| Peer-Reviewed

Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere

Received: 1 February 2017     Accepted: 28 February 2017     Published: 29 March 2017
Views:       Downloads:
Abstract

This paper studies the structure of the hyperbolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions are determined and investigated. Numerical solutions of the equation on graphs and digital n-manifolds are presented.

Published in International Journal of Discrete Mathematics (Volume 2, Issue 3)
DOI 10.11648/j.dmath.20170203.15
Page(s) 88-94
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Hyperbolic PDE, Graph, Solution, Initial Value Problem, Digital Space, Digital Topology

References
[1] Bai, Y., Han, X., Prince, J.(2009) Digital Topology on Adaptive Octree Grids. Journal of Mathematical Imaging and Vision. 34 (2), 165-184.
[2] Borovskikh, A. and Lazarev, K. (2004) Fourth-order differential equations on geometric graphs. Journal of Mathematical Science, 119 (6), 719–738.
[3] Eckhardt, U. and Latecki, L. (2003) Topologies for the digital spaces Z2 and Z3. Computer Vision and Image Understanding, 90, 295-312.
[4] Evako, A. (2016) Solution of a Parabolic Partial Differential Equation on Digital Spaces: A Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band, International Journal of Discrete Mathematics. Vol. 1, No. 1, pp. 5-14. doi: 10.11648/j.dmath.20160101.12.
[5] Evako, A. (01/2017) Properties of Periodic Fibonacci-like Sequences. arXiv:1701.08566, 2017arXiv170108566E.
[6] Evako, A. (2014) Topology preserving discretization schemes for digital image segmentation and digital models of the plane. Open Access Library Journal, 1, e566, http://dx.doi.org/10.4236/oalib.1100566.
[7] Evako, A. (1999) Introduction to the theory of molecular spaces (in Russian language). Publishing House Paims, Moscow.
[8] Evako, A., Kopperman, R. and Mukhin, Y. (1996) Dimensional properties of graphs and digital spaces. Journal of Mathematical Imaging and Vision, 6, 109-119.
[9] Evako, A. (2015) Classification of digital n-manifolds. Discrete Applied Mathematics. 181, 289–296.
[10] Evako, A. (1995) Topological properties of the intersection graph of covers of n-dimensional surfaces. Discrete Mathematics, 147, 107-120.
[11] Ivashchenko, A. (1993) Representation of smooth surfaces by graphs. Transformations of graphs which do not change the Euler characteristic of graphs. Discrete Mathematics, 122, 219-233.
[12] Ivashchenko, A. (1994) Contractible transformations do not change the homology groups of graphs. Discrete Mathematics, 126, 159-170.
[13] Lu, W. T. and Wu, F. Y. (2001) Ising model on nonorientable surfaces: Exact solution for the Moebius strip and the Klein bottle. Phys. Rev., E 63, 026107.
[14] Pokornyi, Y. and Borovskikh, A. (2004) Differential equations on networks (Geometric graphs). Journal of Mathematical Science, 119 (6), 691–718.
[15] Smith, G. D. (1985) Numerical solution of partial differential equations: finite difference methods (3rd ed.). Oxford University Press.
[16] Vol’pert, A. (1972) Differential equations on graphs. Mat. Sb. (N. S.), 88 (130), 578–588.
[17] Xu Gen, Qi. and Mastorakis, N. (2010) Differential equations on metric graph. WSEAS Press.
Cite This Article
  • APA Style

    Alexander V. Evako. (2017). Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere. International Journal of Discrete Mathematics, 2(3), 88-94. https://doi.org/10.11648/j.dmath.20170203.15

    Copy | Download

    ACS Style

    Alexander V. Evako. Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere. Int. J. Discrete Math. 2017, 2(3), 88-94. doi: 10.11648/j.dmath.20170203.15

    Copy | Download

    AMA Style

    Alexander V. Evako. Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere. Int J Discrete Math. 2017;2(3):88-94. doi: 10.11648/j.dmath.20170203.15

    Copy | Download

  • @article{10.11648/j.dmath.20170203.15,
      author = {Alexander V. Evako},
      title = {Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere},
      journal = {International Journal of Discrete Mathematics},
      volume = {2},
      number = {3},
      pages = {88-94},
      doi = {10.11648/j.dmath.20170203.15},
      url = {https://doi.org/10.11648/j.dmath.20170203.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.15},
      abstract = {This paper studies the structure of the hyperbolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions are determined and investigated. Numerical solutions of the equation on graphs and digital n-manifolds are presented.},
     year = {2017}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Solution of the Hyperbolic Partial Differential Equation on Graphs and Digital Spaces: A Klein Bottle a Projective Plane and a 4D Sphere
    AU  - Alexander V. Evako
    Y1  - 2017/03/29
    PY  - 2017
    N1  - https://doi.org/10.11648/j.dmath.20170203.15
    DO  - 10.11648/j.dmath.20170203.15
    T2  - International Journal of Discrete Mathematics
    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
    SP  - 88
    EP  - 94
    PB  - Science Publishing Group
    SN  - 2578-9252
    UR  - https://doi.org/10.11648/j.dmath.20170203.15
    AB  - This paper studies the structure of the hyperbolic partial differential equation on graphs and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions are determined and investigated. Numerical solutions of the equation on graphs and digital n-manifolds are presented.
    VL  - 2
    IS  - 3
    ER  - 

    Copy | Download

Author Information
  • “Dianet”, Laboratory of Digital Technologies, Moscow, Russia

  • Sections