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Mediterranean Journal of Mathematics

, Volume 13, Issue 6, pp 3887–3906 | Cite as

Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System

  • A. Gowrisankar
  • R. Uthayakumar
Article
  • 143 Downloads

Abstract

In recent years, the concept of fractal analysis is the best nonlinear tool towards understanding the complexities in nature. Especially, fractal interpolation has flexibility for approximation of nonlinear data obtained from the engineering and scientific experiments. Random fractals and attractors of some iterated function systems are more appropriate examples of the continuous everywhere and nowhere differentiable (highly irregular) functions, hence fractional calculus is a mathematical operator which best suits for analyzing such a function. The present study deals the existence of fractal interpolation function (FIF) for a sequence of data \({\{(x_n,y_n):n\geq 2\}}\) with countable iterated function system, where \({x_n}\) is a monotone and bounded sequence, \({y_n}\) is a bounded sequence. The integer order integral of FIF for sequence of data is revealed if the value of the integral is known at the initial endpoint or final endpoint. Besides, Riemann–Liouville fractional calculus of fractal interpolation function had been investigated with numerical examples for analyzing the results.

Keywords

Attractor countable iterated function system fractal interpolation function fractional calculus 

Mathematics Subject Classification

28A80 26A33 41A05 

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References

  1. 1.
    Mandelbrot B.B.: The Fractal Geometry of Nature. W.H. Freeman and Company, New York (1983)MATHGoogle Scholar
  2. 2.
    Hutchinson J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 313–747 (1981)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Barnsley M.F.: Fractals Everywhere. 2nd edn. Academic Press, USA (1993)MATHGoogle Scholar
  4. 4.
    Barnsley M.F.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barnsley M.F., Harrington A.N.: The calculus of fractal interpolation functions. J. Approx. Theory 57, 14–34 (1989)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Viswanathan P., Chand A.K.B., Navascues M.A.: Fractal perturbation preserving fundamental shapes: bounds on the scale factors. J. Math. Anal. Appl. 419, 804–817 (2014)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chand A.K.B., Kapoor G.P.: Hidden variable bivariate fractal interpolation surfaces. Fractals 11(3), 277–288 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Secelean N.A.: The fractal interpolation for countable systems of data, Beograd University, Publikacije. Electrotehn. Fak. ser. Matematika 14, 11–19 (2003)MathSciNetMATHGoogle Scholar
  9. 9.
    Secelean N.A.: The existence of the attractor of countable iterated function systems. Mediterr. J. Math. 9, 61–79 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Uthayakumar R., Gowrisankar A.: Fractals in product fuzzy metric space, fractals, wavelets, and their applications. Springer Proc. Math. Stat. 92, 157–164 (2014)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Uthayakumar R., Gowrisankar A.: Generation of fractals via self-similar group of Kannan iterated function system. Appl. Math. Inf. Sci. 9(6), 3245–3250 (2015)MathSciNetGoogle Scholar
  12. 12.
    Massopust, P.R.: Fractal functions. In: Fractal Surfaces and Wavelets. Academic Press, Orlando (1994)Google Scholar
  13. 13.
    Amo E.D., Carrillo M.D., Sanchez J.F.: PCF self-similar sets and fractal interpolation. Math. Comput. Simul. 92, 28–39 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Navascues N.A.: Fractal approximation. Complex Anal. Oper. Theory 4, 953–974 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Miller K.S., Ross B.: An introduction to the fractional calculus and fractional differential equations. Wiley, New York (1993)MATHGoogle Scholar
  16. 16.
    Ross B.: Fractional Calculus and its Applications, Berlin. Springer, Heidelberg (1975)CrossRefGoogle Scholar
  17. 17.
    Tatom F.B.: The relationship between fractional calculus and fractal. Fractals 3(1), 217–229 (1995)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Liang Y.S., Su W.Y.: The relationship between the fractal dimensions of a type of fractal functions and the order of their fractional calculus. Chaos Solitons Fractals 34, 682–692 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ruan H.-J., Su W.-Y., Yao K.: Box dimension and fractional integral of linear fractal interpolation functions. J. Approx. Theory 161, 187–197 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Secelean N.A.: Approximation of the attractor of a countable iterated function system. Gen. Math. 17(3), 221–231 (2009)MathSciNetMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsThe Gandhigram Rural Institute, Deemed UniversityGandhigram, DindigulIndia

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