Skip to main content
SpringerLink
Log in

On the solution of the polynomial systems arising in the discretization of certain ODEs

  • Published:
Computing Aims and scope Submit manuscript

Abstract

We study the positive stationary solutions of a standard finite-difference discretization of the semilinear heat equation with nonlinear Neumann boundary conditions. We prove that, if the diffusion is large enough, then there exists a unique solution of such a discretization, which approximates the unique positive stationary solution of the boundary-value problem under consideration. Furthermore, in this case we exhibit an algorithm computing an ε-approximation of such a solution by means of a homotopy continuation method. The cost of our algorithm is polynomial in the number of nodes involved in the discretization and the logarithm of the number of digits of approximation required.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allgower E, Georg K (1990) Numerical continuation methods: an introduction. Springer Ser. Comput. Math., vol 13. Springer, New York

  2. Allgower E, Bates D, Sommese A, Wampler C (2006) Solution of polynomial systems derived from differential equations. Computing 76(1–2): 1–10

    Article  MATH  MathSciNet  Google Scholar 

  3. Andreu F, Mazon J, Toledo J, Rossi J (2002) Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal 49(4): 541–563

    Article  MATH  MathSciNet  Google Scholar 

  4. Ascher U, Mattheij R, Russell R (1995) Numerical solution of boundary value problems for ordinary differential equations. Classics in applied mathematics, vol 13. SIAM, Philadelphia

    Google Scholar 

  5. Bandle C, Brunner H (1998) Blow-up in diffusion equations: a survey. J Comput Appl Math 97(1–2): 3–22

    Article  MATH  MathSciNet  Google Scholar 

  6. Bebernes J, Eberly D (1989) Mathematical problems from combustion theory. Applied mathematical sciences, vol 83. Springer, New York

  7. Blum L, Cucker F, Shub M, Smale S (1998) Complexity and real computation. Springer, New York

    Google Scholar 

  8. Bonder JF, Rossi J (2001) Blow-up vs. spurious steady solutions. Proc Am Math Soc 129(1): 139–144

    Article  MATH  MathSciNet  Google Scholar 

  9. Cannon J (1984) The one-dimensional heat equation. Cambridge University Press, Cambridge

    MATH  Google Scholar 

  10. Castro D, Giusti M, Heintz J, Matera G, Pardo L (2003) The hardness of polynomial equation solving. Found Comput Math 3(4): 347–420

    Article  MATH  MathSciNet  Google Scholar 

  11. Chipot M, Quittner P (2004) Equilibria, connecting orbits and a priori bounds for semilinear parabolic equations with nonlinear boundary conditions. J Dyn Differ Equ 16(1): 91–138

    Article  MATH  MathSciNet  Google Scholar 

  12. Chipot M, Fila M, Quittner P (1991) Stationary solutions, blow up and convergence to stationary solutions for semilinear parabolic equations with nonlinear boundary conditions. Acta Math Univ Comenian 60(1): 35–103

    MATH  MathSciNet  Google Scholar 

  13. De Leo M, Dratman E, Matera G (2004) On the numerical solution of certain nonlinear systems arising in semilinear parabolic PDEs. Anales JAIIO (Jornadas Argentinas de Informática e Investigación Operativa) 33

  14. De Leo M, Dratman E, Matera G (2005) Numeric vs. symbolic homotopy algorithms in polynomial system solving: a case study. J Complexity 21(4): 502–531

    Article  MATH  MathSciNet  Google Scholar 

  15. Deuflhard P, Bornemann F (2002) Scientific computing with ordinary differential equations. Texts in applied mathematics, vol 42. Springer, New York

  16. Dratman E, Matera G (2008) Discrete vs. continuous stationary solutions for semilinear parabolic equations. In: Busch E et al (eds) Proceedings of the IV Congreso Internacional de Matemática Aplicada a la Ingenierí a, In-Mat 2008, Buenos Aires, August 2008

  17. Dratman E, Matera G, Waissbein A (2009) Robust algorithms for generalized Pham systems. Comput Complex 18(1): 105–154

    Article  Google Scholar 

  18. Duvallet J (1990) Computation of solutions of two-point boundary value problems by a simplicial homotopy algorithm. In: Allgower E, Georg K (eds) Computational solution of nonlinear systems of equations. Lectures Appl Math, vol 26 Amer Math Soc, Providence, pp 135–150

  19. Fiedler B, Gedeon T (1999) A Lyapunov function for tridiagonal competitive-cooperative systems. SIAM J Math Anal 30(3): 469–478

    Article  MATH  MathSciNet  Google Scholar 

  20. Gilding B, Kersner R (2004) Travelling waves in nonlinear diffusion-convection reaction. Birkhäuser, Basel

    MATH  Google Scholar 

  21. Gomez JL, Marquez V, Wolanski N (1993) Dynamic behaviour of positive solutions to reaction-diffusion problems with nonlinear absorption through the boundary. Rev Un Mat Argentina 38: 196–209

    MATH  MathSciNet  Google Scholar 

  22. Grindrod P (1996) The theory and applications of reaction-diffusion equations: patterns and waves. Clarendon Press, Oxford

    MATH  Google Scholar 

  23. Kacewicz B (2002) Complexity of nonlinear two-point boundary-value problems. J Complexity 18: 702–738

    Article  MATH  MathSciNet  Google Scholar 

  24. Keller H (1976) Numerical solution of two point boundary value problems. SIAM, Philadelphia

    Google Scholar 

  25. Levine H (1990) The role of critical exponents in blow up theorems. SIAM Rev 32: 262–288

    Article  MATH  MathSciNet  Google Scholar 

  26. Malajovich G, Rojas J (2004) High probability analysis of the condition number of sparse polynomial systems. Theor Comput Sci 315(2–3): 525–555

    Article  MATH  MathSciNet  Google Scholar 

  27. Meurant G (2000) Gaussian elimination for the solution of linear systems of equations. In: Ciarlet P, Lions J (eds) Handbook of numerical analysis, vol VII. North-Holland, Amsterdam, pp 3–170

    Google Scholar 

  28. Murray J (2002) Mathematical biology, vol 1: an introduction. Interdisciplinary applied mathematics, vol 17. Springer, New York

  29. Ortega J, Rheinboldt W (1970) Iterative solutions of nonlinear equations in several variables. Academic Press, New York

    Google Scholar 

  30. Pao C (1992) Nonlinear parabolic and elliptic equations. Plenum Press, New York

    MATH  Google Scholar 

  31. Pardo L (2000) Universal elimination requires exponential running time. In: Montes A (ed) Computer algebra and applications. Proceedings of EACA–2000, Barcelona, Spain, September 2000, pp 25–51

  32. Pearce C, Pecǎrić J (2000) Inequalities for differentiable mappings with application to special means and quadrature formulæ. Appl Math Lett 13(2): 51–55

    Article  MATH  MathSciNet  Google Scholar 

  33. Quittner P (1993) On global existence and stationary solutions for two classes of semilinear parabolic problems. Comment Math Univ Carolin 34(1): 105–124

    MATH  MathSciNet  Google Scholar 

  34. Rodrí guez-Bernal A, Tajdine A (2001) Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up. J Differ Equ 169(2): 332–372

    Article  Google Scholar 

  35. Rossi J (1998) The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition. Acta Math Univ Comenian (NS) 67(2): 343–350

    MATH  Google Scholar 

  36. Samarskii A, Galaktionov V, Kurdyumov S, Mikhailov A (1995) Blow-up in quasilinear parabolic equations. de Gruyter Exp Math, vol 19. de Gruyter, Berlin

  37. Smillie J (1984) Competitive and cooperative tridiagonal systems of differential equations. SIAM J Math Anal 15(3): 530–534

    Article  MATH  MathSciNet  Google Scholar 

  38. Smith H (1995) Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems. Math. Surveys Monogr., vol 41. Amer Math Soc, Providence, RI

  39. Strang G, Fix G (1973) An analysis of the finite element method. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  40. Walter W (1970) Differential and integral inequalities. Ergeb. Math Grenzgeb, vol 55. Springer, Berlin

  41. Walter W (1998) Ordinary differential equations. Grad. Texts in Math, vol 182. Springer, New York

  42. Watson L (1980) Solving finite difference approximations to nonlinear two-point boundary value problems by a homotopy method. SIAM J Sci Stat Comput 1(4): 467–480

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Instituto de Ciencias, Universidad Nacional de General Sarmiento, Juan M. Gutiérrez 1150, B1613GSX, Los Polvorines, Buenos Aires, Argentina

    Ezequiel Dratman

  2. Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, Juan M. Gutiérrez 1150, B1613GSX, Los Polvorines, Buenos Aires, Argentina

    Guillermo Matera

Authors
  1. Ezequiel Dratman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Guillermo Matera
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Guillermo Matera.

Additional information

Communicated by D. Saupe.

Some of the results presented here were first announced at the IV Congreso Internacional de Matemática Aplicada a la Ingeniería, In-Mat 2008 (Buenos Aires, August 2008) (see [16]).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dratman, E., Matera, G. On the solution of the polynomial systems arising in the discretization of certain ODEs. Computing 85, 301–337 (2009). https://doi.org/10.1007/s00607-009-0046-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00607-009-0046-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation