Year 2018

• Shub’s conjecture for smooth longitudinal maps of S^m

  Publication

  • G. Graff
  • M. Misiurewicz
  • P. Nowak-Przygodzki

  - JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS - Year 2018

  Let f be a smooth map of the m-dimensional sphere Sm to itself, preserving the longitudinal foliation. We estimate from below the number of fixed points of the iterates of f , reduce Shub’s conjecture for longitudinal maps to a lower dimensional classical version, and prove the conjecture in case m = 2 and in a weak form for m = 3.

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Year 2015

• On the convergence of a nonlinear finite-difference discretization of the generalized Burgers–Fisher equation

  Publication

  • A. Szafrańska
  • J. Macías-Díaz

  - JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS - Year 2015

  In this note, we establish analytically the convergence of a nonlinear finite-difference discretization of the generalized Burgers-Fisher equation. The existence and uniqueness of positive, bounded and monotone solutions for this scheme was recently established in [J. Diff. Eq. Appl. 19, 1907{1920 (2014)]. In the present work, we prove additionally that the method is convergent of order one in time, and of order two in space. Some...

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Year 2014

• Existence and uniqueness of monotone and bounded solutions for a finite-difference discretization a` la Mickens of the generalized Burgers–Huxley equation.

  Publication

  • J. E. Macías-Díaz
  • A. Szafrańska

  - JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS - Year 2014

  Departing from a generalized Burgers–Huxley partial differential equation, we provide a Mickens-type, nonlinear, finite-difference discretization of this model. The continuous system is a nonlinear regime for which the existence of travelling-wave solutions has been established previously in the literature. We prove that the method proposed also preserves many of the relevant characteristics of these solutions, such as the positivity,...

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• On the convergence of a finite-difference discretization à la Mickens of the generalized Burgers–Huxley equation

  Publication

  • A. Szafrańska
  • J. E. Macías-Díaz

  - JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS - Year 2014

  In this note, we establish the property of convergence for a finite-difference discretization of a diffusive partial differential equation with generalized Burgers convective law and generalized Hodgkin–Huxley reaction. The numerical method was previously investigated in the literature and, amongst other features of interest, it is a fast and nonlinear technique that is capable of preserving positivity, boundedness and monotonicity....

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