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Regular and Singular Boundary Problems in Maple

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Computer Algebra in Scientific Computing (CASC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6885))

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Abstract

We describe a new maple package for treating boundary problems for linear ordinary differential equations, allowing two-/multi-point 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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References

  1. Korporal, A., Regensburger, G., Rosenkranz, M.: A Maple package for integro-differential operators and boundary problems. ACM Commun. Comput. Algebra 44(3), 120–122 (2010); Also presented as a poster at ISSAC 2010

    MATH  Google Scholar 

  2. Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: A symbolic framework for operations on linear boundary problems. In: Gerdt, V.P., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2009. LNCS, vol. 5743, pp. 269–283. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York (1955)

    Google Scholar 

  4. Stakgold, I.: Green’s functions and boundary value problems. John Wiley & Sons, New York (1979)

    MATH  Google Scholar 

  5. Duffy, D.G.: Green’s functions with applications. Studies in Advanced Mathematics. Chapman & Hall/CRC, Boca Raton, FL (2001)

    Google Scholar 

  6. Agarwal, R.P., O’Regan, D.: An introduction to ordinary differential equations. Universitext. Springer, New York (2008)

    Google Scholar 

  7. Loud, W.S.: Some examples of generalized Green’s functions and generalized Green’s matrices. SIAM Rev. 12, 194–210 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ben-Israel, A., Greville, T.N.E.: Generalized inverses, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  9. Boichuk, A.A., Samoilenko, A.M.: Generalized inverse operators and Fredholm boundary-value problems. VSP, Utrecht (2004)

    Google Scholar 

  10. Nashed, M.Z., Rall, L.B.: Annotated bibliography on generalized inverses and applications. In: Generalized Inverses and Applications, pp. 771–1041. Academic Press, New York (1976)

    Chapter  Google Scholar 

  11. Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. J. Symbolic Comput. 43(8), 515–544 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: Symbolic analysis for boundary problems: From rewriting to parametrized Gröbner bases. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Springer, Wien (to appear, 2011)

    Google Scholar 

  13. Guo, L., Keigher, W.: On differential Rota-Baxter algebras. J. Pure Appl. Algebra 212(3), 522–540 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brown, R.C., Krall, A.M.: Ordinary differential operators under Stieltjes boundary conditions. Trans. Amer. Math. Soc. 198, 73–92 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  15. Regensburger, G., Rosenkranz, M., Middeke, J.: A skew polynomial approach to integro-differential operators. In: May, J.P. (ed.) Proceedings of ISSAC 2009, pp. 287–294. ACM, New York (2009)

    Google Scholar 

  16. Regensburger, G., Rosenkranz, M.: An algebraic foundation for factoring linear boundary problems. Ann. Mat. Pura Appl. (4) 188(1), 123–151 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. Teil I: Gewöhnliche Differentialgleichungen. Akademische Verlagsgesellschaft, Leipzig (1967)

    Google Scholar 

  18. Rosenkranz, M.: A new symbolic method for solving linear two-point boundary value problems on the level of operators. J. Symbolic Comput. 39(2), 171–199 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Djordjević, D.S.: Further results on the reverse order law for generalized inverses. SIAM J. Matrix Anal. Appl. 29(4), 1242–1246 (2007)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  21. Krupchyk, K., Tuomela, J.: The Shapiro-Lopatinskij condition for elliptic boundary value problems. LMS Journal of Computation and Mathematics 9, 287–329 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenberger Str. 69, 4040, Linz, Austria

    Anja Korporal

  2. INRIA Saclay – Île de France, Project DISCO, L2S, Supélec, 91192, Gif-sur-Yvette Cedex, France

    Georg Regensburger

  3. University of Kent, Cornwallis Building, Canterbury, Kent, CT2 7NF, United Kingdom

    Markus Rosenkranz

Authors
  1. Anja Korporal
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  2. Georg Regensburger
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  3. Markus Rosenkranz
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research (JINR), 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Informatik, Lehrstuhl für Effiziente Algorithmen, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Ernst W. Mayr

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    Evgenii V. Vorozhtsov

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Korporal, A., Regensburger, G., Rosenkranz, M. (2011). Regular and Singular Boundary Problems in Maple. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_22

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  • DOI: https://doi.org/10.1007/978-3-642-23568-9_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-23567-2

  • Online ISBN: 978-3-642-23568-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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