Research Article | Open Access
P. Bouboulis, "Construction of Fractal Surfaces via Solutions of Partial Differential Equations", SRX Mathematics, vol. 2010, Article ID 432521, 10 pages, 2010. https://doi.org/10.3814/2010/432521
Construction of Fractal Surfaces via Solutions of Partial Differential Equations
A new construction of fractal interpolation surfaces, using solutions of partial differential equations, is presented. We consider a set of interpolation points placed on a rectangular grid and a specific PDE, such that its Dirichlet's problem is uniquely solvable inside any given orthogonal region. We solve the PDE, using numerical methods, for a number of regions, to construct two functions and , which are then used to produce the fractal surface, as the attractor of an appropriately chosen recurrent iterated function system.
- M. F. Barnsley and S. Demko, “Iterated function systems and the global construction of fractals,” Proceedings of the Royal Society of London A, vol. 399, pp. 243–275, 1985.
- P. R. Massopust, “Fractal surfaces,” The Journal of Mathematical Analysis and Applications, vol. 151, no. 1, pp. 275–290, 1990.
- P. R. Massopust, Fractal Functions, Fractal Surfaces and Wavelets, Academic Press, San Diego, Calif, USA, 1994.
- M. F. Barnsley, “Fractal functions and interpolation,” Constructive Approximation, vol. 2, pp. 303–329, 1986.
- A. K. B. Chand and G. P. Kapoor, “Generalized cubic spline fractal interpolation functions,” SIAM Journal on Numerical Analysis, vol. 44, no. 2, pp. 655–676, 2006.
- M. A. Navascues and M. V. Sebastian, “Generalization of Hermite functions by fractal interpolation,” The Journal of Approximation Theory, vol. 131, pp. 19–29, 2004.
- J. S. Geronimo and D. Hardin, “Fractal interpolation surfaces and a related 2D multiresolutional analysis,” Journal of Mathematical Analysis and Applications, vol. 176, pp. 561–586, 1993.
- N. Zhao, “Construction and application of fractal interpolation surfaces,” The Visual Computer, vol. 12, pp. 132–146, 1996.
- R. Malysz, “The Minkowski dimension of the bivariate fractal interpolation surfaces,” Chaos, Solitons and Fractals, vol. 27, no. 5, pp. 1147–1156, 2006.
- P. Bouboulis, L. Dalla, and V. Drakopoulos, “Construction of recurrent bivariate fractal interpolation surfaces and computation of their box-counting dimension,” The Journal of Approximation Theory, vol. 141, pp. 99–117, 2006.
- P. Bouboulis, “Construction of Orthogonal Multi-Wavelets using Generalized-Affine Fractal Interpolation Functions,” IMA Journal of Applied Mathematics, vol. 74, no. 6, pp. 904–933, 2009.
- P. Bouboulis and L. Dalla, “Closed fractal interpolation surfaces,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 116–126, 2007.
- P. Bouboulis and L. Dalla, “Fractal interpolation surfaces derived from fractal interpolation functions,” Journal of Mathematical Analysis and Applications, vol. 336, no. 2, pp. 919–936, 2007.
- P. Bouboulis and L. Dalla, “A general construction of fractal interpolation functions on grids of ,” European Journal of Applied Mathematics, vol. 18, no. 4, pp. 449–476, 2007.
- M. F. Barnsley, Fractals Everywhere, Academic Press Professional, Boston, Mass, USA, 2nd edition, 1993.
- K. Falconer, Fractal Geometry, John Wiley & Sons, New York, NY, USA, 2003.
Copyright © 2010 P. Bouboulis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.