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Fractional Weierstrass Function by Application of Jumarie Fractional

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Fractional Weierstrass Function by Application of Jumarie Fractional Fractional Weierstrass function by application of Jumarie fractional trigonometric functions and its analysis Uttam Ghosh1 , Susmita Sarkar2 and Shantanu Das 3 1 Department of Mathematics, Nabadwip Vidyasagar College, Nabadwip, Nadia, West Bengal, India; email : [email protected] 2Department of Applied Mathematics, University of Calcutta, Kolkata, India email : [email protected] 3Scientist H+, Reactor Control Systems Design Section E & I Group BARC Mumbai India Senior Research Professor, Dept. of Physics, Jadavpur University Kolkata Adjunct Professor. DIAT-Pune UGC Visiting Fellow. Dept of Appl. Mathematics; Univ. of Calcutta email : [email protected] 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. Keywords Holder exponent, fractional Weierstrass function, Box dimension, Jumarie fractional derivative, Jumarie fractional trigonometric function. 1.0 Introduction Fractional geometry, fractional dimension is an important branch of science to study the irregularity of a function, graph or signals [1-3]. 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 [4-11]. Many authors are trying to relate between the fractional derivative and fractional dimension [1, 12-15]. 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 [12-16] 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 1 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. 2.0 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 f ()x in the interval a to x , with fx()= 0for x < a . x ⎧ 1 −−α 1 ⎪ ()xfd−<τττα (), 0. Γ−()α ∫ ⎪ a ⎪ 1 d x J Dfxαα()=−−(xττ )− f () fad () τ , 0< α< 1 ax[]⎨ ∫ [] ⎪Γ−(1α ) dx a ()m ⎪ ()α −m ⎪()fx() , m≤<+α m1. ⎩⎪ 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 1 of offset function that is f ()xfa− (). Forα >1, we use the third line; that is first we differentiate the offset function with order 0(<−α m )1<, by the formula of second line, and then apply whole m order differentiation to it. Here we chose integer m , just less than the real J α numberα ; that is m≤<α m+1. In this paper we use symbol 0 Dx to denote Jumarie fractional derivative operator, as defined above. In case the start point value f (a) is un- defined there we take finite part of the offset function as f ()xf− (a+ ) ; for calculations. Note α in the above Jumarie definition 0 DCx [ ] = 0 , where C is constant function, otherwise in RL α x−α sense, the fractional derivative of a constant function is 0 DCx [ ] = CΓ−(1 α ) , that is a decaying 2 power-law function. Also we purposely state that fx( )= 0 for x < 0 in order to have initialization function in case of fractional differ-integration to be zero, else results are difficult [9]. b) Mittag-Leffler Function The Mittag-Leffler function [19- 22] of one parameter is denoted by Eα (x) and defined by ∞ xk Exα ()= ∑ (1) k =0 Γ+(1 kα) This function plays a crucial role in classical calculus forα =1, for α =1it becomes the exponential function, that is exp(x )= Ex1 ( ) ∞ xk exp(x ) = ∑ (2) k =0 k! α Like the exponential function; Eα ()x play important role in fractional calculus. The function α α Eα ()x is a fundamental solution of the Jumarie type fractional differential equation 0 Dyy [ ] = y, α where 0 Dx is Jumarie derivative operator as described above. The other important function is the two parameter Mittag-Leffler function is denoted Eαβ, ()x and defined by following series ∞ xk Exαβ, ()= ∑ (3) k =0 Γ+()βαk The functions (1) and (3) play important role in fractional calculus, also we note that ExExαα,1 ()= (). Again from Jumarie definition of fractional derivative we have J Dα 10and J Dxα ⎡⎤β= Γ+(1β ) xβ−α. We now consider the Mittag-Leffler function in the 0 x [ ] = 0 x ⎣⎦Γ+−(1βα ) α following form in infinite series representation for f ()xEx= α ( )for x ≥ 0 and fx( )= 0 for x < 0 as; xxααα23x Ex(α )=+ 1 + + +... α Γ+(1)(12ααα Γ+ )(13Γ+ ) Then taking Jumarie fractional derivative of order 0< α <1term by term for the above series we obtain the following by using the formula J Dxα ⎡⎤ββ= Γ+(1β ) x−αand J Dα 10= 0 x ⎣⎦Γ+−(1βα ) 0 x [ ] ⎡ axααα a22 x a33 x ⎤ JJααα⎡⎤ 00DEaxxx⎣⎦α ( )=+ D⎢ 1 + + +...⎥ ⎣ Γ+(1αα ) Γ+ (1 2 )Γ+ (1 3α ) ⎦ ax23ααax2 ax43α =++0 a + + +... Γ+(1)(12)(13)ααα Γ+ Γ+ α = aEα () ax 3 Again we derive Jumarie derivative of order β for one parameter Mittag-Leffler function α Exα ()and thereby get two parameter Mittag-Leffler function. Here also we use for term by term Jumarie derivative J Dxα ⎡⎤υυ= Γ+(1υ ) x−αand J Dα 10= . 0 x ⎣⎦Γ+−(1υα ) 0 x [ ] ⎡ xxααα23 x⎤ JJβα⎡⎤ β 00DExxx⎣⎦α ( )=+ D⎢ 1 + + +...⎥ ⎣ Γ+(1)(12)(13)ααα Γ+ Γ+ ⎦ xxαβ−−−23αβ xαβ =+0 + + +... Γ+(1)(12)(13)αβ − Γ+ αβ − Γ+ αβ − αβ− α = xEαα,1−+ β () x Jumarie [18] defined the fractional sine and cosine function in the following form def ααα Eixααα()cos()sin()=+ x i x def ∞ 2kα α k x cosα (x )=−∑ ( 1) k=0 Γ+(1 2αk ) def ∞ (2k+ 1)α α k x sinα (x )=−∑ ( 1) k=0 Γ++()1(12)k α 0.7 0.7 1.2 1 0.6 0.6 0.8 0.5 0.5 0.6 0.4 0.4 (x) in S (x) (x) in in 0.4 S 0.3 S 0.3 0.2 0.2 0.2 0 0.1 0.1 -0.2 0 1 2 3 4 5 6 7 8 9 10 x 0 0 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x x For α = 0.8 For α = 0.4 For α = 0.6 1 8 40 0.8 35 6 0.6 30 25 0.4 4 20 0.2 (x) (x) ) 2 15 in in (x S 0 S in S 10 -0.2 0 5 -0.4 0 -2 -0.6 -5 -0.8 -4 -10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 -1 x 0 1 2 3 4 5 6 7 8 9 10 x x For For α = 1.0 α = 1.2 For α = 1.4 α Fig.1 Graph ofsinα ()x 1 1.2 1 0.9 1 0.8 0.7 0.8 0.5 0.6 0.6 s(x) 0.5 s(x) os(x) Co Co 0.4 C 0.4 0 0.3 0.2 0.2 0 0.1 0 -0.2 -0.5 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x x x For α = 0.4 For α = 0.6 For α = 0.8 1 6 20 0.8 4 10 0.6 2 0 0.4 0 -10 0.2 s(x) (x) -2 -20 0 Co s(x) Cos -0.2 Co -4 -30 -0.4 -6 -40 -0.6 -8 -50 -0.8 -10 -60 -1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 x x -12 0 1 2 3 4 5 6 7 8 9 10 x For For α = 1.4 α = 1.0 For α = 1.2 α Fig.2 Graph of cosα (x ) 4 From figure-1 and 2 it is observed that for α <1both the fractional trigonometric functions α α sinα (x ) and cosα (x ) is decaying functions like damped oscillatory motion.
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