Fractional Weierstrass Function by Application of Jumarie Fractional Trigonometric Functions and Its Analysis

Uttam Ghosh,Shantanu Das,Susmita Sarkar

doi:10.4236/apm.2015.512065

Uttam Ghosh, Shantanu Das + Show 1 more

Open Access

https://doi.org/10.4236/apm.2015.512065

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Abstract

The classical example of no-where differentiable but everywhere continuous function is Weierstrass function. In this paper we define the fractional order Weierstrass function in terms of Jumarie fractional trigonometric functions. The Holder exponent and Box dimension of this function are calculated here. It is established that the Holder exponent and Box dimension of this fractional order Weierstrass function are the same as in the original Weierstrass function, independent of incorporating the fractional trigonometric function. This is new development in generalizing the classical Weierstrass function by usage of fractional trigonometric function and obtain its character and also of fractional derivative of fractional Weierstrass function by Jumarie fractional derivative, and establishing that roughness index are invariant to this generalization.

Highlights

The concepts of fractional geometry, fractional dimensions are important branches of science to study the irregularity of a function, graph or signals [1]-[3]
The fractional order derivative of this function has established here. This is a new development in generalizing the classical Weierstrass function by usage of fractional trigonometric functions including the study of its character
We have derived some useful relations of fractional trigonometric functions which shall be used for our further calculations—in characterizing fractional Weierstrass function

Summary

Introduction

The concepts of fractional geometry, fractional dimensions are important branches of science to study the irregularity of a function, graph or signals [1]-[3]. 2) Two Parameter Sine and Cosine Function ( ) ( ) Let us define the two parameter sine and cosine functions cosα,β xα and sinα,β xα as depicted below: With this and with definition of two parameter Mittag-Leffler function (3) with imaginary argument we get the following useful identity ( ) ( ) ( ) ( ) Eα,β ixα. For the continuous function f : R → R , f ( x) satisfies the Lipschitz condition on its domain of definition if f ( x) − f ( y) < C x − y when 0 < x − y < ε , where ε is small positive number, and C > 0 is real constant This function f ( x) has Holder exponent as unity. The Holder and Hurst exponents are equivalent for uni-fractal graphs that has a constant fractional dimension in defined interval [1] [9]

The Fractional Weierstrass Function

The Jumarie Fractional Derivative of Fractional Weierstrass Function

Conclusion