Advertisement

Applications to the Computer-Assisted Proofs in Analysis

  • Mitsuhiro T. Nakao
  • Michael Plum
  • Yoshitaka Watanabe
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 53)

Abstract

This chapter presents other computer-assisted proofs obtained by verification algorithms that introduced previous chapters in Part I. Some applications to the nonlinear parabolic problems are described in the next chapter.

References

  1. 46.
    Cai, S., Nagatou, K., Watanabe, Y.: A numerical verification method for a system of Fitzhugh-Nagumo type. Numer. Funct. Anal. Optim. 33(10), 1195–1220 (2012)MathSciNetzbMATHGoogle Scholar
  2. 86.
    Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys. 68(3), 209–243 (1979)MathSciNetzbMATHGoogle Scholar
  3. 102.
    Hashimoto, K., Abe, R., Nakao, M.T., Watanabe, Y.: A numerical verification method for solutions of singularly perturbed problems with nonlinearity. Japan J. Indust. Appl. Math. 22(1), 111–131 (2005)MathSciNetzbMATHGoogle Scholar
  4. 103.
    Hashimoto, K., Kobayashi, K., Nakao, M.T.: Numerical verification methods for solutions of the free boundary problem. Numer. Funct. Anal. Optim. 26(4–5), 523–542 (2005)MathSciNetzbMATHGoogle Scholar
  5. 112.
    Kawanago, T.: A symmetry-breaking bifurcation theorem and some related theorems applicable to maps having unbounded derivatives. Japan J. Indust. Appl. Math. 21(1), 57–74 (2004)MathSciNetzbMATHGoogle Scholar
  6. 118.
    Kim, M., Nakao, M.T., Watanabe, Y., Nishida, T.: A numerical verification method of bifurcating solutions for 3-dimensional Rayleigh-Bénard problems. Numer. Math. 111(3), 389–406 (2009)MathSciNetzbMATHGoogle Scholar
  7. 153.
    Minamoto, T.: Numerical verification of solutions for nonlinear hyperbolic equations. Appl. Math. Lett. 10(6), 91–96 (1997)MathSciNetzbMATHGoogle Scholar
  8. 155.
    Minamoto, T.: Numerical verification method for solutions of nonlinear hyperbolic equations. In: Symbolic Algebraic Methods and Verification Methods (Dagstuhl, 1999), pp. 173–181. Springer, Vienna (2001)zbMATHGoogle Scholar
  9. 156.
    Minamoto, T.: Numerical method with guaranteed accuracy of a double turning point for a radially symmetric solution of the perturbed Gelfand equation. J. Comput. Appl. Math. 169(1), 151–160 (2004)MathSciNetzbMATHGoogle Scholar
  10. 158.
    Minamoto, T., Yamamoto, N., Nakao, M.T.: Numerical verification method for solutions of the perturbed Gelfand equation. Methods Appl. Anal. 7(1), 251–262 (2000)MathSciNetzbMATHGoogle Scholar
  11. 167.
    Mizutani, A.: On the finite element method for the biharmonic Dirichlet problem in polygonal domains: quasi-optimal rate of convergence. Japan J. Indust. Appl. Math. 22(1), 45–56 (2005)MathSciNetzbMATHGoogle Scholar
  12. 170.
    Nagatou, K.: A numerical method to verify the elliptic eigenvalue problems including a uniqueness property. Computing 63(2), 109–130 (1999)MathSciNetzbMATHGoogle Scholar
  13. 172.
    Nagatou, K.: A computer-assisted proof on the stability of the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math. 169(1), 33–44 (2004)MathSciNetzbMATHGoogle Scholar
  14. 173.
    Nagatou, K.: Validated computation for infinite dimensional eigenvalue problems. In: 12th GAMM – IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006), number E2821, p. 3. IEEE Computer Society (2007)Google Scholar
  15. 174.
    Nagatou, K., Hashimoto, K., Nakao, M.T.: Numerical verification of stationary solutions for Navier-Stokes problems. J. Comput. Appl. Math. 199(2), 445–451 (2007)MathSciNetzbMATHGoogle Scholar
  16. 175.
    Nagatou, K., Morifuji, T.: An enclosure method for complex eigenvalues of ordinary differential operators. Nonlinear Theory and Its Applications, IEICE 2(1), 111–122 (2011)Google Scholar
  17. 177.
    Nagatou, K., Nakao, M.T., Wakayama, M.: Verified numerical computations for eigenvalues of non-commutative harmonic oscillators. Numer. Funct. Anal. Optim. 23(5–6), 633–650 (2002)MathSciNetzbMATHGoogle Scholar
  18. 179.
    Nagatou, K., Yamamoto, N., Nakao, M.T.: An approach to the numerical verification of solutions for nonlinear elliptic problems with local uniqueness. Numer. Funct. Anal. Optim. 20(5–6), 543–565 (1999)MathSciNetzbMATHGoogle Scholar
  19. 180.
    Nagatou, K.: Validated computations for fundamental solutions of linear ordinary differential equations. In: Inequalities and Applications. Volume 157 of International Series of Numerical Mathematics, pp. 43–50. Birkhäuser, Basel (2009)Google Scholar
  20. 181.
    Nagatou, K., Plum, M., Nakao, M.T.: Eigenvalue excluding for perturbed-periodic one-dimensional Schrödinger operators. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468(2138), 545–562 (2012)MathSciNetzbMATHGoogle Scholar
  21. 184.
    Nakao, M.T., Lee, S.H., Ryoo, C.S.: Numerical verification of solutions for elasto-plastic torsion problems. Comput. Math. Appl. 39(3–4), 195–204 (2000)MathSciNetzbMATHGoogle Scholar
  22. 186.
    Nakao, M.T., Yamamoto, N., Nagatou, K.: Numerical verifications for eigenvalues of second-order elliptic operators. Japan J. Indust. Appl. Math. 16(3), 307–320 (1999)MathSciNetzbMATHGoogle Scholar
  23. 190.
    Nakao, M.T.: Solving nonlinear parabolic problems with result verification. I. One-space-dimensional case. In: Proceedings of the International Symposium on Computational Mathematics (Matsuyama, 1990), vol. 38, pp. 323–334 (1991)Google Scholar
  24. 193.
    Nakao, M.T.: Numerical verifications of solutions for nonlinear hyperbolic equations. Interval Comput./Interval. Vychisl. 4, 64–77 (1994). SCAN-93, Vienna, 1993Google Scholar
  25. 196.
    Nakao, M.T., Hashimoto, K.: A numerical verification method for solutions of nonlinear parabolic problems. J. Math-for-Ind. 1, 69–72 (2009)MathSciNetzbMATHGoogle Scholar
  26. 197.
    Nakao, M.T., Hashimoto, K., Kobayashi, K.: Verified numerical computation of solutions for the stationary Navier-Stokes equation in nonconvex polygonal domains. Hokkaido Math. J. 36(4), 777–799 (2007)MathSciNetzbMATHGoogle Scholar
  27. 202.
    Nakao, M.T., Ryoo, C.S.: Numerical verifications of solutions for variational inequalities using Newton-like method. Information 2(1), 27–35 (1999). Industrial and Applied Mathematics, Okayama, 1997/1998Google Scholar
  28. 203.
    Nakao, M.T., Ryoo, C.S.: Numerical verification methods for solutions of free boundary problems. In: Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Yamaguchi, 2000. Volume 19 of Lecture Notes in Computational Science and Engineering, pp. 195–208. Springer, Berlin (2002)Google Scholar
  29. 204.
    Nakao, M.T., Watanabe, Y.: On computational proofs of the existence of solutions to nonlinear parabolic problems. In: Proceedings of the Fifth International Congress on Computational and Applied Mathematics (Leuven, 1992), vol. 50, pp. 401–410 (1994)MathSciNetzbMATHGoogle Scholar
  30. 209.
    Nakao, M.T., Watanabe, Y., Yamamoto, N.: Verified numerical computations for an inverse elliptic eigenvalue problem with finite data. Japan J. Indust. Appl. Math. 18(2), 587–602 (2001). Recent topics in mathematics moving toward science and engineeringMathSciNetzbMATHGoogle Scholar
  31. 210.
    Nakao, M.T., Watanabe, Y., Yamamoto, N., Nishida, T.: Some computer assisted proofs for solutions of the heat convection problems. In: Proceedings of the Validated Computing 2002 Conference (Toronto), vol. 9, pp. 359–372 (2003)Google Scholar
  32. 215.
    Nakao, M.T., Yamamoto, N., Nishimura, Y.: Numerical verification of the solution curve for some parametrized nonlinear elliptic problem. In: Proceedings of Third China-Japan Seminar on Numerical Mathematics, Dalian, 1997, pp. 238–245. Science Press, Beijing (1998)Google Scholar
  33. 216.
    Nakao, M.T., Yamamoto, N., Watanabe, Y.: Constructive L 2 error estimates for finite element solutions of the Stokes equations. Reliab. Comput. 4(2), 115–124 (1998)MathSciNetzbMATHGoogle Scholar
  34. 217.
    Nakao, M.T., Yamamoto, N., Watanabe, Y.: A posteriori and constructive a priori error bounds for finite element solutions of the Stokes equations. J. Comput. Appl. Math. 91(1), 137–158 (1998)MathSciNetzbMATHGoogle Scholar
  35. 218.
    Nakao, M.T., Nagatou. K., Hashimoto, K.: Numerical enclosure of solutions for two dimensional driven cavity problems. In: Proceedings of European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004) (2004)Google Scholar
  36. 221.
    Nishida, T., Ikeda, T., Yoshihara, H.: Pattern formation of heat convection problems. In: Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Yamaguchi, 2000. Volume 19 of Lecture Notes in Computational Science and Engineering, pp. 209–218. Springer, Berlin (2002)Google Scholar
  37. 251.
    Ryoo, C.S.: Numerical verification of solutions for a simplified Signorini problem. Comput. Math. Appl. 40(8–9), 1003–1013 (2000)MathSciNetzbMATHGoogle Scholar
  38. 255.
    Ryoo, C.S., Song, H., Kim, S.D.: Numerical verification of solutions for some unilateral boundary value problems. Comput. Math. Appl. 44(5–6), 787–797 (2002)MathSciNetzbMATHGoogle Scholar
  39. 265.
    Takayasu, A., Liu, X., Oishi, S.: Verified computations to semilinear elliptic boundary value problems on arbitrary polygonal domains. Nonlinear Theory Appl. IEICE 4(1), 34–61 (2013)Google Scholar
  40. 268.
    Toyonaga, K., Nakao, M.T., Watanabe, Y.: Verified numerical computations for multiple and nearly multiple eigenvalues of elliptic operators. J. Comput. Appl. Math. 147(1), 175–190 (2002)MathSciNetzbMATHGoogle Scholar
  41. 269.
    Tsuchiya, T., Nakao, M.T.: Numerical verification of solutions of parametrized nonlinear boundary value problems with turning points. Japan J. Indust. Appl. Math. 14(3), 357–372 (1997)MathSciNetzbMATHGoogle Scholar
  42. 279.
    Watanabe, Y.: A computer-assisted proof for the Kolmogorov flows of incompressible viscous fluid. J. Comput. Appl. Math. 223(2), 953–966 (2009)MathSciNetzbMATHGoogle Scholar
  43. 282.
    Watanabe, Y.: An efficient numerical verification method for the Kolmogorov problem of incompressible viscous fluid. J. Comput. Appl. Math. 302, 157–170 (2016)MathSciNetzbMATHGoogle Scholar
  44. 284.
    Watanabe, Y., Nagatou, K., Nakao, M.T., Plum, M.: A computer-assisted stability proof for the Orr-Sommerfeld problem with Poiseuille flow. Nonlinear Theory Appl. IEICE 2(1), 123–127 (2011)Google Scholar
  45. 285.
    Watanabe, Y., Nagatou, K., Plum, M., Nakao, M.T.: Verified computations of eigenvalue exclosures for eigenvalue problems in Hilbert spaces. SIAM J. Numer. Anal. 52(2), 975–992 (2014)MathSciNetzbMATHGoogle Scholar
  46. 288.
    Watanabe, Y., Nakao, M.T.: Numerical verification method of solutions for elliptic equations and its application to the Rayleigh-Bénard problem. Japan J. Indust. Appl. Math. 26(2–3), 443–463 (2009)MathSciNetzbMATHGoogle Scholar
  47. 290.
    Watanabe, Y., Nakao, M.T., Nagatou, K.: On the compactness of a nonlinear operator related to stream function-vorticity formulation for the Navier-Stokes equations. JSIAM Lett. 9, 77–80 (2017)MathSciNetzbMATHGoogle Scholar
  48. 291.
    Watanabe, Y., Plum, M., Nakao, M.T.: A computer-assisted instability proof for the Orr-Sommerfeld problem with Poiseuille flow. ZAMM Z. Angew. Math. Mech. 89(1), 5–18 (2009)MathSciNetzbMATHGoogle Scholar
  49. 294.
    Watanabe, Y., Yamamoto, N., Nakao, M.T., Nishida, T.: A numerical verification of nontrivial solutions for the heat convection problem. J. Math. Fluid Mech. 6(1), 1–20 (2004)MathSciNetzbMATHGoogle Scholar
  50. 299.
    Wieners, C.: Numerical enclosures for solutions of the Navier-Stokes equation for small Reynolds numbers. In: Numerical Methods and Error Bounds, Oldenburg, 1995. Volume 89 of Mathematical Research, pp. 280–286. Akademie Verlag, Berlin (1996)Google Scholar
  51. 309.
    Yamamoto, N., Komori, T.: An application of Taylor models to the Nakao method on ODEs. Japan J. Ind. Appl. Math. 26(2–3), 365–392 (2009)MathSciNetzbMATHGoogle Scholar
  52. 312.
    Yamamoto, N., Nakao, M.T., Watanabe, Y.: Validated computation for a linear elliptic problem with a parameter. In: Advances in Numerical Mathematics; Proceedings of the Fourth Japan-China Joint Seminar on Numerical Mathematics, Chiba, 1998. Volume 12 of Gakuto International Series/Mathematical Sciences and Applications, pp. 155–162. Gakkōtosho, Tokyo (1999)Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
  • Michael Plum
    • 2
  • Yoshitaka Watanabe
    • 3
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Faculty of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan

Personalised recommendations