Section9.46

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## 9.46 LinearOrdinaryDifferentialOperator2

LinearOrdinaryDifferentialOperator2(A, M) is the domain of linear ordinary differential operators with coefficients in the differential ring A and operating on M, an A-module. This includes the cases of operators which are polynomials in D acting upon scalar or vector expressions of a single variable. The coefficients of the operator polynomials can be integers, rational functions, matrices or elements of other domains.

### 9.46.1 Differential Operators with Constant Coefficients

This example shows differential operators with rational number coefficients operating on univariate polynomials.

We begin by making type assignments so we can conveniently refer to univariate polynomials in x over the rationals.

Q := Fraction Integer

FractionInteger

Type: Domain
PQ := UnivariatePolynomial('x, Q)

UnivariatePolynomial(x,FractionInteger)

Type: Domain
x: PQ := 'x

x

Type: UnivariatePolynomial(x,Fraction Integer)

Now we assign Dx to be the differential operator DDLinearOrdinaryDifferentialOperator2 corresponding to d/dx.

Dx: LODO2(Q, PQ) := D()

D

Type: LinearOrdinaryDifferentialOperator2( Fraction Integer, UnivariatePolynomial(x,Fraction Integer))

New operators are created as polynomials in D().

a := Dx + 1

D+1

Type: LinearOrdinaryDifferentialOperator2( Fraction Integer, UnivariatePolynomial(x,Fraction Integer))
b := a + 1/2*Dx**2 - 1/2

1 2 D 2+D+1 2

Type: LinearOrdinaryDifferentialOperator2( Fraction Integer, UnivariatePolynomial(x,Fraction Integer))

To apply the operator a to the value p the usual function call syntax is used.

p := 4*x**2 + 2/3

4 x 2+2 3

Type: UnivariatePolynomial(x,Fraction Integer)
a p

4 x 2+8 x+2 3

Type: UnivariatePolynomial(x,Fraction Integer)

Operator multiplication is defined by the identity (a*b) p = a(b(p))

(a * b) p = a b p

2 x 2+12 x+37 3=2 x 2+12 x+37 3

Type: Equation UnivariatePolynomial(x,Fraction Integer)

Exponentiation follows from multiplication.

c := (1/9)*b*(a + b)**2

1 72 D 6+5 36 D 5+13 24 D 4+19 18 D 3+79 72 D 2+7 12 D+1 8

Type: LinearOrdinaryDifferentialOperator2( Fraction Integer, UnivariatePolynomial(x,Fraction Integer))

Finally, note that operator expressions may be applied directly.

(a**2 - 3/4*b + c) (p + 1)

3 x 2+44 3 x+541 36

Type: UnivariatePolynomial(x,Fraction Integer)

### 9.46.2 Differential Operators with Matrix Coefficients Operating on Vectors

This is another example of linear ordinary differential operators with non-commutative multiplication. Unlike the rational function case, the differential ring of square matrices (of a given dimension) with univariate polynomial entries does not form a field. Thus the number of operations available is more limited.

In this section, the operators have three by three matrix coefficients with polynomial entries.

PZ := UnivariatePolynomial(x,Integer)

UnivariatePolynomial(x, Integer)

Type: Domain
x:PZ := 'x

x

Type: UnivariatePolynomial(x,Integer)
Mat := SquareMatrix(3,PZ)

SquareMatrix(3, UnivariatePolynomial(x, Integer))

Type: Domain

The operators act on the vectors considered as a Mat-module.

Vect := DPMM(3, PZ, Mat, PZ)

DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3,UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))

Type: Domain
Modo := LODO2(Mat, Vect)

LinearOrdinaryDifferentialOperator2( SquareMatrix(3,UnivariatePolynomial(x,Integer)), DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3,UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer)))

Type: Domain

The matrix m is used as a coefficient and the vectors p and q are operated upon.

m:Mat := matrix [ [x**2,1,0],[1,x**4,0],[0,0,4*x**2] ]

[x 2 1 0 1 x 4 0 0 0 4 x 2]

Type: SquareMatrix(3,UnivariatePolynomial(x,Integer))
p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]

[3 x 2+1, 2 x, 7 x 3+2 x]

Type: DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3,UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))
q: Vect := m * p

[3 x 4+x 2+2 x, 2 x 5+3 x 2+1, 28 x 5+8 x 3]

Type: DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3,UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))

Now form a few operators.

Dx : Modo := D()

D

Type: LinearOrdinaryDifferentialOperator2( SquareMatrix(3,UnivariatePolynomial(x,Integer)), DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3,UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer)))
a : Modo := Dx + m

D+[x 2 1 0 1 x 4 0 0 0 4 x 2]

Type: LinearOrdinaryDifferentialOperator2( SquareMatrix(3,UnivariatePolynomial(x,Integer)), DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer)))
b : Modo := m*Dx + 1

[x 2 1 0 1 x 4 0 0 0 4 x 2]D+[1 0 0 0 1 0 0 0 1]

Type: LinearOrdinaryDifferentialOperator2( SquareMatrix(3, UnivariatePolynomial(x,Integer)), DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer)))
c := a*b

[x 2 1 0 1 x 4 0 0 0 4 x 2]D 2+[x 4+2 x+2 x 4+x 2 0 x 4+x 2 x 8+4 x 3+2 0 0 0 16 x 4+8 x+1]D+[x 2 1 0 1 x 4 0 0 0 4 x 2]

Type: LinearOrdinaryDifferentialOperator2( SquareMatrix(3, UnivariatePolynomial(x,Integer)), DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer)))

These operators can be applied to vector values.

a p

[3 x 4+x 2+8 x, 2 x 5+3 x 2+3, 28 x 5+8 x 3+21 x 2+2]

Type: DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))
b p

[6 x 3+3 x 2+3, 2 x 4+8 x, 84 x 4+7 x 3+8 x 2+2 x]

Type: DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))
(a + b + c) (p + q)

[10 x 8+12 x 7+16 x 6+30 x 5+85 x 4+94 x 3+40 x 2+40 x+17, 10 x 12+10 x 9+12 x 8+92 x 7+6 x 6+32 x 5+72 x 4+28 x 3+49 x 2+32 x+19, 2240 x 8+224 x 7+1280 x 6+3508 x 5+492 x 4+751 x 3+98 x 2+18 x+4]

Type: DirectProductMatrixModule(3, UnivariatePolynomial(x,Integer), SquareMatrix(3, UnivariatePolynomial(x,Integer)), UnivariatePolynomial(x,Integer))
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