Partial Integro-Differential Operators [email protected]School of Mathematics, Statistics and Actuarial Science University of Kent at Canterbury CT1 2DX, United Kingdom Joint work with G. Regensburger, L. Tec and B. Buchberger ACA 2010 Applications of Computer Algebra Vlora, Albania, 24 June 2010 M. Rosenkranz Partial Integro-Differential Operators
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Partial Integro-Differential Operators · M. Rosenkranz Partial Integro-Differential Operators. Da capo: Partial BndProb Consider a domain ˆR 2: Tu = f ... Da capo: Partial BndProb
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In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
Multiplying and Factoring BndProb
DefinitionA boundary problem is a pair (T ,B), where T is an epimorphismof a vector space V and B ≤ V∗ an orthogonally closed subspace.
In the above monoid, the regular boundary problems form a submonoiddually isomorphic to the monoid of Green’s operators.
TheoremLet (T ,B) be a regular boundary problem. A given factorizationT = T1T2 can be lifted to a factorization (T ,B) = (T1,B1) · (T2,B2) ofregular boundary problems with B2 ≤ B.
M. Rosenkranz Partial Integro-Differential Operators
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators
Partial Integro-Differential Operators
DefinitionThe partial integro-differential operators are defined as the algebra inthe following indeterminates given with their respective action on afunction f(x, y) ∈ C∞(R2).
Name Indeterminates Action
Differential operators ∂x , ∂y fx(x, y), fy(x, y)
Integral operatorsr x ,
r y r x0f(ξ, y) dξ,
r y0f(x, η) dη
Evaluation operators Lx , Ly f(0, y), f(x, 0)
Substitution operators(
a bc d
)∗∈ GL2(R) f(ax + by, cx + dy)
Selected Rewrite Rules:
Univariate: All rules of F [∂,r
] copied twice.
Chain Rule: ∂xM = a M∂x + c M∂y
Substitution Rule:r xM = 1
a (1 − Lx)Mr x
M. Rosenkranz Partial Integro-Differential Operators