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Solving and factoring boundary problems for linear ordinary differential equations in differential algebras

Authors:

Markus Rosenkranz,

Georg RegensburgerAuthors Info & Claims

Journal of Symbolic Computation, Volume 43, Issue 8

Pages 515 - 544

https://doi.org/10.1016/j.jsc.2007.11.007

Published: 01 August 2008 Publication History

Abstract

We present a new approach for expressing and solving boundary problems for linear ordinary differential equations in the language of differential algebras. Starting from an algebra with a derivation and integration operator, we construct an algebra of linear integro-differential operators that is expressive enough for specifying regular boundary problems with arbitrary Stieltjes boundary conditions as well as their solution operators. On the basis of these structures, we define a new multiplication on regular boundary problems in such a way that the resulting Green's operator is the reverse composition of the constituent Green's operators. We provide also a method for lifting any factorization of the underlying differential operator to the level of boundary problems. Since this method only needs the computation of initial value problems, it can be used as an effective alternative for computing Green's operators in the case where one knows how to factor the given differential operators.

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Cited By

Lange-Hegermann MRobertz D(2022)On Boundary Conditions Parametrized by Analytic FunctionsComputer Algebra in Scientific Computing10.1007/978-3-031-14788-3_13(225-245)Online publication date: 22-Aug-2022
https://dl.acm.org/doi/10.1007/978-3-031-14788-3_13
Boreale M(2020)Automatic Pre- and Postconditions for Partial Differential EquationsQuantitative Evaluation of Systems10.1007/978-3-030-59854-9_15(193-210)Online publication date: 31-Aug-2020
https://dl.acm.org/doi/10.1007/978-3-030-59854-9_15
Regensburger GAbramov SZima EGao X(2016)Symbolic Computation with Integro-Differential OperatorsProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930942(17-18)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930942
Show More Cited By

Index Terms

Solving and factoring boundary problems for linear ordinary differential equations in differential algebras
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
    2. Numerical analysis
      1. Computations on matrices

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Published In

cover image Journal of Symbolic Computation

Journal of Symbolic Computation Volume 43, Issue 8

August, 2008

97 pages

ISSN:0747-7171

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Copyright © Elsevier Ltd © 2007.

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Academic Press, Inc.

United States

Publication History

Published: 01 August 2008

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Cited By

Lange-Hegermann MRobertz D(2022)On Boundary Conditions Parametrized by Analytic FunctionsComputer Algebra in Scientific Computing10.1007/978-3-031-14788-3_13(225-245)Online publication date: 22-Aug-2022
https://dl.acm.org/doi/10.1007/978-3-031-14788-3_13
Boreale M(2020)Automatic Pre- and Postconditions for Partial Differential EquationsQuantitative Evaluation of Systems10.1007/978-3-030-59854-9_15(193-210)Online publication date: 31-Aug-2020
https://dl.acm.org/doi/10.1007/978-3-030-59854-9_15
Regensburger GAbramov SZima EGao X(2016)Symbolic Computation with Integro-Differential OperatorsProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930942(17-18)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930942
Hossein Poor JRaab CRegensburger GAbramov SZima EGao X(2016)Algorithmic Operator Algebras via Normal Forms for TensorsProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930900(397-404)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930900
Zhang SGuo LKeigher W(2016)Monads and distributive laws for Rota-Baxter and differential algebrasAdvances in Applied Mathematics10.1016/j.aam.2015.09.01472:C(139-165)Online publication date: 1-Jan-2016
https://dl.acm.org/doi/10.1016/j.aam.2015.09.014
Rosenkranz MSerwa NRobertz DLinton SYokoyama K(2015)Green's Functions for Stieltjes Boundary ProblemsProceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2755996.2756681(315-321)Online publication date: 24-Jun-2015
https://dl.acm.org/doi/10.1145/2755996.2756681
Rosenkranz MLiu JMaletzky ABuchberger B(2015)Two-Point Boundary Problems with One Mild Singularity and an Application to Graded Kirchhoff PlatesProceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 930110.1007/978-3-319-24021-3_30(406-423)Online publication date: 14-Sep-2015
https://dl.acm.org/doi/10.1007/978-3-319-24021-3_30
Kredel H(2015)Parametric Solvable Polynomial Rings and ApplicationsProceedings of the 17th International Workshop on Computer Algebra in Scientific Computing - Volume 930110.1007/978-3-319-24021-3_21(275-291)Online publication date: 14-Sep-2015
https://dl.acm.org/doi/10.1007/978-3-319-24021-3_21
Middeke J(2014)Annihilating monomials with the integro-differential Weyl algebraACM Communications in Computer Algebra10.1145/2576802.257682147:3/4(116-117)Online publication date: 28-Jan-2014
https://dl.acm.org/doi/10.1145/2576802.2576821
Korporal ARegensburger GRosenkranz M(2013)ISSAC 2012 software demonstrationsACM Communications in Computer Algebra10.1145/2429135.242916946:3/4(154-156)Online publication date: 15-Jan-2013
https://dl.acm.org/doi/10.1145/2429135.2429169
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