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

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1 Advances in Pure Mahemaics, 25, 5, Published Online Ocober 25 in SciRes. hp:// hp://d.doi.org/.4236/apm Fracional Weiersrass Funcion by Applicaion of umarie Fracional Trigonomeric Funcions and Is Analysis Uam Ghosh, Susmia Sarar 2, Shananu Das 3 Deparmen of Mahemaics, Nabadwip Vidyasagar College, Nabadwip, India 2 Deparmen of Applied Mahemaics, Universiy of Calcua, Kolaa, India 3 Reacor Conrol Sysems Design Secion, E & I Group, BARC, Mumbai, India uam_mah@yahoo.co.in, susmia62@yahoo.co.in, shananu@barc.gov.in Received 3 Augus 25; acceped Ocober 25; published 3 Ocober 25 Copyrigh 25 by auhors and Scienific Research Publishing Inc. This wor is licensed under he Creaive Commons Aribuion Inernaional License (CC BY. hp://creaivecommons.org/licenses/by/4./ Absrac The classical eample of no-where differeniable bu everywhere coninuous funcion is Weiersrass funcion. In his paper we have defined fracional order Weiersrass funcion in erms of umarie fracional rigonomeric funcions. The Hölder eponen and Bo dimension of his new funcion have been evaluaed here. I has been esablished ha he values of Hölder eponen and Bo dimension of his fracional order Weiersrass funcion are he same as in he original Weiersrass funcion. This new developmen in generalizing he classical Weiersrass funcion by use of fracional rigonomeric funcion analysis and fracional derivaive of fracional Weiersrass funcion by umarie fracional derivaive, esablishes ha roughness indices are invarian o his generalizaion. Keywords Hölder Eponen, Fracional Weiersrass Funcion, Bo Dimension, umarie Fracional Derivaive, umarie Fracional Trigonomeric Funcion. Inroducion The conceps of fracional geomery, fracional dimensions are imporan branches of science o sudy he irregulariy of a funcion, graph or signals []-[3]. On he oher hand fracional calculus is anoher developing mahemaical ool o sudy he coninuous bu non-differeniable funcions (signals where he convenional calculus fails [4]-[]. Many auhors are rying o relae he fracional derivaive and fracional dimension [] How o cie his paper: Ghosh, U., Sarar, S. and Das, S. (25 Fracional Weiersrass Funcion by Applicaion of umarie Fracional Trigonomeric Funcions and Is Analysis. Advances in Pure Mahemaics, 5, hp://d.doi.org/.4236/apm

2 [2]-[5]. The funcions which are coninuous bu non-differeniable in ineger order calculus can be characerized in erms of fracional calculus and especially hrough Holder eponen [] [6]. To sudy he no-where differeniable funcions auhors in [2]-[6] used differen ypes of fracional derivaives. umarie [7] defined he fracional rigonomeric funcions in erms of Miag-Leffler funcion and esablished differen useful fracional rigonomeric formulas. The fracional order derivaives of hose funcions were esablished in-erms of he umarie [7] [8] modified fracional order derivaives. In his paper we have defined he fracional order Weiersrass funcions in erms of he fracional order sine funcion. The Hölder eponen and bo-dimension (fracional dimension of graph of his funcion have been obained here. The fracional order derivaive of his funcion has also esablished here. This is a new developmen in generalizing he classical Weiersrass funcion by usage of fracional rigonomeric funcions including he sudy of is characer. The paper is organized as: Secion 2 deals wih descripion of umarie fracional derivaive, Miag-Leffler funcion of one and wo parameer ypes; fracional rigonomeric funcion of one and wo parameer ypes and derivaion of umarie fracional derivaives of hose funcions. In his secion we also have derived some useful relaions of fracional rigonomeric funcions which shall be used for our furher calculaions in characerizing fracional Weiersrass funcion. We have coninued his secion by inroducing Lipschiz Hölder eponen (LHE is definiion, is relaion o Hurs eponen and fracional dimension and also definiion of Hölder coninuiy. The classical Weiersrass funcion has also been defined here. These Lipschiz Hölder eponen, Hurs eponen, and fracional dimension are basic parameers o indicae roughness inde of a funcion or a graph. In Secion 3 we have described he fracional Weiersrass funcion by generalizing he classical Weiersrass funcion by use of fracional sine rigonomeric funcion. Subsequenly we apply derived ideniies of fracional rigonomeric funcions o evaluae he properies of his new fracional Weiersrass funcion. In Secion 4 we have done derivaion of properies of fracional derivaives of fracional Weiersrass funcion, and concluded he paper wih conclusion and references. 2. umarie Fracional Order Derivaive and Miag-Leffler Funcion a Fracional Order Derivaive of umarie Type umarie [7] defined he fracional order derivaive by modifying he Lef Riemann-Liouvellie (RL fracional f in he inerval a o, wih f ( for < a. derivaive in he following form for he funcion a ( τ f ( τ d τ, <. Γ ( a d D f ( τ f ( τ f ( a d τ, ( d < < Γ ( a ( ( m m f, m < m+. In he above definiion, he firs epression is jus he Riemann-Liouvelli fracional inegraion; he second line is Riemann-Liouvelli fracional derivaive of order < < of offse funcion ha is f f ( a. For >, we use he hird line; ha is firs we differeniae he offse funcion wih order < ( m <, by he formula of second line, and hen apply whole m order differeniaion o i. Here we chose ineger m, jus less han he real number ; ha is m < m+. In his paper we use symbol D o denoe umarie fracional derivaive operaor, as defined above. In case he sar poin value f ( a is un-defined, here we ae finie par of he offse funcion as f f ( a + ; for calculaions. Noe in he above umarie definiion D [ C], where C is consan funcion, oherwise in RL sense, he fracional derivaive of a consan funcion is D [ C] C, ha is a decaying power-law funcion. Also we purposely sae ha f ( for Γ( < in order o have iniializaion funcion in case of fracional differ-inegraion o be zero, else resuls are difficul [9]. b Miag-Leffler Funcion and Is umarie Type Fracional Derivaive: One and Two Parameer Type One Parameer Miag-Leffler Funcion 78

3 The Miag-Leffler funcion [9]-[22] of one parameer is denoed by E E ( and defined by (2 Γ + This funcion plays a crucial role in classical calculus for, for i becomes he eponenial func- E ep ion, ha is ep (3! We now consider he Miag-Leffler funcion in he following form in infinie series represenaion for f E for and E f for < as; ( (4 Γ + Γ + Γ + ( ( 2 ( 3 Then aing umarie fracional derivaive of order < < erm by erm for he above series we obain he β Γ ( + β β following by using he formula D and D [ ] Γ + β a a a E ( a D Γ ( + Γ ( + 2 Γ ( a a a + a (5 Γ + Γ + Γ + ( ( 2 ( 3 E ( a ae ( a Lie he eponenial funcion; E ( play imporan role in fracional calculus. The funcion E ( a fundamenal soluion of he umarie ype fracional differenial equaion [ ] is D y y, where D is umarie derivaive operaor as described above. umarie in [8] esablished E i + y E i E iy. We reproduce he Proof of he above relaion. Le us consider a funcion f ( which saisfies he condiion f ( ( f y f ( + y. Differeniaing boh side wih respec o and y of -order respecively we ge he following. Firs consider y a consan, and we fracionally differeniae w.r.. by umarie derivaive f ( f ( y f ( ( y ( y + + D [ + y] f ( f ( f ( ( + y f ( ( + y Γ ( + Γ ( + ( + y f ( ( f ( y f ( + y ( D + y Γ + Γ ( + Now we consider as consan and do he following seps y y y [ ] ( f Dy f y Dy y Dy f + y Dy + y Dy + y f y f y D f + y f + y [ ] Γ + y Γ + + y f ( f ( y ( f ( ( + y ( Dy + y Γ + Γ + [ ] 79

4 Here we pu equivalence of Dy [ y] Dy [ y] Du [ u C] are aen as consan he funcion form of hese wo quaniies ges equivalen ha is equivalen o D [ u] + + +, wih C as consan; ha is when or y umarie fracional derivaive of consan is zero. Therefore he RHS of above wo epressions are equal, from ha we ge he following f ( f ( y f ( f ( y f ( f ( y f ( f ( y The above wo may be equaed o a consan say. Then we have f ( f ( f ( f ( Leffler funcion we now ha E ( a ae ( a ; we imply ha he soluion of f ( f ( is f ( E (. Therefore E ( saisfies he condiion f ( f ( + y, or E ( E ( y E ( + y. Considering i can wrie he following ideniy u as, or we wrie. From he propery of Miag-Leffler funcion and umarie derivaive of he Miag- ( (( + Using definiion E ( i cos ( i sin ( E i y E ( i E ( i ( y. + we epand he above as depiced below ( + y + i ( + y ( + i ( ( y + i ( y cos ( cos ( y sin ( y sin ( + isin ( cos ( y + sin ( y cos (. cos sin cos sin cos sin Comparing real and imaginary par in above derived relaion we ge he following ( + y ( ( y + ( y ( ( + y ( ( y ( y ( sin sin cos sin cos cos cos cos sin sin, we herefore This is very useful relaion as in conjugaion wih classical rigonomeric funcions, and we will be using hese relaions in our analysis of fracional Weiersrass funcion and is fracional derivaive. 2 Two Parameer Miag-Leffler Funcion The oher imporan funcion is he wo parameer Miag-Leffler funcion denoed by Eβ, and defined by, E β, (6 Γ + ( β The funcions (2 and (6 play imporan role in fracional calculus, also we noe ha E, E Again from umarie definiion of fracional derivaive we have D β Γ ( + β β [ ] and. Γ ( + β Again we derive umarie derivaive of order β for one parameer Miag-Leffler funcion E (. and hereby ge wo parameer Miag-Leffler funcion. For finding erm by erm umarie derivaive we use υ Γ ( + υ υ and D [ ]. Γ + υ 72

5 where E, ( β 2 3 E ( D Γ ( + Γ ( + 2 Γ ( + 3 β β β 2 β 3 β E Γ + Γ + Γ + ( β ( 2 β ( 3 β + is wo parameer Miag-Leffler funcion. β, β+ c umarie Definiion of Fracional Sine and Cosine Funcion and Their Fracional Derivaive: Boh One Parameer and Two Parameer Type One Parameer Sine and Cosine Funcion umarie [8] defined he one parameer fracional sine and cosine funcion in he following form, ( def cos ( sin ( E i + i (8a def 2 cos ( ( (8b Γ + def ( ( sin ( 2 ( 2 + (8c Γ ( From Figure and Figure 2 i is observed ha for < boh he fracional rigonomeric funcions sin ( and cos ( is decaying funcions lie damped oscillaory moion. For i is lie simple harmonic moion wih susained oscillaions; and for > i grows while i oscillaes infiniely; lie unsable oscillaor. f cos for and f ( for < is following The series represenaion of Taing erm by erm umarie derivaive we ge, a a a cos ( a + + Γ + Γ + Γ + sin ( 2 ( 4 ( Γ Γ Γ + 6 cos ( a a + a a + Γ + 2 Γ + Γ + 4 Γ + 3 Γ + 6 Γ a a a + (9 Γ ( + Γ ( + 3 cos a a a The series presenaion of f ( sin ( sin ( a, for Taing erm by erm umarie derivaive we ge wih f for < is a a a a + + Γ + Γ + Γ + Γ + cos ( ( 3 ( 5 ( a Γ + Γ + 3 sin ( a a a Γ + Γ + Γ + 3 Γ + 3 sin a a Γ Γ a + Γ + 5 Γ + 5 Γ + 7 Γ a a a a + + Γ ( + 2 Γ ( + 4 Γ ( + 6 (7 ( 72

6 Sin( (a.6.5 Sin( (b.8 Sin( (c 722

7 Sin( (d 6 4 Sin( (e Sin( (f Figure. Graph of sin (. ; (f For.2 ; (g For.4.. (a For.4 ; (b For.6 ; (c For.8 ; (d For 723

8 Cos( (a.8 Cos( (b.5 Cos( (c 724

9 Cos( (d Cos( (e - Cos( Figure 2. Graph of cos ( (f. (a For.4 ; (b For.6 ; (c For.8 ; (d For. ; (e For.2 ; (f For

10 Thus we ge cos a a sin 2 Two Parameer Sine and Cosine Funcion and ( a a ( sin cos. Le us define he wo parameer sine and cosine funcions cos, ( β and sin, ( def cosβ, ( ( + Γ + Γ Γ + Γ + ( β 2 ( β ( 2 β ( 4 β ( ( β ( 2 Γ ( + β Γ ( 3 + β Γ ( 5 + β β as depiced below: def sinβ, ( ( + + Γ + + Now wih his and wih definiion of wo parameer Miag-Leffler funcion (3 wih imaginary argumen we ge he following useful ideniy E β, ( i ( ( isinβ( ( i 2 3 i i i Γ β + Γ β Γ + β Γ 2 + β Γ 3 + β i ( β ( 2 β ( 4 β Γ Γ + Γ + Γ ( + β Γ ( 3 + β Γ ( 5 + β cos + β,, β Now for β >, we do he umarie derivaive of order on he funcion f cosβ, ( as depiced in following seps, wih formula β β β, Γ ( + υ ( υ Γ + β υ υ β, ( D. and [ ] 2 4 cos ( D ( + Γ( β Γ ( 2 + β Γ ( 4 + β Thus we ge a very useful relaion Similarly i can be shown ha β β+ β+ 3 + Γ( β Γ β + Γ β β+ 3 β ( + Γ( β Γ( β + 2 Γ( β + 4 cos β cos cos. β β β,, β sin sin. β β β,, Now we calculae he umarie ype fracional order derivaive of ep E υ Γ ( + υ υ and D [ ]. Γ ( + υ by using he formula a a a ep ( a E ( a D Γ( 2 Γ( 3 Γ( a a a ae Γ Γ Γ ( 2 ( 3 ( 4 lie we did for E (,2 ( a 726

11 On he oher hand he umarie ype fracional order derivaive of cos( a is following, as we did for ( υ Γ ( + υ υ by using he formula and D [ ]. Γ ( + υ cos We obain a a a cos ( a D Γ( 3 Γ( 5 Γ( a a a + + Γ Γ Γ a ( 3 ( 5 ( 7 sin ( a,2 cos a a sin a.,2 Similarly he umarie ype fracional order derivaive of sin ( is ( a a ( a sin cos.,2 2.. Definiion of Some Useful Roughness Indices a Lipschiz Hölder Eponen (LHE A funcion is said o have LHE [] i saisfies he following condiion f f y y < y < ε where ε is a small posiive number. The propery LHE defined above corresponds o local propery. The glob- ab, is denoed by and is defined by al LHE in inerval [ ] unless inf [ ab, ] f is a consan funcion,. The Lipschiz Holder eponen is someimes named as Holder eponen. For he coninuous funcion f : R R, f ( saisfies he Lipschiz condiion on is domain of definiion if f f( y < C y when < y < ε, where ε is small posiive number, and C > is real consan. This funcion f ( has Holder eponen as uniy. Consider he funcion: f : f sin f f y sin sin y < C y when < y < ε such ha hen is a funcion wih Holder eponen. In a way i saes ha he coninuous funcion in consideraion is one-whole differeniable and he value of differeniaion is bounded, ha is f f y < C for y < y < ε. b Holder Coninuiy A coninuous funcion f eponen if which is non-differeniable in classical sense is said o holder coninuous wih f f y < C y < y < ε where C > is a real consan and ε >. c Fracional Dimension Fracional dimension (d or bo dimension [] of a funcion or graph is local propery, denoes he degree of roughness of a funcion or graph. Le he graph of a funcion is f ( for [ ab, ] can be covered by lim r he fracional dimension of he graph is defined as, N-squares of size r hen wih d log ( N ( r lim r log 727

12 Again if H be he Hurs eponen hen he relaion beween he above Holder eponens are H d 2 H 2 [] [9]. The Holder and Hurs eponens are equivalen for uni-fracal graphs ha has a consan fracional dimension in defined inerval [] [9]. 3. The Fracional Weiersrass Funcion In 872 K. Weiersrass [23]-[25] proposed his famous eample of an everywhere coninuous bu no-where dif- W on he real line wih wo parameers b a > in he following form fereniable funcion sin W a b where b is odd-ineger. He proved ha his funcion is coninuous for all R and is non-differeniable for all 3π real values of provided ab > +. Considering b a consan say b a consan and assuming, and 2 log a s 2 anoher presenaion of he Weiersrass funcion [3] can be obained which is log b W sin > < s < 2 ( ( s 2 In reference [3] Falconer esablished he fracional dimension of Weiersrass funcion defined in ( is s and he corresponding Holder eponen is 2 s. We define he fracional Weiersrass Funcion in erms of umarie [28] fracional sine funcion, ha is sin in he following form for ( s 2 W sin > < s < 2 (2 i reduces he original Weiersrass Funcion, and a condiion ha W ( where, < <, and for for <. We only are saing some lemmas which will be used o characerize he fracional Weiersrass funcion and is fracional derivaive. Lemma : Le f be funcion coninuous in inerval [ ] Suppose f f( y C y < y< s, and s [2]-[4]. hen he dimension [2]-[4] of he graph f is d 2 s. 2 Suppose δ >. For every [,], and δ δ s f f ( y Cδ < < here eiss [,] hen he dimension [2]-[4] of he graph f is d 2 s. Theorem : The Holder eponen of fracional Weiersrass funcion W ( y such ha y < δ and wih < < is 2 s and,. consequenly he Hausdorff dimension or fracional dimension is s over any finie inerval suppose i is [ ] Proof: We calculae W ( + h W sin ( a ( + y sin ( a cos ( ay + cos ( a sin ( ay in following seps where we have used our derived epression ( s 2 s 2 ( + sin ( + sin W h W h ( s 2 ( s 2 sin cos cos sin sin ( s 2 sin ( ( cos ( h cos ( sin ( h ( h + h + 728

13 From he series epansion of sin ( and cos ( is clear ha for small, sin ( and cos ( also boh sin ( cos ( is less han or equal o. Therefore, wih above observaion ha is for small h, sin ( h cos ( h and for large h, cos ( h we wrie he following W + h W and also from he Figure and Figure 2, i ( s 2 sin ( cos ( h cos ( sin ( h + ( s 2 min ( h, and h, ( m+ m Choose < h < hen one can find posiive ineger m such ha h hen divide he summa- ( s 2 ion ha is min ( h, ino wo pars. Firs par for o m hen sin ( and for oher values of maimum value of he epression in hird brace is equal o. We use he geomeric series formulas m m+ m a a a a and a a,for in he following derivaion. m+ a Wih ( m+ m, ha is h W + h W h + m s s ( 2 ( 2 m+ m ( s 2 h + m+ m ( s 2( m+ ( s 2 h + ( m+ ( s 2( m+ h + ( s 2 ( m+ m we ge he following h ( s 2 h h ( + + ( s 2 W h W h + h Ch ( s 2 2 s 2 s where he consan C +. From definiion of Holderian funcion and he above discus- ( s 2 sion i is clear ha fracional Weiersrass funcion is also Holder coninuous wih Holder eponen ( 2 s, a fracional number. This shows (by Lemma- ha Hausdorff dimension of graph of fracional Weiersrass funcion is 2 ( 2 s s. Thus he Hausdorff dimension of fracional Weiersrass funcion and original Weiersrass funcion is same, is independen of fracional eponen as defined in (. 4. The umarie Fracional Derivaive of Fracional Weiersrass Funcion Many auhors found he fracional derivaive of he coninuous bu nowhere differeniable funcion ha is Weiersrass Funcion []-[7] using differen ype definiions of fracional derivaives. Here we consider uma- W is of order rie ype fracional order derivaive of ( ( sin cos. s 2 s 2 D W D 729

14 We used in above derivaion he ideniy sin ( a a cos ( a we obain he following, (. Therefore from above derivaion W cos. (5 Since if will be ( 2 bounded funcion if s + ( s 2 is convergen. Since + is a geomeric series will be conver gen if s 2+ < implying < 2 s. Hence he fracional derivaive of order wih < < of he Weiersrass Funcion will eiss when < 2 s. Again if > hen cos ( and sin ( for, 2, 3, are unbounded funcions (Figure and Figure 2 and will grow by oscillaing wihou bound o ± for. Since < s < 2 and > ( s 2 implying s + 2> herefore + is a divergen series. Therefore < < hen cos ( is a bounded funcion and herefore W ( cos ( W is a divergen series for >. We wrie following observaion This shows ha -order ( Bounded for < 2 s W ( Unbounded for 2 s < < umarie fracional derivaive of he fracional Weiersrass funcion eiss when < 2 s and for 2 s i does no eis. Thus we can sae a heorem in he following form < < umarie fracional derivaive of he fracional Weiersrass funcion Theorem 2: -order ( s 2 W sin > < s < 2 eiss when < 2 s and for 2 s i does no eis. Theorem 3: The Holder eponen of -order fracional derivaive of fracional Weiersrass funcion W (, < < is 2 s and consequenly he Hausdorff dimension or fracional dimension is s + over any finie inerval [, ]. Proof: Le ( cos ( W W denoes -order fracional umarie derivaive of fracional Weiersrass funcion. Then using he ideniy ( a ( + y ( a ( ay ( a ( ay cos cos cos sin sin we ge he following ( s 2 s ( + cos ( + cos W h W h cos ( cos ( sin ( sin ( cos cos ( ( cos ( h sin ( sin ( h ( h h From he series epansion of sin ( and cos ( is clear ha for small, sin ( and cos ( also boh sin ( cos ( sin ( h h, and also from he Figure and Figure 2 i and is less han or equal o. Therefore, wih above observaion ha is for small h, 73

15 ( h and for large h, ( h cos cos we wrie he following U. Ghosh e al. s 2+ ( + cos cos + sin sin W h W h h min ( h, Choose < h < hen one can find posiive ineger m such ha derivaion for W ( Wih where C we do he following seps m ( + + ( m+ W h W h ( m+ m, ha is h m 2 h + ( m+ ( m+ m h hen as per our earlier 2 m ( m+ 2 2 h + 2( m+ ( m+ h + 2 ( m+ m we ge he following h 2( m+ ( m+ ( W h W h 2 ( s ( s h h h + + Ch s 2 h 2 s +. From definiion of Holderian funcion and above discussion i is clear < < fracional derivaive of fracional Weiersrass funcion is also Holder coninuous ha -order wih Holder eponen 2 s. This shows ha Hausdorff dimension of graph of fracional Weiersrass funcion is 2 ( 2 s s+ (by lemma-. The graph dimension increased by fracional order for fracional derivaive of Weiersrass funcion by amoun of fracional derivaive-he graph becomes rougher. 5. Conclusion The fracional Weiersrass funcion is a coninuous funcion for all real values of he argumens, and is bo dimension and Holder eponen are independen of fracional order ha incorporaes o he fracional Weiersrass funcions. Again he Bo dimension of fracional derivaive of he fracional Weiersrass increases wih increase of order of fracional derivaive. This invarian naure of he roughness inde of fracional Weiersrass funcion when generalized wih fracional rigonomeric funcion is remarable. The oher embodimen in similar lines as in his paper o ge differen fracional Weiersrass funcion is under developmen. Acnowledgemens Acnowledgmens are o Board of Research in Nuclear Science (BRNS, Deparmen of Aomic Energy Governmen of India for financial assisance received hrough BRNS research projec no. 37(3/4/46/24-BRNS wih BSC BRNS, ile Characerizaion of unreachable (Holderian funcions via Local Fracional Derivaive 73

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