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

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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.

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Authors and Affiliations

  1. Department of Mathematics, The Gandhigram Rural Institute, Deemed University, Gandhigram, Dindigul, Tamil Nadu, 624 302, India

    A. Gowrisankar & R. Uthayakumar

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  1. A. Gowrisankar
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  2. R. Uthayakumar
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Correspondence to A. Gowrisankar.

Additional information

The research work has been supported by University Grants Commission, Government of India, New Delhi, India, under the scheme of UGC-MRP with Grant No.: F.No. 42-21/2013 (SR)/dated 12.03.2013.

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Gowrisankar, A., Uthayakumar, R. Fractional Calculus on Fractal Interpolation for a Sequence of Data with Countable Iterated Function System. Mediterr. J. Math. 13, 3887–3906 (2016). https://doi.org/10.1007/s00009-016-0720-x

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  • DOI: https://doi.org/10.1007/s00009-016-0720-x

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