The uniformly continuous theorem of fractal interpolation surface function and its proof

Xuezai Pan; Minggang Wang; Xuezai Pan; Minggang Wang

doi:10.3934/math.2024529

AIMS Mathematics

2024, Volume 9, Issue 5: 10858-10868. doi: 10.3934/math.2024529

Previous Article Next Article

Research article

The uniformly continuous theorem of fractal interpolation surface function and its proof

Xuezai Pan ^{1
,
,},
Minggang Wang ²

1.
School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224003, China
2.
School of Mathematics, Nanjing Normal University, Nanjing 210023, China

Received: 29 January 2024 Revised: 01 March 2024 Accepted: 05 March 2024 Published: 19 March 2024
MSC : 33E99

In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.
- fractal geometry,
- affine mapping,
- fractal interpolation,
- surface function,
- accumulation principle,
- uniform continuity of a function of two variables
Citation: Xuezai Pan, Minggang Wang. The uniformly continuous theorem of fractal interpolation surface function and its proof[J]. AIMS Mathematics, 2024, 9(5): 10858-10868. doi: 10.3934/math.2024529

Related Papers:

Abstract

In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area.

References

[1]	B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science, 156 (1967), 636–638. https://doi.org/10.1126/science.156.3775.636 doi: 10.1126/science.156.3775.636
[2]	B. B. Mandelbrot, J. W. V. Ness, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10 (1968), 422–437. https://doi.org/10.1137/1010093 doi: 10.1137/1010093
[3]	B. B. Mandelbrot, The fractal geometry of nature, New York: Macmillan Press, 1983.
[4]	B. B. Mandelbrot, D. E. Passoja, A. J. Paulley, Fractal character of fracture surfaces of metals, Nature, 308 (1984), 721–722. https://doi.org/10.1038/308721a0 doi: 10.1038/308721a0
[5]	B. B. Mandelbrot, M Aizenman, Fractals: form, chance, and dimension, Phys. Today, 32 (1979), 65–66. https://doi.org/10.1063/1.2995555 doi: 10.1063/1.2995555
[6]	H. P. Xie, Fractals in rock mechanics, London: CRC Press, 1993. https://doi.org/10.1201/9781003077626
[7]	H. P. Xie, H. Q. Sun, Y. Ju, Study on generation of rock surfaces by using fractal interpolation, Int. J. Solids Struct., 38 (2001), 5765–5787. https://doi.org/10.1016/S0020-7683(00)00390-5 doi: 10.1016/S0020-7683(00)00390-5
[8]	M. Nasehnejad, G. Nabiyouni, M. G. Shahraki, Thin film growth by 3D multi-particle diffusion limited aggregation model: Anomalous roughening and fractal analysis, Physica A, 493 (2018), 135–147. https://doi.org/10.1016/j.physa.2017.09.099 doi: 10.1016/j.physa.2017.09.099
[9]	S. Bianchi, F. Massimiliano, Fractal stock markets: International evidence of dynamical (in) efficiency, Chaos, 27 (2017), 071102. https://doi.org/10.1063/1.4987150 doi: 10.1063/1.4987150
[10]	S. S. Yang, H. H. Fu, B. M. Yu, Fractal analysis of flow resistance in tree-like branching networks with roughened microchannels, Fractals, 25 (2017), 1750008. https://doi.org/10.1142/S0218348X17500086 doi: 10.1142/S0218348X17500086
[11]	Z. G. Feng, Variation and Minkowski dimension of fractal interpolation surface, J. Math. Anal. Appl., 345 (2008), 322–334. https://doi.org/10.1016/j.jmaa.2008.03.075 doi: 10.1016/j.jmaa.2008.03.075
[12]	X. B. Zhang, H. L. Zhu, H. X. Yao, Analysis of a new three-dimensional chaotic system, Nonlinear Dyn., 67 (2012), 335–343. https://doi.org/10.1007/s11071-011-9981-x doi: 10.1007/s11071-011-9981-x
[13]	M. F. Barnsley, Fractal functions and interpolation, Constr. Approx., 2 (1986), 303–329. https://doi.org/10.1007/BF01893434 doi: 10.1007/BF01893434
[14]	M. F. Barnsley, M. Hegland, P. Massopust, Numerics and fractals, Bull. Inst. Math. Acad., 9 (2014), 389–430.
[15]	P. R. Massopust, Fractal surfaces, J. Math. Anal. Appl., 151 (1990), 275–290. https://doi.org/10.1016/0022-247X(90)90257-G doi: 10.1016/0022-247X(90)90257-G
[16]	P. R. Massopust, Fractal functions, fractal surfaces, and wavelets, Orlando: Academic Press, 1994. https://doi.org/10.1016/c2009-0-21290-6
[17]	M. Barnsley, A. J. Hurd, Fractals everywhere, Am. J. Phys., 57 (1989), 1053. https://doi.org/10.1119/1.15823 doi: 10.1119/1.15823
[18]	X. Z. Pan, X. D. Shang, M. G. Wang, F. Zuo, The cantor set's multi-fractal spectrum formed by different probability factors in mathematical experiment, Fractals, 25 (2017), 1750002. https://doi.org/10.1142/S0218348X17500025 doi: 10.1142/S0218348X17500025
[19]	S. Iqbal, M. Idrees, A. M. Siddiqui, A. R. Ansari, Some solutions of the linear and nonlinear Klein-Gordon equations using the optimal homotopy asymptotic method, Appl. Math. Comput., 216 (2010), 22898–2909. https://doi.org/10.1016/j.amc.2010.04.001 doi: 10.1016/j.amc.2010.04.001
[20]	B. Wang, Q. X. Zhu, Stability analysis of semi-Markov switched stochastic systems, Automatica, 94 (2018), 72–80. https://doi.org/10.1016/j.automatica.2018.04.016 doi: 10.1016/j.automatica.2018.04.016
[21]	X. Z. Pan, Fractional calculus of fractal interpolation function on [0, b] (b > 0), Abstr. Appl. Anal., 2014 (2014), 640628. http://doi.org/10.1155/2014/640628 doi: 10.1155/2014/640628
[22]	B. J. Hong, D. C. Lu, Modified fractional variational iteration method for solving the generalized time-space fractional schrdinger equation, Sci. World J., 2014 (2014), 964643. http://doi.org/:10.1155/2014/964643 doi: 10.1155/2014/964643
[23]	X. Z. Pan, R. F. Xu, X. D. Shang, M. G. Wang, The properties of fractional order calculus of fractal interpolation function of broken line segments, Fractals, 26 (2018), 1850031. http://doi.org/10.1142/S0218348X18500317 doi: 10.1142/S0218348X18500317
[24]	Y. Chang, Y. Q. Wang, F. E. Alsaadi, G. D. Zong, Adaptive fuzzy output-feedback tracking control for switched stochastic pure-feedback nonlinear systems, Int. J. Adapt. Control, 33 (2019), 1567–1582. http://doi.org/10.1002/acs.3052 doi: 10.1002/acs.3052
[25]	W. Rubin, Principles of mathematical analysis, New York: McGraw-Hill Publishing Company, 1976.
[26]	Y. Y. Shan, Remarks about two theorems in principles of mathematical analysis, Journal of Mathematics Research, 1 (2009), 61–63. http://doi.org/10.5539/jmr.v1n2p61 doi: 10.5539/jmr.v1n2p61
[27]	H. P. Xie, H. Q. Sun, The theory of fractal interpolated surface and its applications, Appl. Math. Mech., 19 (1998), 321–331. https://doi.org/10.1007/BF02457536 doi: 10.1007/BF02457536
[28]	H. P. Xie, H. Q. Sun, The study on bivariate fractal interpolation functions and creation of fractal interpolation surfaces, Fractals, 5 (1997), 625–634. http://doi.org/10.1142/S0218348X97000504 doi: 10.1142/S0218348X97000504

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)