A maple package for integro-differential operators and boundary problems

Abstract. Stieltjes boundary problems generalize the customary class of well-posed two-point boundary value problems in three independent directions, regarding the specification of the boundary conditions: (1) They allow more than two evaluation points. (2) They allow derivatives of arbitrary order. (3) Global terms in the form of definite integrals are allowed. Assuming the Stieltjes boundary problem is regular (a unique solution exists for every forcing function), there are symbolic methods for computing the associated Green's operator.In the classical case of well-posed two-point boundary value problems, it is known how to transform the Green's operator into the so-called Green's function, the representation usually preferred by physicists and engineers. In this paper we extend this transformation to the whole class of Stieltjes boundary problems. It turns out that the extension (1) leads to more case distinction, (2) implies ill-posed problems and hence distributional terms, (3) has apparently no effect on the structure of the Green's function.

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“…The above theorem is constructive, and we plan to implement the underlying algorithm on top of the Maple package IntDiffOp [12].…”

Section: Extracting Green's Functionsmentioning

confidence: 99%

Green's Functions for Stieltjes Boundary Problems

Rosenkranz

Serwa

2015

Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation

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“…• In the first case we apply σ as a postprocessing step after computing the normal forms with respect to Table 1. We have chosen this approach in the upcoming Maple implementation [49]. • In the combined approach, σ may be used at any point during a reduction along the rules of Table 1.…”

Section: Basis Expansionmentioning

confidence: 99%

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

Rosenkranz

Regensburger

Tec³

et al. 2011

Texts &Amp; Monographs in Symbolic Computation

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We review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the TH∃OREM∀ system; some code fragments and sample computations are included.

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“…In this paper, we describe a Maple package with algorithms for regular as well as singular boundary problems for linear ordinary differential equations (LODEs). While a first version of the package with functions for regular boundary problems was presented in [1], the methods and the implementation for singular problems are new. A prototype implementation for regular boundary problems in the TH∃OREM∀ system was described in [2] as part of a general symbolic framework for boundary problems, including also some first steps towards linear partial differential equations (LPDEs).…”

Section: Introductionmentioning

confidence: 99%

“…],[1]) can be factored as G =G • ∂ whereG is the standard Green's operator of the boundary problem (∂ 3 , [E 0 ∂, E 1 ∂, 1 0 ]). HenceG = G • x 0 and, assuming (6) for the composition, also (∂ 3 , [E 0 ∂, E 1 ∂, ], [0]) = (∂, [E 0 ], [0]) • (∂ 2 , [E 0 ∂, E 1 ∂, ], [1]),…”

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Regular and Singular Boundary Problems in Maple

Korporal

Regensburger

Rosenkranz

2011

Computer Algebra in Scientific Computing

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Abstract. We describe a new Maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multipoint as well as Stieltjes boundary conditions. For expressing differential operators, boundary conditions, and Green's operators, we employ the algebra of integro-differential operators. The operations implemented for regular boundary problems include computing Green's operators as well as composing and factoring boundary problems. Our symbolic approach to singular boundary problems is new; it provides algorithms for computing compatibility conditions and generalized Green's operators.

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A maple package for integro-differential operators and boundary problems

Cited by 8 publications

References 6 publications

Green's Functions for Stieltjes Boundary Problems

Green's Functions for Stieltjes Boundary Problems

Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Gröbner Bases

Regular and Singular Boundary Problems in Maple

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