Integro-differential Operators via Normal Forms in
M APLE
Anja Korporal
Research Institute for Symbolic Computation
Johannes Kepler University
Linz, Austria
Applications of Computer Algebra
Vlora, June 24, 2010
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
1 / 17
Outline
Available features
We show how our M APLE package IntDiffOp can
perform arithmetic operations in the algebra of integro-differential
operators,
compute solution operators (Green’s operators) for boundary
problems for LODEs from a given fundamental system,
multiply boundary problems in a way corresponding to
multiplication of their solution operators,
lift given factorizations of differential operators to boundary
problems.
Download
http://www.risc.jku.at/people/akorpora/index.html
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
2 / 17
Integro-differential Algebras
Definition
r
An integro-differential algebra (F, ∂, ) consists of a commutative
differential algebra (F, ∂) over a field K and
r
a K -linear right inverse of ∂
r
r
r
r
r
( f 0 )( g 0 ) = ( f 0 )g + f ( g 0 ) − (fg) (differential Baxter axiom)
r
We always have the evaluation E = 1 − ∂.
Example
r
rx
C ∞ (R) together with usual derivation and integration = 0 .
In this case, the formulae above and the evaluation are just
fundamental theorem of calculus
a version of integration by parts
E : f 7→ f (0).
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
3 / 17
Integro-differential Operators
Definition
r
The integro-differential operators F[∂, ] is the free K -algebra in
r
the symbols ∂ and ,
the functions f ∈ F,
a collection of multiplicative functionals Φ ⊆ F ∗ ,
modulo the Noetherian and confluent rewrite system given by
fg
→
f •g
ϕψ
→
ψ
ϕf
r r
f
r
f∂
r
fϕ
→
Anja Korporal (RISC)
→
→
→
∂f
→
∂ • f + f∂
∂ϕ → 0
r
(ϕ • f ) ϕ ∂
→ 1
r
r r r
( • f) − ( • f)
r
f − (∂ • f ) − (E •f ) E
r
( • f)ϕ
Integro-differential Operators in M APLE
ACA 2010/06/24
4 / 17
Implementation in General
General setting
As F, we use “C ∞ (R)” functions, that are representable in M APLE.
r
∂, are the usual derivation and integration.
Φ ⊂ F ∗ consists of arbitrary point evaluations.
Notation
The function parameter is denoted by x.
The differential operator ∂ is represented by D.
r
Rx
The integral operator = 0 is represented by A.
Point evaluations are denoted by E[c], where c ∈ R is the
evaluation point.
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
5 / 17
Normal forms of Integro-differential Operators
Basic strategy
We implemented arithmetic operations directly in normal forms.
Normal forms
Direct sum of a differential, an integral, and a so-called Stieltjes
boundary operator.
Differential operators are sums of terms of the form f ∂ i with f ∈ F.
r
Integral Operators are sums of terms of the form f g with f , g ∈ F.
Data Structures
INTDIFFOP(DIFFOP(. . .), INTOP(. . .), BOUNDOP(. . .))
DIFFOP(f0 , f1 , f2 , ...), where fi are the coefficients of Di
INTOP(INTTERM(f1 , g1 ), INTTERM(f2 , g2 ), ...), where
INTTERM(f , g) represents the term f A g
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
6 / 17
Normal forms of Integro-differential Operators
Normal Forms of Boundary Operators
r
Boundary operators are sums of terms having the form f ϕ∂ i or f ϕ g
with ϕ ∈ Φ and f , g ∈ F.
Data Structures for Boundary Operators
BOUNDOP(EVOP, EVOP, ...) as a tuple of evaluations at different
points
EVOP(c, EVDIFFOP(. . .), EVINTOP(. . .)) where c is the evaluation
point
EVDIFFOP and EVINTOP similar to DIFFOP and INTOP
EVDIFFOP(f0 , f1 , f2 , ...), with fi the coefficients of E Di
EVINTOP(EVINTTERM(f1 , g1 ), ...), for f E A g + . . ..
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
7 / 17
Example
Arithmetic of Integro-differential Operators
The following M APLE session shows the multiplication rule for integral
operators,
r r
r
r r r
· f = ( • f ) − ( • f ),
with • meaning the application of the integral operator.
>
>
>
>
with(IntDiffOp):
op1 := INTOP(INTTERM(1,1)):
op2 := INTOP(INTTERM(f(x), 1)):
multiply(op1, op2);
Rx
Rx
0 f (x) dx . A − A . 0 f (x) dx
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
8 / 17
Boundary Problems
Example
u 00 = f
u(0) = u(1) = 0
General notation
Given
a forcing function f ∈ F,
a monic differential operator T ∈ F[∂] of order n,
n linear functionals β1 , . . . , βn ∈ F ∗ .
Find u ∈ F such that
Tu = f
β1 u = · · · = βn u = 0
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
9 / 17
Representation of Boundary Problems
In General
A boundary problem (T , [β1 , . . . , βn ]) is represented by
BP(DIFFOP(T ), BC(. . .))
BC(BOUNDOP(β1 ), . . . BOUNDOP(βn ))
Example
Back to the previous example.
>
>
>
>
T := DIFFOP(0,0,1):
b1 := BOUNDOP(EVOP(0, EVDIFFOP(1), EVINTOP())):
b2 := BOUNDOP(EVOP(1, EVDIFFOP(1), EVINTOP())):
Bp := BP(T, BC(b1, b2));
Bp := BP(D2 , BC(E[0], E[1]))
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
10 / 17
Green’s Operators
Definition
A boundary problem is called regular, if it has a unique solution for
each forcing function. The operator G : F → F, f 7→ u is called Green’s
operator.
Computation
> Bp := BP(T, BC(b1, b2));
Bp := BP(D2 , BC(E[0], E[1]))
> greens_op(Bp);
(x. A) − (A .x) − ((x E[1]). A) + ((x E[1]). A .x)
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
11 / 17
Advantages of our package
Example
Consider the boundary problem
u 000 − (ex + 2)u 00 − u 0 + (ex + 2)u = f
u(0) = u(1) = u 0 (1) + u 00 (0) = 0
M APLE can solve the differential equation and
solve it together with some easy boundary conditions, but
M APLE cannot solve this boundary problem systemetically.
The Green’s operator computed by our package is given in our
example worksheet.
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
12 / 17
Composition and Factorization of Boundary Problems
Definition
The composition of two boundary problems (T1 , [β1 , . . . , βn ]) and
(T2 , [γ1 , . . . , γm ]) is given by
(T1 T2 , [β1 T2 , . . . , βn T2 , γ1 , . . . , γm ]).
Theorem
For a regular boundary problem (T , [β1 , . . . , βn ]) every factorization of
the differential operator T = T1 T2 can be lifted to a factorization into
regular lower-order boundary problems.
Note
For factoring the differential operator, we use the function DFactor by
Mark van Hoeij in the M APLE package DEtools.
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
13 / 17
Example
Easy Example
u 00 = f
u(0) = u(1) = 0
=
u0 = f
u0 = f
R1
◦
u(0) = 0
0 u(ξ)dξ = 0
Sample Computation
> Bp := BP(T, BC(b1, b2));
Bp := BP(D2 , BC(E[0], E[1]))
> f1, f2 := factor_bp(Bp);
f1 , f2 := BP(D, BC(E[1]. A)), BP(D, BC(E[0]))
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
14 / 17
Outlook
Future Work
Computation of Green’s functions
Extension for singular boundary problems
Extension for multivariate functions
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
15 / 17
Thank you for your attention!
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
16 / 17
Download
http://www.risc.jku.at/people/akorpora/index.html
References
A. Korporal, G. Regensburger, and M. Rosenkranz.
Integro-differential operators via normal forms and boundary problems in Maple (extended
abstract).
In ISSAC 2010 (submitted).
G. Regensburger, M. Rosenkranz, and J. Middeke.
A skew polynomial approach to integro-differential operators.
In J. P. May, editor, Proceedings of ISSAC ’09, pages 287–294. ACM Press, 2009.
M. Rosenkranz and G. Regensburger.
Integro-differential polynomials and operators.
In D. Jeffrey, editor, Proceedings of ISSAC ’08, pages 261–268. ACM Press, 2008.
M. Rosenkranz and G. Regensburger.
Solving and factoring boundary problems for linear ordinary differential equations in
differential algebras.
J. Symbolic Comput., 43(8):515–544, 2008.
Anja Korporal (RISC)
Integro-differential Operators in M APLE
ACA 2010/06/24
17 / 17