Partial Integro-Differential Operators

Markus.Rosenkranz@oeaw.ac.at

School of Mathematics, Statistics and Actuarial ScienceUniversity of Kent at Canterbury

CT1 2DX, United Kingdom

Joint work with G. Regensburger, L. Tec and B. Buchberger

ACA 2010

Applications of Computer AlgebraVlora, Albania, 24 June 2010

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene: Geometry.

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene:

Geometry.

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene:

Geometry.

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene:

Geometry.

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene:

Geometry.

M. Rosenkranz Partial Integro-Differential Operators

(Integro-)Differential Operators

Ordinary differentialoperators F [∂]

⊆Partial differentialoperators F [∂x , ∂y ]⊇ ⊇

Ordinary integro-differentialoperators F [∂,

r]

⊆Partial integro-differentialoperators F [∂x , ∂y ,

r x,r y

]

But how should F [∂x , ∂y ,r x,r y

] be defined?

Consider F = C∞(R2) for simplicity.

Need more than ∂x , ∂y ,r x,r y and f ∈ F and ϕ ∈ F ∗.

A new player enters the scene: Geometry.

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebra

T ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)

β1, . . . , βn ∈ F∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Recap: Ordinary BndProb

Given f ∈ F , find u ∈ F such that:

Tu = fβ1(u) = · · · = βn(u) = 0

Role of f as symbolic parameter

F = C∞[a, b] Function algebraT ∈ F [∂] Differential operator (monic of degree n)β1, . . . , βn ∈ F

∗ Boundary functionals

Classical two-point boundary functionals (ai , bi ∈ C):

β(u) = a0 u(a) + · · ·+ an−1u(n−1)(a) + b0 u(b) + · · ·+ bn−1u(n−1)(b)

Other boundary functionals for Stieltjes boundary conditions

Invariant description via B = [β1, . . . , βn] ≤ F ∗

Regular boundary problem: ∀f ∃!u

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):

Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):

Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):

Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):

Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗

, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Da capo: Partial BndProb

Consider a domain Ω ⊂ R2:

Tu = fβiu = 0 (i ∈ I)

T ∈ F [∂x , ∂y ]βi ∈ F

∗

Typical boundary conditions (I = ∂Ω):Dirichlet conditions: βξ = u(ξ)

Neumann conditions: βξ = ∂u∂n (ξ)

Robin conditions: βξ = a u(ξ) + b ∂u∂n (ξ)

Invariant description via B = [βi]i∈I ≤ F∗, regularity as before

Simple example (more details later):

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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.

LODEs / LPDEs, systems→ regularity, Green’s operator. . .

Multiplication: (T1,B1) · (T2,B2) , (T1T2,B1T2 + B2)

Cascade integration:((T1,B1) · (T2,B2)

)−1= (T2,B2)−1 · (T1,B1)−1

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

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Factorization Example of Partial BndProb

Unbounded wave equation:

(Dtt − Dxx , [Lt , Lt Dt ]) = (Dt − Dx , [Lt ]) · (Dt + Dx , [Lt ])

or utt − uxx = fu(x, 0) = ut (x, 0) = 0 =

ut − ux = fu(x, 0) = 0 ·

ut + ux = fu(x, 0) = 0

x

t

uHx,0L=utHx,0L=0

uH0,tL=0 uH1,tL=0 Bounded wave equation:(Dtt − Dxx , [Lt , Lt Dt , Lx ,Rx ]) = (Dt − Dx , [Lt ,S]) · (Dt + Dx , [Lt , Lx ])

or utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

ut − ux = fu(x, 0) =

r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

M. Rosenkranz Partial Integro-Differential Operators

Geometric Interpretation for Bounded Wave Equation

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

t=0x=0 x=1

Hx,tL

H0,t-xL

H1,t+x-1LH1-x,t-1L

+

-

+

-

+

+

-

+

-

+

±

±

1

M. Rosenkranz Partial Integro-Differential Operators

Geometric Interpretation for Bounded Wave Equation

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 =

t=0x=0 x=1

Hx,tL

H0,t-xL

H1,t+x-1LH1-x,t-1L

+

-

+

-

+

+

-

+

-

+

±

±

1

M. Rosenkranz Partial Integro-Differential Operators

Geometric Interpretation for Bounded Wave Equation

utt − uxx = fu(x, 0) = ut (x, 0) = u(0, t) = u(1, t) = 0 = ut − ux = f

u(x, 0) =r 1(1−t)+

u(ξ, ξ + t − 1) dξ = 0 ·ut + ux = fu(x, 0) = u(0, t) = 0

t=0x=0 x=1

Hx,tL

H0,t-xL

H1,t+x-1LH1-x,t-1L

+

-

+

-

+

+

-

+

-

+

±

±

1

t=0x=0 x=1

Hx,tL

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

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)

M. Rosenkranz Partial Integro-Differential Operators

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)

M. Rosenkranz Partial Integro-Differential Operators

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)

M. Rosenkranz Partial Integro-Differential Operators

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)

M. Rosenkranz Partial Integro-Differential Operators

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)

M. Rosenkranz Partial Integro-Differential Operators

Back to Unbounded Wave Equation

Constant-coefficient first-order boundary problem:

a ux + b ut = fu(kt + c, t) = 0

Solution by standard methods:

u(x, y) =1a

∫ x

Xf(ξ, b

a (ξ − x) + t) dξ with X =ac+(at−bx)k

a−bk

Partial integro-differential operator (c = 0):

Ga,b ,k =(

1/K −k/K−b/L a/L

) r x (a kL/Kb L/K

)with K = a − bk , L = a2 + b2

Green’s Operator for Unbounded Wave Equation:

Gf(x, t) =r t

0

r τ0f(ξ, 2τ − ξ + x − t) dξ dτ

G = G1,−1,0G1,1,0 =(

1 0−1 1

) r x( 1 02 1

) r x( 1 0−1 1

)M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators

Future (and Current) Work

A long way to “real” partial boundary problems:

Complete the collection of rules to a Grobner basis

Generalize from linear to affine substitutions

Include evaluations by admitting singular matrices

Generalize from R2 to Rn

Allow convex polyhedra as integration domains

Generalize from linear to polynomial substitutions

Use them as coordinate charts for manifolds

That’s all folks. . .

THANK YOU

M. Rosenkranz Partial Integro-Differential Operators