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Symbolic Computation with Integro-Differential Operators

Author:

Georg RegensburgerAuthors Info & Claims

ISSAC '16: Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation

Pages 17 - 18

https://doi.org/10.1145/2930889.2930942

Published: 20 July 2016 Publication History

Abstract

The algebraic and algorithmic study of integro-differential algebras and operators has only started in the past decade. Integro-differential operators allow us in particular to study initial value and boundary problems for linear ODEs from an algebraic point of view. Differential operators already provide a rich algebraic structure with a wealth of results and algorithmic methods. Adding integral operators and evaluations, many new phenomena appear, including zero devisors and non-finitely generated ideals. In this tutorial, we give an introduction to symbolic methods for integro-differential operators and boundary problems developed over the last years. In particular, we discuss normal forms, basic algebraic properties, and the computation of polynomial solutions for ordinary integro-differential equations with polynomial coefficients. We will also outline methods for manipulating and solving linear boundary problems and illustrate them with an implementation.

References

[1]

S. A. Abramov, M. Bronstein, and M. Petkovsek. On polynomial solutions of linear operator equations. In Proceedings of ISSAC '95, pages 290--296, New York, NY, USA, 1995. ACM.

Digital Library

[2]

R. P. Agarwal. Boundary value problems for higher order differential equations. World Scientific Publishing Co. Inc., Teaneck, NJ, 1986.

[3]

M. A. Barkatou. On rational solutions of systems of linear differential equations. J. Symbolic Comput., 28:547--567, 1999.

Digital Library

[4]

V. V. Bavula. Generalized Weyl algebras and their representations. Algebra i Analiz, 4:75--97, 1992.

[5]

V. V. Bavula. The algebra of integro-differential operators on a polynomial algebra. J. Lond. Math. Soc. (2), 83:517--543, 2011.

[6]

V. V. Bavula. The algebra of integro-differential operators on an affine line and its modules. J. Pure Appl. Algebra, 217:495--529, 2013.

[7]

A. Bostan, T. Cluzeau, and B. Salvy. Fast algorithms for polynomial solutions of linear differential equations. In Proceedings of ISSAC '05, pages 45--52, New York, NY, USA, 2005. ACM.

Digital Library

[8]

F. Chyzak, A. Quadrat, and D. Robertz. Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Engrg. Comm. Comput., 16:319--376, 2005.

Digital Library

[9]

F. Chyzak and B. Salvy. Non-commutative elimination in Ore algebras proves multivariate identities. J. Symbolic Comput., 26:187--227, 1998.

Digital Library

[10]

S. C. Coutinho. A primer of algebraic $D$-modules. Cambridge University Press, Cambridge, 1995.

[11]

D. G. Duffy. Green's functions with applications. Chapman & Hall/CRC, Boca Raton, FL, 2001.

[12]

X. Gao, L. Guo, and M. Rosenkranz. Free integro-differential algebras and Gröbner-Shirshov bases. J. Algebra, 442:354--396, 2015.

[13]

X. Gao, L. Guo, and S. Zheng. Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases. J. Algebra Appl., 13:1350160, 38, 2014.

[14]

J. Gómez-Torrecillas. Basic module theory over non-commutative rings with computational aspects of operator algebras. In AADIOS 2012, volume 8372 of LNCS, pages 23--82. Springer, 2014. With an appendix by V. Levandovskyy.

[15]

L. Guo. An introduction to Rota-Baxter algebra. International Press, Somerville, MA, 2012.

[16]

L. Guo and W. Keigher. On differential Rota-Baxter algebras. J. Pure Appl. Algebra, 212:522--540, 2008.

[17]

L. Guo, G. Regensburger, and M. Rosenkranz. On integro-differential algebras. J. Pure Appl. Algebra, 218:456--473, 2014.

[18]

J. Hossein Poor, C. G. Raab, and G. Regensburger. Algorithmic operator algebras via normal forms for tensors. In Proceedings of ISSAC '16, this volume, New York, NY, USA, 2016. ACM.

Digital Library

[19]

A. Korporal and G. Regensburger. Composing and factoring generalized Green's operators and ordinary boundary problems. In AADIOS 2012, volume 8372 of LNCS, pages 116--134. Springer, 2014.

[20]

A. Korporal and G. Regensburger. On the product of projectors and generalized inverses. Linear Multilinear Algebra, 62:1567--1582, 2014.

[21]

A. Korporal, G. Regensburger, and M. Rosenkranz. Symbolic computation for ordinary boundary problems in Maple. ACM Commun. Comput. Algebra, 46:154--156, 2012. Software demonstration ISSAC '12.

Digital Library

[22]

Ø. Ore. Theory of non-commutative polynomials. Ann. Math., 34:480--508, 1933.

[23]

A. Quadrat and G. Regensburger. Polynomial solutions and annihilators of ordinary integro-differential operators. In Proceedings of SSSC 2013, pages 308--313. IFAC, 2013.

[24]

G. Regensburger and M. Rosenkranz. An algebraic foundation for factoring linear boundary problems. Ann. Mat. Pura Appl. (4), 188:123--151, 2009.

[25]

G. Regensburger, M. Rosenkranz, and J. Middeke. A skew polynomial approach to integro-differential operators. In Proceedings of ISSAC '09, pages 287--294, New York, NY, USA, 2009. ACM.

Digital Library

[26]

M. Rosenkranz. The Green's Algebra: A Polynomial Approach to Boundary Value Problems. PhD thesis, RISC, Johannes Kepler University Linz, 2003.

[27]

M. Rosenkranz. A new symbolic method for solving linear two-point boundary value problems on the level of operators. J. Symbolic Comput., 39:171--199, 2005.

Digital Library

[28]

M. Rosenkranz, J. Liu, A. Maletzky, and B. Buchberger. Two-point boundary problems with one mild singularity and an application to graded Kirchhoff plates. In Proceedings of CASC 2015, volume 9301 of LNCS, pages 406--423. Springer, 2015.

Digital Library

[29]

M. Rosenkranz and N. Phisanbut. A symbolic approach to boundary problems for linear partial differential equations. In Proceedings of CASC 2013, volume 8136 of LNCS, pages 301--314. Springer, 2013.

Digital Library

[30]

M. Rosenkranz and G. Regensburger. Integro-differential polynomials and operators. In Proceedings of ISSAC '08, pages 261--268, New York, NY, USA, 2008. ACM.

Digital Library

[31]

M. Rosenkranz and G. Regensburger. Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput., 43:515--544, 2008.

Digital Library

[32]

M. Rosenkranz, G. Regensburger, L. Tec, and B. Buchberger. Symbolic analysis for boundary problems: From rewriting to parametrized Gröbner bases. In Numerical and Symbolic Scientific Computing: Progress and Prospects, pages 273--331. Springer Vienna, 2012.

[33]

M. Rosenkranz and N. Serwa. Green's functions for Stieltjes boundary problems. In Proceedings of ISSAC '15, pages 315--321, New York, NY, USA, 2015. ACM.

Digital Library

[34]

I. Stakgold. Green's functions and boundary value problems. John Wiley & Sons, New York, 1979.

[35]

M. van der Put and M. F. Singer. Galois theory of linear differential equations. Springer-Verlag, 2003.

Cited By

Cluzeau TPoor JQuadrat ARaab CRegensburger G(2018)Symbolic Computation for Integro-Differential-Time-Delay Operators with Matrix CoefficientsIFAC-PapersOnLine10.1016/j.ifacol.2018.07.21551:14(153-158)Online publication date: 2018
https://doi.org/10.1016/j.ifacol.2018.07.215
Chen SKauers M(2017)Some open problems related to creative telescopingJournal of Systems Science and Complexity10.1007/s11424-017-6202-930:1(154-172)Online publication date: 14-Feb-2017
https://doi.org/10.1007/s11424-017-6202-9

Index Terms

Symbolic Computation with Integro-Differential Operators
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Mathematics of computing
  1. Mathematical software

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Information

Published In

cover image ACM Conferences

ISSAC '16: Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation

July 2016

434 pages

ISBN:9781450343800

DOI:10.1145/2930889

General Chairs:
Sergei Abramov
Russian Academy of Sciences, Russia
,
Eugene Zima
Wilfrid Laurier University, Canada
,
Program Chair:
Xiao-Shan Gao
Chinese Academy of Sciences, China

Copyright © 2016 Owner/Author.

Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 20 July 2016

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Austrian Science Fund (FWF)

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ISSAC '16

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SIGSAM

ISSAC '16: International Symposium on Symbolic and Algebraic Computation

July 20 - 22, 2016

ON, Waterloo, Canada

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Cited By

Cluzeau TPoor JQuadrat ARaab CRegensburger G(2018)Symbolic Computation for Integro-Differential-Time-Delay Operators with Matrix CoefficientsIFAC-PapersOnLine10.1016/j.ifacol.2018.07.21551:14(153-158)Online publication date: 2018
https://doi.org/10.1016/j.ifacol.2018.07.215
Chen SKauers M(2017)Some open problems related to creative telescopingJournal of Systems Science and Complexity10.1007/s11424-017-6202-930:1(154-172)Online publication date: 14-Feb-2017
https://doi.org/10.1007/s11424-017-6202-9

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