?(r, y := ky::F))
or (d := degree(p := univariate(numer f, ky))) < 2
or not one? degree(pp := reductum p) or reductum(pp) ~= 0
or (not freeOf?(a := (leadingCoefficient(pp)::F), y))
or (not freeOf?(b := (leadingCoefficient(p)::F), y)) => "failed"
[a / r, b / r, d]
-- solves the equation dy/dx + rec.coef1 y + rec.coefn y^rec.exponent = 0
-- the change of variable v = y^{1-n} transforms the above equation to
-- dv/dx + (1 - n) p v + (1 - n) q = 0
solveBernoulli(rec, y, x, yx) ==
n1 := 1 - rec.exponent::Integer
deq := differentiate(yx, x) + n1 * rec.coef1 * yx + n1 * rec.coefn
sol := solve(deq, y, x)::SOL -- can always solve for order 1
-- if v = vp + c v0 is the general solution of the linear equation, then
-- the general first integral for the Bernoulli equation is
-- (y^{1-n} - vp) / v0 = c for any constant c
(yx**n1 - sol.particular) / first(sol.basis)
-- check whether the equation is of the form
-- dy/dx + q0(x) + q1(x)y + q2(x)y^2 = 0
-- i.e. whether m/n is a quadratic polynomial in y.
-- returns the list [q0, q1, q2] if the equation is in that form
checkRiccati(m, n, ky) ==
q := denom(f := m / n)::F
(not freeOf?(q, y := ky::F)) or degree(p := univariate(numer f, ky)) > 2
or (not freeOf?(a0 := (coefficient(p, 0)::F), y))
or (not freeOf?(a1 := (coefficient(p, 1)::F), y))
or (not freeOf?(a2 := (coefficient(p, 2)::F), y)) => "failed"
[a0 / q, a1 / q, a2 / q]
-- solves the equation dy/dx + l.1 + l.2 y + l.3 y^2 = 0
solveRiccati(l, y, x, yx) ==
-- get first a particular solution
(u := partSolRiccati(l, y, x, yx)) case "failed" => "failed"
-- once a particular solution yp is known, the general solution is of the
-- form y = yp + 1/v where v satisfies the linear 1st order equation
-- v' - (l.2 + 2 l.3 yp) v = l.3
deq := differentiate(yx, x) - (l.2 + 2 * l.3 * u::F) * yx - l.3
gsol := solve(deq, y, x)::SOL -- can always solve for order 1
-- if v = vp + c v0 is the general solution of the above equation, then
-- the general first integral for the Riccati equation is
-- (1/(y - yp) - vp) / v0 = c for any constant c
(inv(yx - u::F) - gsol.particular) / first(gsol.basis)
-- looks for a particular solution of dy/dx + l.1 + l.2 y + l.3 y^2 = 0
partSolRiccati(l, y, x, yx) ==
-- we first do the change of variable y = z / l.3, which transforms
-- the equation into dz/dx + l.1 l.3 + (l.2 - l.3'/l.3) z + z^2 = 0
q0 := l.1 * (l3 := l.3)
q1 := l.2 - differentiate(l3, x) / l3
-- the equation dz/dx + q0 + q1 z + z^2 = 0 is transformed by the change
-- of variable z = w'/w into the linear equation w'' + q1 w' + q0 w = 0
lineq := differentiate(yx, x, 2) + q1 * differentiate(yx, x) + q0 * yx
-- should be made faster by requesting a particular nonzero solution only
(not((gsol := solve(lineq, y, x)) case SOL))
or empty?(bas := (gsol::SOL).basis) => "failed"
differentiate(first bas, x) / (l3 * first bas)
@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
-- - Redistributions of source code must retain the above copyright
-- notice, this list of conditions and the following disclaimer.
--
-- - Redistributions in binary form must reproduce the above copyright
-- notice, this list of conditions and the following disclaimer in
-- the documentation and/or other materials provided with the
-- distribution.
--
-- - Neither the name of The Numerical ALgorithms Group Ltd. nor the
-- names of its contributors may be used to endorse or promote products
-- derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>
-- Compile order for the differential equation solver:
-- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad odeef.spad
<<package NODE1 NonLinearFirstOrderODESolver>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
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