International Workshop on Computer Algebra in Scientific Computing

Computer Algebra in Scientific Computing pp 406-423 | Cite as

Two-Point Boundary Problems with One Mild Singularity and an Application to Graded Kirchhoff Plates

  • Markus Rosenkranz
  • Jane Liu
  • Alexander Maletzky
  • Bruno Buchberger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)

Abstract

We develop a new theory for treating boundary problems for linear ordinary differential equations whose fundamental system may have a singularity at one of the two endpoints of the given interval. Our treatment follows an algebraic approach, with (partial) implementation in the Theorema software system (which is based on Mathematica). We study an application to graded Kirchhoff plates for illustrating a typical case of such boundary problems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebras. Journal of Symbolic Computation 43(8), 515–544 (2008)Google Scholar
  2. 2.
    Regensburger, G., Rosenkranz, M.: An algebraic foundation for factoring linear boundary problems. Ann. Mat. Pura Appl. (4) 188(1), 123–151 (2009). doi:10.1007/s10231-008-0068-3Google Scholar
  3. 3.
    Rosenkranz, M., Regensburger, G., Tec, L., Buchberger, B.: Symbolic analysis of boundary problems: from rewriting to parametrized Gröbner bases. In: Langer, U., Paule, P. (eds.) Numerical and Symbolic Scientific Computing: Progress and Prospects. Springer, pp. 273–331 (2012)Google Scholar
  4. 4.
    Yosida, K.: Lectures on Differential and Integral Equations. Dover (1991)Google Scholar
  5. 5.
    Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. McGraw-Hill Book Company Inc, New York-Toronto-London (1955)Google Scholar
  6. 6.
    Rosenkranz, M., Buchberger, B., Engl, H.W.: Solving linear boundary value problems via non-commutative Gröbner bases. Appl. Anal. 82, 655–675 (2003)Google Scholar
  7. 7.
    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)Google Scholar
  8. 8.
    Buchberger, B., Craciun, A., Jebelean, T., Kovacs, L., Kutsia, T., Nakagawa, K., Piroi, F., Popov, N., Robu, J., Rosenkranz, M., Windsteiger, W.: Theorema: Towards computer-aided mathematical theory exploration. Journal of Applied Logic 4(4), 359–652 (2006). ISSN 1570–8683Google Scholar
  9. 9.
    Korporal, A., Regensburger, G., Rosenkranz, M.: Regular and Singular Boundary Problems in Maple. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2011. LNCS, vol. 6885, pp. 280–293. Springer, Heidelberg (2011)Google Scholar
  10. 10.
    Guo, L.: An Introduction to Rota-Baxter Algebras. International Press (2012)Google Scholar
  11. 11.
    Korporal, A.: Symbolic Methods for Generalized Green’s Operators and Boundary Problems. Ph.D. thesis, Johannes Kepler University, Linz, Austria (November 2012). Abstracted in ACM Communications in Computer Algebra, vol. 46, no. 4, issue 182, December 2012Google Scholar
  12. 12.
    Rosenkranz, M., Serwa, N.: Green’s functions for Stieltjes boundary problems. In: ISSAC (to appear, 2015)Google Scholar
  13. 13.
    Rosenkranz, M., Liu, J., Maletzky, A., Buchberger, B.: Two-point boundary problems with one mild singularity and an application to graded kirchhoff plates (May 2015). Preprint on http://arxiv.org/abs/1505.01956 http://arxiv.org/abs/1505.01956
  14. 14.
    Reddy, J.: Theory and analysis of elastic plates and shells. CRC Press, Taylor and Francis (2007)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Markus Rosenkranz
    • 1
  • Jane Liu
    • 3
  • Alexander Maletzky
    • 2
  • Bruno Buchberger
    • 2
  1. 1.University of KentCanterbury, KentUnited Kingdom
  2. 2.Research Institute for Symbolic Computation (RISC)HagenbergAustria
  3. 3.Tennessee Tech UniversityCookevilleUSA

Personalised recommendations