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

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.


Introduction
Fractional geometry, fractional dimension is an important branch of science to study the irregularity of a function, graph or signals [1][2][3]. On the other hand fractional calculus is another developing mathematical tool to study the continuous but non-differentiable functions (signals) where the conventional calculus fails [4][5][6][7][8][9][10][11]. Many authors are trying to relate between the fractional derivative and fractional dimension [1,[12][13][14][15]. The functions which are continuous but non-differentiable in integer order calculus can be characterized in terms of fractional calculus and especially through Holder exponent [10,16]. To study the no-where differentiable functions authors in [12][13][14][15][16] used different type of fractional derivatives. Jumarie [17] defines the fractional trigonometric functions in terms of Mittag-Leffler function and established different useful fractional trigonometric formulas. The fractional order derivatives of those functions were established in-terms the Jumarie [17][18] modified fractional order derivatives. In this paper we define the fractional order Weierstrass functions in terms the fractional order sine function. The Holder exponent, box-dimension (Fractional dimension) of graph of this function is obtained here; also the fractional order derivative of this function is established here. This is new development in generalizing the classical Weierstrass function by usage of fractional trigonometric function and obtain its character. The paper is organized as sections; with section-2 dealing with describing Jumarie fractional derivative and Mittag-Leffler function of one and two parameter type, fractional trigonometric function of one and two parameter type and their Jumarie fractional derivatives are derived. In this section we derived useful relations of fractional trigonometric function that we shall be using for our calculations-in characterizing fractional Weierstrass function. We continue this section by introducing Lipschitz Holder exponent (LHE)its definition, its relation to Hurst exponent and fractional dimension and definition of Holder continuity, and we define here the classical Weierstrass function. These parameters are basic parameter to indicate roughness index of a function or graph. In section-3 we describe the fractional Weierstrass function by generalizing the classical Weierstrass function by use of fractional sine trigonometric function. Subsequently we apply derived identities of fractional trigonometric functions to evaluate the properties of this new fractional Weierstrass function. In section-4 we do derivation of properties of fractional derivative of fractional Weierstrass function, and conclude the paper with conclusion and references.

Jumarie fractional order derivative and Mittag-Leffler Function a) Fractional order derivative of Jumarie Type
Jumarie [17] defined the fractional order derivative modifying the Left Riemann-Liouvellie (RL) fractional derivative in the form for the function ( ) In the above definition, the first expression is just Riemann-Liouvelli fractional integration; the second line is Riemann-Liouvelli fractional derivative of order 0 α < < of offset , that is a decaying power-law function. Also we purposely state that ( ) 0 f x = for in order to have initialization function in case of fractional differ-integration to be zero, else results are difficult [9].

) Mittag-Leffler Function
The Mittag-Leffler function [19][20][21][22] of one parameter is denoted by and defined by ( This function plays a crucial role in classical calculus for , for 1 α = α = it becomes the exponential function, that is is a fundamental solution of the Jumarie type fractional differential equation where 0 x D α is Jumarie derivative operator as described above. The other important function is the two parameter Mittag-Leffler function is denoted and defined by following series . We now consider the Mittag-Leffler function in the following form in infinite series representation for ( ) ) ( Then taking Jumarie fractional derivative of order 0 1 α < < term by term for the above series we obtain the following by using the formula    [18] defined the fractional sine and cosine function in the following form The series presentation of ( ) cos ( ) Taking term by term Jumarie derivative we get The series presentation of ( ) sin ( ) at a t a t a t at Taking term by term Jumarie derivative we get (1 ) . Proof of the above relation we reproduce. Let us consider a function f x which satisfies the condition Differentiating both side with respect to x and y of α − order respectively we get the following First consider y a constant, and we fractionally differentiate w.r.t. x by Jumarie derivative Now we consider x as constant and do the following steps Here we put equivalence of as Jumarie fractional derivative of constant is zero. Therefore the RHS of above two expressions are equal, from that we get the following The above two may be equated to a constant say λ . Then we have ( , or we From the property of Mittag-Leffler function and Jumarie derivative of the Mittag-Leffler function we know that 0 Considering i λ = , we therefore can write the following identity Comparing real and imaginary part in above derived relation we get the following sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) cos ( ) cos ( ) cos ( ) sin ( ) sin ( ) x y x y y x x y x y y This is very useful relation as in conjugation with classical trigonometric functions, and we will be using these relations in our analysis of fractional Weierstrass function and its fractional derivative. Let us define Now with this and with definition of two parameter Mittag-Leffler (3) function with imaginary argument we get the following useful identity   ) ) ...
Thus we get a very useful relation Similarly it can be shown that Now we calculate the Jumarie type fractional order derivative of [ ]

Some definitions of roughness index a) Lipschitz Holder exponent(LHE)
A function is said to have LHE [1] as α if the following condition is satisfied Where ε is a small positive number. The property LHE is a local property. The global LHE in where ε is small positive number, and is real constant. This function 0 C > ( ) f x has Holder exponent as1.
Consider the function x y ε < − < is a function with Holder exponent . In a way it states that the continuous function in consideration is one-whole differentiable and the value of differentiation is bounded, that is

b) Holder Continuity
A continuous function ( ) f x which is non-differentiable in classical sense is said to holder , [9]. The Holder and Hurst exponents are equivalent for unifractal graphs that has a constant fractional dimension in defined interval [1], [9].

d) Weierstrass function
In 1872 K. Weierstrass [23][24][25] proposed his famous example of an always continuous but nowhere differentiable function on the real line with two parameters in the following form Where is odd-integer number. He proved that this function is continuous for all real number and is non-differentiable for all real values of x if

The fractional Weierstrass Function
The original Weierstrass Function (4) is defined in the following form  We only are stating some lemmas which will be used to characterize the fractional Weierstrass function and its fractional derivative.