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Priyanka, T. M. C.; Agathiyan, A.; Gowrisankar, A.

Weyl-Marchaud fractional derivative of a vector valued fractal interpolation function with function contractivity factors. (English) Zbl 1524.28018

J. Anal. 31, No. 1, 657-689 (2023).
Summary: This article explores the idea of Weyl-Marchaud fractional derivative on the vector-valued fractal interpolation function with function contractivity factors. Initially, the Weyl-Marchaud fractional derivative of a hidden variable fractal interpolation function (HFIF) with function contractivity factors is differentiated and proved as HFIF when the fractional order meets the necessary condition. Further, a new HFIF called the quadratic hidden variable fractal interpolation function (QHFIF) is introduced and its Weyl-Marchaud fractional derivative is investigated with function contractivity factors which generalizes the fractional derivative of QHFIF with constant contractivity factors. The variables in the HFIF have been chosen as functions that influence the fractal characteristics of the fractal functions, in order to maximize their effectiveness on fractal functions.

Cited in 6 Documents

MSC:

28A80 Fractals
26A33 Fractional derivatives and integrals
41A05 Interpolation in approximation theory

Keywords:

iterated function system; hidden variable fractal interpolation function; function contractivity factors; Weyl-Marchaud fractional derivative
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References:

[1] Mandelbrot, BB, The fractal geometry of nature (1983), New York: W.H. Freeman and Company, New York · doi:10.1119/1.13295
[2] Barnsley, MF, Fractals Everywhere (1993), Cambridge: Academic Press, Cambridge
[3] Barnsley, MF, Fractal functions and interpolation, Constructive Approximation, 2, 1, 303-329 (1986) · Zbl 0606.41005 · doi:10.1007/BF01893434
[4] Santo Banerjee, MK; Hassan, Sayan Mukherjee; Gowrisankar, A., Fractal patterns in nonlinear dynamics and applications (2020), Baco Raton: CRC Press, Baco Raton · Zbl 1445.28001 · doi:10.1201/9781315151564
[5] Pasupathi, R.; Chand, AKB; Navascués, MA, Cyclic Meir-Keeler contraction and its fractals, Numerical Functional Analysis and Optimization, 42, 9, 1053-1072 (2021) · Zbl 1494.47090 · doi:10.1080/01630563.2021.1937215
[6] Easwaramoorthy, D.; Uthayakumar, R., Analysis on fractals in fuzzy metric spaces, Fractals, 19, 3, 379-386 (2011) · Zbl 1231.26030 · doi:10.1142/S0218348X11005543
[7] Viswanathan, P.; Navascués, MA; Chand, AKB, Fractal polynomials and maps in approximation of continuous functions, Numerical Functional Analysis and Optimization, 37, 1, 106-127 (2016) · Zbl 1339.41024 · doi:10.1080/01630563.2015.1078816
[8] Verma, S.; Viswanathan, P., Parameter identification for a class of bivariate fractal interpolation functions and constrained approximation, Numerical Functional Analysis and Optimization, 41, 9, 1109-1148 (2020) · Zbl 1447.28013 · doi:10.1080/01630563.2020.1738458
[9] Santo Banerjee, D., and Easwaramoorthy, and A. Gowrisankar. 2021. Fractal functions, dimensions and signal analysis. Cham: Springer. · Zbl 1458.94072
[10] Wang, Hong-Yong; Jia-Shan, Yu, Fractal interpolation functions with variable parameters and their analytical properties, Journal of Approximation Theory, 175, 1-18 (2013) · Zbl 1303.28014 · doi:10.1016/j.jat.2013.07.008
[11] Barnsley, MF; Elton, J.; Hardin, D.; Massopust, P., Hidden variable fractal interpolation functions, SIAM Journal on Mathematical Analysis, 20, 5, 1218-1242 (1989) · Zbl 0704.26009 · doi:10.1137/0520080
[12] Bouboulis, P.; Dalla, L., Hidden variable vector valued fractal interpolation functions, Fractals, 13, 3, 227-232 (2005) · Zbl 1100.41011 · doi:10.1142/S0218348X05002854
[13] Chand, AKB; Kapoor, GP, Spline coalescence hidden variable fractal interpolation function, Journal of Applied Mathematics (2006) · Zbl 1146.41003 · doi:10.1155/JAM/2006/36829
[14] Wang, Hong-Yong, Sensitivity analysis for hidden variable fractal interpolation functions & their moments, Fractals, 17, 2, 161-170 (2009) · Zbl 1178.28018 · doi:10.1142/S0218348X09004429
[15] Yun, Chol-Hui, Hidden variable recurrent fractal interpolation functions with function contractivity factors, Fractals, 27, 7, 1950113 (2019) · Zbl 1434.28042 · doi:10.1142/S0218348X19501135
[16] Yun, Chol-Hui; Ri, Mi Gyong, Box-counting dimension and analytic properties of hidden variable fractal interpolation functions with function contractivity factors, Chaos, Solitons & Fractals, 134 (2020) · Zbl 1483.41001 · doi:10.1016/j.chaos.2020.109700
[17] Kim, Jinmyong; Kim, Hyonjin; Mun, Hakmyong, Construction of nonlinear hidden variable fractal interpolation functions and their stability, Fractals, 27, 6, 1950103 (2019) · Zbl 1435.28012 · doi:10.1142/S0218348X19501032
[18] Tatom, FB, The relationship between fractional calculus and fractals, Fractals, 3, 1, 217-229 (1995) · Zbl 0877.28009 · doi:10.1142/S0218348X95000175
[19] Yao, K.; Su, WY; Zhou, SP, On the connection between the order of fractional calculus and the dimensions of a fractal function, Chaos, Solitons & Fractals, 23, 2, 621-629 (2005) · Zbl 1062.28012 · doi:10.1016/j.chaos.2004.05.037
[20] Ruan, Huo-Jun; Wei-Yi, Su; Yao, Kui, Box dimension and fractional integral of linear fractal interpolation functions, Journal of Approximation Theory, 161, 1, 187-197 (2009) · Zbl 1181.41005 · doi:10.1016/j.jat.2008.08.012
[21] He, Ji-Huan, Fractal calculus and its geometrical explanation, Results in Physics, 10, 272-276 (2018) · doi:10.1016/j.rinp.2018.06.011
[22] Liang, Y.-S., and Q. Zhang. 2016. A type of fractal interpolation functions and their fractional calculus. Fractals 24 (2): 1650026. · Zbl 1351.28016
[23] Gowrisankar, A.; Uthayakumar, R., Fractional calculus on fractal interpolation function for a sequence of data with countable iterated function system, Mediterranean Journal of Mathematics, 13, 6, 3887-3906 (2016) · Zbl 1349.28011 · doi:10.1007/s00009-016-0720-x
[24] Yao, Kui; Haotian Chen, WL; Peng, Zekun Wang; Yao, Jia; Yipeng, Wu, A new method on box dimension of Weyl-Marchaud fractional derivative of Weierstrass function, Chaos, Solitons & Fractals, 142 (2021) · Zbl 1496.26007 · doi:10.1016/j.chaos.2020.110317
[25] Zhang, Qi; Liang, Y-S, The Weyl-Marchaud fractional derivative of a type of self-affine functions, Applied Mathematics and Computation, 218, 17, 8695-8701 (2012) · Zbl 1250.26008 · doi:10.1016/j.amc.2012.01.077
[26] Gowrisankar, A.; Prasad, MGP, Riemann-Liouville calculus on quadratic fractal interpolation function with variable scaling factors, The Journal of Analysis, 27, 2, 347-363 (2019) · Zbl 1423.28021 · doi:10.1007/s41478-018-0133-2
[27] Ri, Mi-Gyong; Yun, Chol-Hui, Riemann Liouville fractional integral of hidden variable fractal interpolation function, Chaos, Solitons & Fractals, 140 (2020) · Zbl 1495.26012 · doi:10.1016/j.chaos.2020.110126
[28] Priyanka, TMC; Gowrisankar, A., Analysis on Wely-Marchaud fractional derivative of types of fractal interpolation function with fractal dimension, Fractals, 29, 7, 2150215 (2021) · Zbl 1511.28007 · doi:10.1142/S0218348X21502157
[29] Priyanka, TMC; Gowrisankar, A., Riemann-Liouville fractional integral of non-affine fractal interpolation function and its fractional operator, The European Physical Journal Special Topics, 230, 3789-3805 (2021) · doi:10.1140/epjs/s11734-021-00315-6
[30] Peng, WL; Yao, Kui; Zhang, Xia; Yao, Jia, Box dimension of Weyl-Marchaud fractional derivative of linear fractal interpolation functions, Fractals, 27, 4, 1950058 (2019) · Zbl 1433.28025 · doi:10.1142/S0218348X19500580
[31] Fausto Ferrari. 2018. Weyl and Marchaud derivatives: a forgotten history. Mathematics 6(1) · Zbl 1473.26006
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