( F , ) + Georg Regensburger joint work Markus Rosenkranz Radon - PowerPoint PPT Presentation

Integro-Differential Algebras, Operators, and Polynomials � ( F , ∂ ) + Georg Regensburger joint work Markus Rosenkranz Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Linz, Austria DART IV Academy of Mathematics and Systems Science 27–30 October 2010, Beijing, China

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

Motivation Algebraic approaches and Symbolic Computation for Boundary problems Importance of boundary problems in applications and Scientific Computing Almost exclusively in numerical segment Algebraic structures for manipulating boundary problems for LODEs: Differential operators Boundary conditions (evaluations) Integral operators (Green’s operators)

The Simplest Boundary Problem Given f ∈ C ∞ [ 0 , 1 ] , find u ∈ C ∞ [ 0 , 1 ] such that u ′′ = f , u ( 0 ) = u ( 1 ) = 0 Solution: Green’s operator G : C ∞ [ 0 , 1 ] → C ∞ [ 0 , 1 ] , f �→ u Green’s Operator G via Green’s Function g : � 1  ( x − 1 ) ξ for x ≥ ξ   Gf ( x ) = g ( x , ξ ) f ( ξ ) d ξ g ( x , ξ ) =   ξ ( x − 1 ) for x ≤ ξ 0   Green’s operator as integro-differential operator: G = XAX + XBX − AX − BX , � x � 1 A = 0 u ( ξ ) d ξ , B = x u ( ξ ) d ξ , and X the multiplication operator, � x XAX f ( x ) = x 0 ξ f ( ξ ) d ξ

The Simplest Boundary Problem Given f ∈ C ∞ [ 0 , 1 ] , find u ∈ C ∞ [ 0 , 1 ] such that u ′′ = f , u ( 0 ) = u ( 1 ) = 0 Solution: Green’s operator G : C ∞ [ 0 , 1 ] → C ∞ [ 0 , 1 ] , f �→ u Green’s Operator G via Green’s Function g : � 1  ( x − 1 ) ξ for x ≥ ξ   Gf ( x ) = g ( x , ξ ) f ( ξ ) d ξ g ( x , ξ ) =   ξ ( x − 1 ) for x ≤ ξ 0   Green’s operator as integro-differential operator: G = XAX + XBX − AX − BX , � x � 1 A = 0 u ( ξ ) d ξ , B = x u ( ξ ) d ξ , and X the multiplication operator, � x XAX f ( x ) = x 0 ξ f ( ξ ) d ξ

The Simplest Boundary Problem Given f ∈ C ∞ [ 0 , 1 ] , find u ∈ C ∞ [ 0 , 1 ] such that u ′′ = f , u ( 0 ) = u ( 1 ) = 0 Solution: Green’s operator G : C ∞ [ 0 , 1 ] → C ∞ [ 0 , 1 ] , f �→ u Green’s Operator G via Green’s Function g : � 1  ( x − 1 ) ξ for x ≥ ξ   Gf ( x ) = g ( x , ξ ) f ( ξ ) d ξ g ( x , ξ ) =   ξ ( x − 1 ) for x ≤ ξ 0   Green’s operator as integro-differential operator: G = XAX + XBX − AX − BX , � x � 1 A = 0 u ( ξ ) d ξ , B = x u ( ξ ) d ξ , and X the multiplication operator, � x XAX f ( x ) = x 0 ξ f ( ξ ) d ξ

The Simplest Boundary Problem Given f ∈ C ∞ [ 0 , 1 ] , find u ∈ C ∞ [ 0 , 1 ] such that u ′′ = f , u ( 0 ) = u ( 1 ) = 0 Solution: Green’s operator G : C ∞ [ 0 , 1 ] → C ∞ [ 0 , 1 ] , f �→ u Green’s Operator G via Green’s Function g : � 1  ( x − 1 ) ξ for x ≥ ξ   Gf ( x ) = g ( x , ξ ) f ( ξ ) d ξ g ( x , ξ ) =   ξ ( x − 1 ) for x ≤ ξ 0   Green’s operator as integro-differential operator: G = XAX + XBX − AX − BX , � x � 1 A = 0 u ( ξ ) d ξ , B = x u ( ξ ) d ξ , and X the multiplication operator, � x XAX f ( x ) = x 0 ξ f ( ξ ) d ξ

The Simplest Boundary Problem Given f ∈ C ∞ [ 0 , 1 ] , find u ∈ C ∞ [ 0 , 1 ] such that u ′′ = f , u ( 0 ) = u ( 1 ) = 0 Solution: Green’s operator G : C ∞ [ 0 , 1 ] → C ∞ [ 0 , 1 ] , f �→ u Green’s Operator G via Green’s Function g : � 1  ( x − 1 ) ξ for x ≥ ξ   Gf ( x ) = g ( x , ξ ) f ( ξ ) d ξ g ( x , ξ ) =   ξ ( x − 1 ) for x ≤ ξ 0   Green’s operator as integro-differential operator: G = XAX + XBX − AX − BX , � x � 1 A = 0 u ( ξ ) d ξ , B = x u ( ξ ) d ξ , and X the multiplication operator, � x XAX f ( x ) = x 0 ξ f ( ξ ) d ξ

� Integro-Differential Algebras ( F , ∂, ) � x Example: C ∞ ( R ) , ∂ usual derivation, � : f �→ a f ( ξ ) d ξ Leibniz Rule, Fundamental Theorem of Calculus, Integration by Parts Definition � ( F , ∂, ) is an integro-differential algebra if is a K -linear section of ∂ = ′ , ( F , ∂ ) is a differential K -algebra and � f ) ′ = f , such that the differential Baxter axiom � i.e. ( ( fg ) ′ = ( f ′ )( g ′ ) + f ′ ) g + f ( g ′ ) ( � � � � � holds. cf. R-R ’08, Guo-Keigher ’08 (Exponential) polynomials, holonomic functions x k = x k + 1 / ( k + 1 ) � K [ x ] or K [[ x ]] with Q ≤ K usual ∂ and � n � Hurwitz series ( Keigher-Pritchard ’00 ): ( a n ) · ( b n ) = ( � n a i b n − i ) n i = 0 i � ∂ ( a 0 , a 1 , a 2 , . . . ) = ( a 1 , a 2 , . . . ) ( a 0 , a 1 , . . . ) = ( 0 , a 0 , a 1 , . . . )

� Integro-Differential Algebras ( F , ∂, ) � x Example: C ∞ ( R ) , ∂ usual derivation, � : f �→ a f ( ξ ) d ξ Leibniz Rule, Fundamental Theorem of Calculus, Integration by Parts Definition � ( F , ∂, ) is an integro-differential algebra if is a K -linear section of ∂ = ′ , ( F , ∂ ) is a differential K -algebra and � f ) ′ = f , such that the differential Baxter axiom � i.e. ( ( fg ) ′ = ( f ′ )( g ′ ) + f ′ ) g + f ( g ′ ) ( � � � � � holds. cf. R-R ’08, Guo-Keigher ’08 (Exponential) polynomials, holonomic functions x k = x k + 1 / ( k + 1 ) � K [ x ] or K [[ x ]] with Q ≤ K usual ∂ and � n � Hurwitz series ( Keigher-Pritchard ’00 ): ( a n ) · ( b n ) = ( � n a i b n − i ) n i = 0 i � ∂ ( a 0 , a 1 , a 2 , . . . ) = ( a 1 , a 2 , . . . ) ( a 0 , a 1 , . . . ) = ( 0 , a 0 , a 1 , . . . )

� Integro-Differential Algebras ( F , ∂, ) � x Example: C ∞ ( R ) , ∂ usual derivation, � : f �→ a f ( ξ ) d ξ Leibniz Rule, Fundamental Theorem of Calculus, Integration by Parts Definition � ( F , ∂, ) is an integro-differential algebra if is a K -linear section of ∂ = ′ , ( F , ∂ ) is a differential K -algebra and � f ) ′ = f , such that the differential Baxter axiom � i.e. ( ( fg ) ′ = ( f ′ )( g ′ ) + f ′ ) g + f ( g ′ ) ( � � � � � holds. cf. R-R ’08, Guo-Keigher ’08 (Exponential) polynomials, holonomic functions x k = x k + 1 / ( k + 1 ) � K [ x ] or K [[ x ]] with Q ≤ K usual ∂ and � n � Hurwitz series ( Keigher-Pritchard ’00 ): ( a n ) · ( b n ) = ( � n a i b n − i ) n i = 0 i � ∂ ( a 0 , a 1 , a 2 , . . . ) = ( a 1 , a 2 , . . . ) ( a 0 , a 1 , . . . ) = ( 0 , a 0 , a 1 , . . . )