Greens Functions for Stieltjes Boundary Problems Markus Rosenkranz - PowerPoint PPT Presentation

Green’s Functions for Stieltjes Boundary Problems Markus Rosenkranz Nitin Serwa School of Mathematics, Statistics & Act. Sci. University of Kent ISSAC, 2015 Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 1 / 21

Outline Motivation 1 Introduction 2 Stieltjes Boundary Conditions 3 Equitable Integro-Differential Operators 4 Extracting Green’s Function 5 Examples 6 Conclusion 7 Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 2 / 21

Example of four point Boundary Problem. Given f ∈ C ∞ [ a , b ], find u ∈ C ∞ [ a , b ] such that − u ′′ = f , u (0) + u (1 / 3) = u (1) + u (2 / 3) = 0 . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 3 / 21

Example of four point Boundary Problem. Given f ∈ C ∞ [ a , b ], find u ∈ C ∞ [ a , b ] such that − u ′′ = f , u (0) + u (1 / 3) = u (1) + u (2 / 3) = 0 . � − D 2 , [ ⌊ 0 ⌋ + ⌊ 1 / 3 ⌋ , ⌊ 1 ⌋ + ⌊ 2 / 3 ⌋ ] � Represented by pair . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 3 / 21

Example of four point Boundary Problem. Given f ∈ C ∞ [ a , b ], find u ∈ C ∞ [ a , b ] such that − u ′′ = f , u (0) + u (1 / 3) = u (1) + u (2 / 3) = 0 . � − D 2 , [ ⌊ 0 ⌋ + ⌊ 1 / 3 ⌋ , ⌊ 1 ⌋ + ⌊ 2 / 3 ⌋ ] � Represented by pair . Evaluation functionals: ⌊ ⌋ Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 3 / 21

Green’s Operator Similarly: Regular boundary problem ( T , B ) for LODE. Meaning ∃ unique solution u ∈ C ∞ [ a , b ] for all f ∈ C ∞ [ a , b ]: Tu = f , β ( u ) = 0 ( β ∈ B ) . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 4 / 21

Green’s Operator Similarly: Regular boundary problem ( T , B ) for LODE. Meaning ∃ unique solution u ∈ C ∞ [ a , b ] for all f ∈ C ∞ [ a , b ]: Tu = f , β ( u ) = 0 ( β ∈ B ) . Algorithm to compute Green’s operator G : f �→ u . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 4 / 21

Green’s Operator Similarly: Regular boundary problem ( T , B ) for LODE. Meaning ∃ unique solution u ∈ C ∞ [ a , b ] for all f ∈ C ∞ [ a , b ]: Tu = f , β ( u ) = 0 ( β ∈ B ) . Algorithm to compute Green’s operator G : f �→ u . However, sometimes we want Green’s functions. Why? Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 4 / 21

Green’s Operator Similarly: Regular boundary problem ( T , B ) for LODE. Meaning ∃ unique solution u ∈ C ∞ [ a , b ] for all f ∈ C ∞ [ a , b ]: Tu = f , β ( u ) = 0 ( β ∈ B ) . Algorithm to compute Green’s operator G : f �→ u . However, sometimes we want Green’s functions. Why? - Nice intuition (see below). - Standard form for solutions. - Useful for communicating with engineers. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 4 / 21

Green’s Operator Similarly: Regular boundary problem ( T , B ) for LODE. Meaning ∃ unique solution u ∈ C ∞ [ a , b ] for all f ∈ C ∞ [ a , b ]: Tu = f , β ( u ) = 0 ( β ∈ B ) . Algorithm to compute Green’s operator G : f �→ u . However, sometimes we want Green’s functions. Why? - Nice intuition (see below). - Standard form for solutions. - Useful for communicating with engineers. How to extract Green’s function from Green’s operators? Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 4 / 21

r r r r Intuition behind Green’s Function. Consider again general boundary problem for LODE Tu = f , β ( u ) = 0 ( β ∈ B ) . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 5 / 21

r r r r Intuition behind Green’s Function. Consider again general boundary problem for LODE Tu = f , β ( u ) = 0 ( β ∈ B ) . Green’s function g ξ is solution for f = δ ξ . Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 5 / 21

r r r r Intuition behind Green’s Function. Consider again general boundary problem for LODE Tu = f , β ( u ) = 0 ( β ∈ B ) . Green’s function g ξ is solution for f = δ ξ . Note that g ( x , ξ ) = g ξ ( x ) and “ δ ( x , ξ ) = δ ξ ( x )”. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 5 / 21

Intuition behind Green’s Function. Consider again general boundary problem for LODE Tu = f , β ( u ) = 0 ( β ∈ B ) . Green’s function g ξ is solution for f = δ ξ . Note that g ( x , ξ ) = g ξ ( x ) and “ δ ( x , ξ ) = δ ξ ( x )”. = r g ξ f ( ξ ) d ξ u = ⇒ = r Tg ξ f ( ξ ) d ξ = r δ ξ f ( ξ ) d ξ = f Tu β ( u ) = r β ( g ξ ) f ( ξ ) d ξ = 0 Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 5 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: More than two evaluation points → multipoint BP. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: More than two evaluation points → multipoint BP. Derivatives of arbitrary order → ill-posed BP. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: More than two evaluation points → multipoint BP. Derivatives of arbitrary order → ill-posed BP. Global terms in the form of definite integrals → nonlocal BP. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: More than two evaluation points → multipoint BP. Derivatives of arbitrary order → ill-posed BP. Global terms in the form of definite integrals → nonlocal BP. Are there Green’s functions g ( x , ξ ) for such Stieltjes BPs? Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

Green’s Functions for Stieltjes Boundary Problems Extraction of Green’s functions: G � g ( x , ξ ) Well known for “classical case” [3]. Sometimes one needs Stieltjes boundary conditions: More than two evaluation points → multipoint BP. Derivatives of arbitrary order → ill-posed BP. Global terms in the form of definite integrals → nonlocal BP. Are there Green’s functions g ( x , ξ ) for such Stieltjes BPs? Can we extract it from G ? Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 6 / 21

r r r Integro-Differential Operators Recall algebraic setting for boundary problems [3]: Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 7 / 21

r r Integro-Differential Operators Recall algebraic setting for boundary problems [3]: Ordinary integro-differential K -algebra ( F , ∂, r ). Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 7 / 21

r r Integro-Differential Operators Recall algebraic setting for boundary problems [3]: Ordinary integro-differential K -algebra ( F , ∂, r ). Characters on F : multiplicative linear functionals. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 7 / 21

r r Integro-Differential Operators Recall algebraic setting for boundary problems [3]: Ordinary integro-differential K -algebra ( F , ∂, r ). Characters on F : multiplicative linear functionals. Ring of integro-differential operators over F with characters Φ. Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 7 / 21

r Integro-Differential Operators Recall algebraic setting for boundary problems [3]: Ordinary integro-differential K -algebra ( F , ∂, r ). Characters on F : multiplicative linear functionals. Ring of integro-differential operators over F with characters Φ. Standard integro-differential operator ring F Φ [ ∂, r ] Markus Rosenkranz, Nitin Serwa (Uni. of Kent) Green’s Functions ISSAC, 2015 7 / 21