partial integro-differential operators · m. rosenkranz partial integro-differential operators. da...
TRANSCRIPT
![Page 1: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/1.jpg)
Partial Integro-Differential Operators
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
![Page 2: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/2.jpg)
(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
![Page 3: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/3.jpg)
(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
![Page 4: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/4.jpg)
(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
![Page 5: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/5.jpg)
(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
![Page 6: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/6.jpg)
(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
![Page 7: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/7.jpg)
(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
![Page 8: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/8.jpg)
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
![Page 9: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/9.jpg)
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
![Page 10: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/10.jpg)
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
![Page 11: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/11.jpg)
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
![Page 12: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/12.jpg)
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
![Page 13: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/13.jpg)
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
![Page 14: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/25.jpg)
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
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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
![Page 27: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/27.jpg)
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
![Page 28: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/28.jpg)
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
![Page 29: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/33.jpg)
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
![Page 34: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/35.jpg)
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
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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
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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
![Page 38: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/38.jpg)
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
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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
![Page 40: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/40.jpg)
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
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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
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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
![Page 43: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/43.jpg)
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
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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
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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
![Page 46: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/46.jpg)
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
![Page 47: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/47.jpg)
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
![Page 48: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/48.jpg)
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
![Page 49: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/49.jpg)
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
![Page 50: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/50.jpg)
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
![Page 51: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/51.jpg)
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
![Page 52: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/52.jpg)
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
![Page 53: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/53.jpg)
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
![Page 54: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/54.jpg)
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
![Page 55: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/55.jpg)
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
![Page 56: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/56.jpg)
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
![Page 57: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/57.jpg)
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
![Page 58: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/58.jpg)
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
![Page 59: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/59.jpg)
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
![Page 60: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/60.jpg)
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
![Page 61: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/61.jpg)
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
![Page 62: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/62.jpg)
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
![Page 63: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/63.jpg)
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
![Page 64: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/64.jpg)
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
![Page 65: 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](https://reader031.vdocuments.site/reader031/viewer/2022020104/5b84dc897f8b9aea498d2766/html5/thumbnails/65.jpg)
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