Fractional Weierstrass function by application of Jumarie fractional trigonometric functions and its analysis

TL;DR

This paper introduces a fractional order Weierstrass function using Jumarie fractional trigonometric functions, analyzing its properties and showing invariance of roughness indices compared to the classical version.

Contribution

It generalizes the classical Weierstrass function with fractional trigonometric functions and demonstrates that key fractal properties remain unchanged.

Findings

Holder exponent remains the same as classical Weierstrass function

Box dimension is invariant under fractional generalization

Fractional derivatives preserve the function's roughness index

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.