.. status: ok



9.60 OrderlyDifferentialPolynomial
----------------------------------

Many systems of differential equations may be transformed to equivalent
systems of ordinary differential equations where the equations are
expressed polynomially in terms of the unknown functions. In FriCAS, the
domain constructors OrderlyDifferentialPolynomial (abbreviated ODPOL)
and SequentialDifferentialPolynomial (abbreviation SDPOL) implement two
domains of ordinary differential polynomials over any differential ring.
In the simplest case, this differential ring is usually either the ring
of integers, or the field of rational numbers. However, FriCAS can
handle ordinary differential polynomials over a field of rational
functions in a single indeterminate.

The two domains ODPOL and SDPOL are almost identical, the only
difference being the choice of a different ranking, which is an ordering
of the derivatives of the indeterminates. The first domain uses an
orderly ranking, that is, derivatives of higher order are ranked higher,
and derivatives of the same order are ranked alphabetically. The second
domain uses a sequential ranking, where derivatives are ordered first
alphabetically by the differential indeterminates, and then by order. A
more general domain constructor,
DifferentialSparseMultivariatePolynomial (abbreviation DSMP) allows both
a user-provided list of differential indeterminates as well as a
user-defined ranking. We shall illustrate ODPOL(FRAC INT), which
constructs a domain of ordinary differential polynomials in an arbitrary
number of differential indeterminates with rational numbers as
coefficients.


.. spadInput
::

	dpol:= ODPOL(FRAC INT)


.. spadMathAnswer
.. spadMathOutput
.. math::

+------------------------------------------------+
| OrderlyDifferentialPolynomialFractionInteger   |
+------------------------------------------------+




.. spadType

:sub:`Type: Domain`



A differential indeterminate w may be viewed as an infinite sequence of
algebraic indeterminates, which are the derivatives of w. To facilitate
referencing these, FriCAS provides the operation
makeVariablemakeVariableOrderlyDifferentialPolynomial to convert an
element of type Symbol to a map from the natural numbers to the
differential polynomial ring.


.. spadInput
::

	w := makeVariable('w)$dpol


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------------+
| theMap(...)   |
+---------------+




.. spadType

:sub:`Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction`
Integer)




.. spadInput
::

	z := makeVariable('z)$dpol


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------------+
| theMap(...)   |
+---------------+




.. spadType

:sub:`Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction`
Integer)



The fifth derivative of w can be obtained by applying the map w to the
number 5. Note that the order of differentiation is given as a subscript
(except when the order is 0).


.. spadInput
::

	w.5


.. spadMathAnswer
.. spadMathOutput
.. math::

+------+
| w5   |
+------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`




.. spadInput
::

	w 0


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| w   |
+-----+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The first five derivatives of z can be generated by a list.


.. spadInput
::

	[z.i for i in 1..5]


.. spadMathAnswer
.. spadMathOutput
.. math::

+--------------------+
| [z1,z2,z3,z4,z5]   |
+--------------------+




.. spadType

:sub:`Type: List OrderlyDifferentialPolynomial Fraction Integer`



The usual arithmetic can be used to form a differential polynomial from
the derivatives.


.. spadInput
::

	f:= w.4 - w.1 * w.1 * z.3


.. spadMathAnswer
.. spadMathOutput
.. math::

+------------+
| w4-w12z3   |
+------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`




.. spadInput
::

	g:=(z.1)^3 * (z.2)^2 - w.2


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------------+
| z13z22-w2   |
+-------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The operation DDOrderlyDifferentialPolynomial computes the derivative of
any differential polynomial.


.. spadInput
::

	D(f)


.. spadMathAnswer
.. spadMathOutput
.. math::

+--------------------+
| w5-w12z4-2w1w2z3   |
+--------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The same operation can compute higher derivatives, like the fourth
derivative.


.. spadInput
::

	D(f,4)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------------------------------------------------------------------------------+
| w8-w12z7-8w1w2z6+(-12w1w3-12w22)z5-2w1z3w5+(-8w1w4-24w2w3)z4-8w2z3w4-6w32z3   |
+-------------------------------------------------------------------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The operation makeVariablemakeVariableOrderlyDifferentialPolynomial
creates a map to facilitate referencing the derivatives of f, similar to
the map w.


.. spadInput
::

	df:=makeVariable(f)$dpol


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------------+
| theMap(...)   |
+---------------+




.. spadType

:sub:`Type: (NonNegativeInteger -> OrderlyDifferentialPolynomial Fraction`
Integer)



The fourth derivative of f may be referenced easily.


.. spadInput
::

	df.4


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------------------------------------------------------------------------------+
| w8-w12z7-8w1w2z6+(-12w1w3-12w22)z5-2w1z3w5+(-8w1w4-24w2w3)z4-8w2z3w4-6w32z3   |
+-------------------------------------------------------------------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The operation orderorderOrderlyDifferentialPolynomial returns the order
of a differential polynomial, or the order in a specified differential
indeterminate.


.. spadInput
::

	order(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| 2   |
+-----+




.. spadType

:sub:`Type: PositiveInteger`




.. spadInput
::

	order(g, 'w)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| 2   |
+-----+




.. spadType

:sub:`Type: PositiveInteger`



The operation
differentialVariablesdifferentialVariablesOrderlyDifferentialPolynomial
returns a list of differential indeterminates occurring in a
differential polynomial.


.. spadInput
::

	differentialVariables(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------+
| [z,w]   |
+---------+




.. spadType

:sub:`Type: List Symbol`



The operation degreedegreeOrderlyDifferentialPolynomial returns the
degree, or the degree in the differential indeterminate specified.


.. spadInput
::

	degree(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+----------+
| z22z13   |
+----------+




.. spadType

:sub:`Type: IndexedExponents OrderlyDifferentialVariable Symbol`




.. spadInput
::

	degree(g, 'w)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| 1   |
+-----+




.. spadType

:sub:`Type: PositiveInteger`



The operation weightsweightsOrderlyDifferentialPolynomial returns a list
of weights of differential monomials appearing in differential
polynomial, or a list of weights in a specified differential
indeterminate.


.. spadInput
::

	weights(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------+
| [7,2]   |
+---------+




.. spadType

:sub:`Type: List NonNegativeInteger`




.. spadInput
::

	weights(g,'w)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------+
| [2]   |
+-------+




.. spadType

:sub:`Type: List NonNegativeInteger`



The operation weightweightOrderlyDifferentialPolynomial returns the
maximum weight of all differential monomials appearing in the
differential polynomial.


.. spadInput
::

	weight(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| 7   |
+-----+




.. spadType

:sub:`Type: PositiveInteger`



A differential polynomial is isobaric if the weights of all differential
monomials appearing in it are equal.


.. spadInput
::

	isobaric?(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------+
| false   |
+---------+




.. spadType

:sub:`Type: Boolean`



To substitute differentially, use evalevalOrderlyDifferentialPolynomial.
Note that we must coerce 'w to Symbol, since in ODPOL, differential
indeterminates belong to the domain Symbol. Compare this result to the
next, which substitutes algebraically (no substitution is done since w.0
does not appear in g).


.. spadInput
::

	eval(g,['w::Symbol],[f])


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------------------------------------------+
| -w6+w12z5+4w1w2z4+(2w1w3+2w22)z3+z13z22   |
+-------------------------------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`




.. spadInput
::

	eval(g,variables(w.0),[f])


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------------+
| z13z22-w2   |
+-------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Since OrderlyDifferentialPolynomial belongs to PolynomialCategory, all
the operations defined in the latter category, or in packages for the
latter category, are available.


.. spadInput
::

	monomials(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+----------------+
| [z13z22,-w2]   |
+----------------+




.. spadType

:sub:`Type: List OrderlyDifferentialPolynomial Fraction Integer`




.. spadInput
::

	variables(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+--------------+
| [z2,w2,z1]   |
+--------------+




.. spadType

:sub:`Type: List OrderlyDifferentialVariable Symbol`




.. spadInput
::

	gcd(f,g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----+
| 1   |
+-----+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`




.. spadInput
::

	groebner([f,g])


.. spadMathAnswer
.. spadMathOutput
.. math::

+------------------------+
| [w4-w12z3,z13z22-w2]   |
+------------------------+




.. spadType

:sub:`Type: List OrderlyDifferentialPolynomial Fraction Integer`



The next three operations are essential for elimination procedures in
differential polynomial rings. The operation
leaderleaderOrderlyDifferentialPolynomial returns the leader of a
differential polynomial, which is the highest ranked derivative of the
differential indeterminates that occurs.


.. spadInput
::

	lg:=leader(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+------+
| z2   |
+------+




.. spadType

:sub:`Type: OrderlyDifferentialVariable Symbol`



The operation separantseparantOrderlyDifferentialPolynomial returns the
separant of a differential polynomial, which is the partial derivative
with respect to the leader.


.. spadInput
::

	sg:=separant(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+----------+
| 2z13z2   |
+----------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



The operation initialinitialOrderlyDifferentialPolynomial returns the
initial, which is the leading coefficient when the given differential
polynomial is expressed as a polynomial in the leader.


.. spadInput
::

	ig:=initial(g)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-------+
| z13   |
+-------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Using these three operations, it is possible to reduce f modulo the
differential ideal generated by g. The general scheme is to first reduce
the order, then reduce the degree in the leader. First, eliminate z.3
using the derivative of g.


.. spadInput
::

	g1 := D g


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----------------------+
| 2z13z2z3-w3+3z12z23   |
+-----------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Find its leader.


.. spadInput
::

	lg1:= leader g1


.. spadMathAnswer
.. spadMathOutput
.. math::

+------+
| z3   |
+------+




.. spadType

:sub:`Type: OrderlyDifferentialVariable Symbol`



Differentiate f partially with respect to this leader.


.. spadInput
::

	pdf:=D(f, lg1)


.. spadMathAnswer
.. spadMathOutput
.. math::

+--------+
| -w12   |
+--------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Compute the partial remainder of f with respect to g.


.. spadInput
::

	prf:=sg * f- pdf * g1


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----------------------------+
| 2z13z2w4-w12w3+3w12z12z23   |
+-----------------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Note that high powers of lg still appear in prf. Compute the leading
coefficient of prf as a polynomial in the leader of g.


.. spadInput
::

	lcf:=leadingCoefficient univariate(prf, lg)


.. spadMathAnswer
.. spadMathOutput
.. math::

+-----------+
| 3w12z12   |
+-----------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`



Finally, continue eliminating the high powers of lg appearing in prf to
obtain the (pseudo) remainder of f modulo g and its derivatives.


.. spadInput
::

	ig * prf - lcf * g * lg


.. spadMathAnswer
.. spadMathOutput
.. math::

+---------------------------------+
| 2z16z2w4-w12z13w3+3w12z12w2z2   |
+---------------------------------+




.. spadType

:sub:`Type: OrderlyDifferentialPolynomial Fraction Integer`