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Solution of a Parabolic Partial Differential Equation on Digital Spaces: A Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band

Received: 8 October 2016     Accepted: 17 November 2016     Published: 27 December 2016
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Abstract

This paper studies a parabolic partial differential equation on digital spaces and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions of equations are determined and investigated. Numerical solutions of the equation on a Klein bottle, a projective plane, a 4D sphere and a Moebius strip are presented.

Published in International Journal of Discrete Mathematics (Volume 1, Issue 1)
DOI 10.11648/j.dmath.20160101.12
Page(s) 5-14
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), 2016. Published by Science Publishing Group

Keywords

Digital Surface, Graph, Parabolic PDE, Digital Topology, Moebius Strip, Klein Bottle, Projective Plane

References
[1] Gantmakher, F. R. (1959) The theory of matrices. New York, Chelsea Pub. Co.
[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. (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.
[5] Evako, A. (1999) Introduction to the theory of molecular spaces (in Russian language). Publishing House Paims, Moscow.
[6] Evako, A., Kopperman, R. and Mukhin, Y. (1996) Dimensional properties of graphs and digital spaces. Journal of Mathematical Imaging and Vision, 6, 109-119.
[7] Evako, A. (2006) Topological properties of closed digital spaces. One method of constructing digital models of closed continuous surfaces by using covers. Computer Vision and Image Understanding. 102, 134-144.
[8] Evako, A. (2015) Classification of digital n-manifolds. Discrete Applied Mathematics. 181, 289–296.
[9] Evako, A. (1995) Topological properties of the intersection graph of covers of n-dimensional surfaces. Discrete Mathematics, 147, 107-120.
[10] 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.
[11] Ivashchenko, A. (1994) Contractible transformations do not change the homology groups of graphs. Discrete Mathematics, 126, 159-170.
[12] Lankaster, P. (1969) Theory of matrices. Academic Press, New-York-London.
[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.
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  • APA Style

    Alexander V. Evako. (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, 1(1), 5-14. https://doi.org/10.11648/j.dmath.20160101.12

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    ACS Style

    Alexander V. Evako. Solution of a Parabolic Partial Differential Equation on Digital Spaces: A Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band. Int. J. Discrete Math. 2016, 1(1), 5-14. doi: 10.11648/j.dmath.20160101.12

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    AMA Style

    Alexander V. Evako. Solution of a Parabolic Partial Differential Equation on Digital Spaces: A Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band. Int J Discrete Math. 2016;1(1):5-14. doi: 10.11648/j.dmath.20160101.12

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  • @article{10.11648/j.dmath.20160101.12,
      author = {Alexander V. Evako},
      title = {Solution of a Parabolic Partial Differential Equation on Digital Spaces: A Klein Bottle, a Projective Plane, a 4D Sphere and a Moebius Band},
      journal = {International Journal of Discrete Mathematics},
      volume = {1},
      number = {1},
      pages = {5-14},
      doi = {10.11648/j.dmath.20160101.12},
      url = {https://doi.org/10.11648/j.dmath.20160101.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20160101.12},
      abstract = {This paper studies a parabolic partial differential equation on digital spaces and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions of equations are determined and investigated. Numerical solutions of the equation on a Klein bottle, a projective plane, a 4D sphere and a Moebius strip are presented.},
     year = {2016}
    }
    

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    AB  - This paper studies a parabolic partial differential equation on digital spaces and digital n-dimensional manifolds, which are digital models of continuous n-manifolds. Conditions for the existence of solutions of equations are determined and investigated. Numerical solutions of the equation on a Klein bottle, a projective plane, a 4D sphere and a Moebius strip are presented.
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Author Information
  • Laboratory of Digital Technologies, Moscow, Russia

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