A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F F ER EN TIAL IN THR EE- DI M ENSIONAL S PAC E Ξ

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1 Appled Mathematcs and Mechancs ( Englsh Edton, Vol 24, No 3, Mar 2003) Publshed by Shangha Unversty, Shangha, Chna Artcle ID : (2003) A P PL I CA TIONS OF FRACTIONAL EXTERIOR DI F F ER EN TIAL IN THR EE- DI M ENSIONAL S PAC E Ξ CHEN Yong ( ), YAN Zhen- ya ( ), ZHANG Hong- qng ( ) ( Department of Appled Mathematcs, Dalan Unversty of Technology, Dalan , P. R. Chna) ( Contrbuted by ZHANG Hong- qng) Abst ract : A bref survey of fractonal calculus and fractonal dfferental forms was frstly gven. The fractonal exteror transton to curvlnear coordnate at the orgn were dscussed and the two coordnate transformatons for the fractonal dfferentals for three- dmensonal Cartesan coordnates to sphercal and cylndrcal coordnates are obtaned, respectvely. In partcular, for v = m = 1, the usual exteror transformatons, between the sphercal coordnate and Cartesan coordnate, as well as the cylndrcal coordnate and Cartesan coordnate, are found respectvely, from fractonal exteror transformaton. Key wor ds : fractonal dfferental form ; Cartesan coordnate ; sphercal coordnate ; cylndrcal coordnate Ch nese L brary Classfcaton : O175 Docume nt code : A 2000 M R S ubject Classfcaton : 26A33 ; 53C26 I nt rod ucton In generalzed ntegraton and dfferentaton the queston of extenson of meanng s : can the meanng of dervatves of ntegral order d n y/ d x n be extended to have meanng where n s any number (e. g., rratonal, fracton or complex)? In 1695 Lebnz nvented above notaton. Eular and Fourer mentoned dervatves of arbtrary order but they gave no applcatons or examples. So the honor of makng the frst applcaton belongs to Abel n Abel appled the fractonal calculus n the soluton of an ntegral equaton whch arses n the formulaton of the Tautochrone problem. Abel s soluton was so elegant that t attempted the attenton of Louvlle who made the frst major attempt to gve a logcal defnton of a fractonal dervatve n Remann n 1847 whle a student wrote a paper publshed posthumously n whch he gave a defnton of a fractonal operaton. A defnton named n honor of Remann and Louvlle s d x = 6 n =1 (5 ( x - a ) ) v. Ξ Receve d dat e : ; Revse d dat e : Foundaton t e ms : the Natonal Natural Scence Foundaton of Chna ( ) ; the Natonal Key Basc Research Development Project Program of Chna ( G ) Bograp hy : CHEN Yong ( ), Doctor ( E- mal : edu. cn) Tsnghua Tongfang Optcal Dsc Co., Ltd. All rghts reserved.

2 Applcatons of Fractonal Exteror Dfferental 257 By now scentsts and appled mathematcans have found the fractonal calculus useful n varous felds : rheology, quanttatve bology, electrochemstry, scatterng theory, dffuson, transport theory, probablty potental theory and elastcty [1]. In recent years exteror calculus has been generalzed by basng t on varous graded algebras [2,3 ]. Other attempts at generalzaton are based on nonassocatve geometres [4,5 ]. Recently, Cottrll-Shepherd and Naber gave the defnton of a fractonal exteror dervatve [6 ] and found that fractonal dfferental formal space generates new vector spaces of fnte and nfnte dmenson, the defnton of closed and exact forms are extended to the new fractonal form spaces wth closure and ntegrablty condton worked out for a specal case. Coordnate transformaton rules are also computed. 1 Tra nston to Curvl nea r Coor d nates a n d Two I mp orta nt Exa mp les In the paper the Rmmann-Louvlle defnton of fractonal ntegraton and dfferentaton wll be used. ( q) s the gamma functon ( generalzed factoral) of the parameter q [. e., ( n + 1) = n! for all whole number, n ], 5 q f ( x) (5 ( x - a) ) q = 1 ( - 5 q f ( x) (5 ( x - a) ) q = 5 n 5 x n 1 ( n - q) x f ( ) d a ( x - q) x a ) q+1 (Re ( q) < 0), (1) f ( ) d Re ( q) Ε 0 ( x - ) q- n +1 ( n > q ( n s whole) ). (2) The parameter q s the order of the ntegral or dervatve and allowed to be complex. Postve real values of q represent dervatves and negatve real values represent ntegrals. Eq. ( 1) s a fractonal ntegral and Eq. ( 2) s a fractonal dervatve. In ths paper, only real and postve values of q wll be consdered. If the partal dervatve are allowed to assume fractonal orders, a fractonal exteror dervatve can be defned (5 ( x - a ) ) v. (3) =1 Note that the subscrpt denotes the coordnate number, the superscrpt v denotes the order of the fractonal coordnate dfferental, and a s the ntal pont of the dervatve. For convenence, the ntal pont a for the fractonal dervatve s taken to the orgn. Let { x } and { y } be two coordnate systems wth a one to one mappng between them n some neghborhood of p E n. Take { x } agan to be Cartesan coordnates and { y } to be curvlnear coordnates. Assume the { x } can be wrtten smoothly n terms of the { y }, x = x ( y). (4) The exteror dervatve s then appled to Eq. (4) gvng the followng : d x = 6 n =1 d y l 5 x 5 y l. (5) In the two coordnate systems, the fractonal exteror dervatve d v takes the followng forms : =1 5 x v (6) and whch gves rse to =1 d y 5 y v (7) Tsnghua Tongfang Optcal Dsc Co., Ltd. All rghts reserved.

3 258 CHEN Yong, YAN Zhen-ya and ZHANG Hong-qng 6 n =1 d x 5 x v = 6 n =1 d y 5 y v. (8) Consder a functon f k that maps ponts n E n nto the complex numbers It s easly seen that and (1) f k = ( 7 n =1/ k f k (5 x ) v = 0 ( k,. e., f k Ker x ) v - m x v k. (9) (5 x ) v ) (10) f k (5 x ) v = 1 ( = k). (11) Applyng the fractonal exteror dervatve ( 8) to both sdes of ( 9) n two dfferent coordnate systems the followng coordnate transformaton rule can be obtaned d x v k = 6 n =1 d y v (5 y ) v ( ( 7 n j =1/ j k x j ( y) ) v - m x k ( y) v ). (12) In what follows we consder the coordnate transformaton for three- dmensonal Cartesan to cylndrcal coordnates and sphercal coordnates. Exa mple 1 Sphercal coordnate Consder the coordnate transformaton of sphercal coordnate = rsn( ) cos( <) x 1 x 2 x 3 = rsn( ) ( <) = rcos( ) ( r 0, 0, 0 < ). (13) The coordnate transformatons for the fractonal dfferentals are then d x v 1 = d x v 2 = d x v 3 = (2 v - 2 m + 1) cos v - m ( ) cos v ( <) sn m - 2 ) sn m - v ( <) r2 v - 2 m d r cos v - m ( ) cos v ( <) ) v sn m - 2 d v ) sn m - ( <) cos v - m ( (5 <) v sn m - 2 (2 v - 2 m + 1) ) cos v ( <) ) sn m - v ( <) d < v, (14) cos v - m ( ) cos v - m ( <) sn v ( <) sn m - 2 v r 2 v - 2 m d r ( ) cos v - m ( ) cos v - m ( <) sn v ( <) ) v sn m - 2 d ) cos v - m ( (5 <) v (2 v - 2 m + 1) ) cos v - m ( <) sn v ( <) sn m - 2 ) d < v, (15) cos v - m ( <) cos ) sn m - v ( <) sn 2 m - 2 ) r2 v - 2 m d r cos v - m ( <) cos ) ) v sn m - v ( <) sn 2 m - 2 d ) cos v - m ( <) cos ) (5 <) v ) sn m - v ( <) sn 2 m - 2 v d < ( v. (16) Tsnghua Tongfang Optcal Dsc Co., Ltd. All rghts reserved.

4 Applcatons of Fractonal Exteror Dfferental 259 that s, For v = m = 1, we from (14) - (16) have = = sn ( ) cos ( <) d r + rcos( ) cos( <) d - rsn ( ) sn ( <) d <, = sn ( ) sn ( <) d r + rcos( ) sn ( <) d + rsn( ) cos( <) d <, = cos ( ) d r - rsn ( ) d, sn ( ) cos ( <) rcos( ) cos ( <) - rsn ( ) sn ( <) sn ( ) sn ( <) rcos ( ) sn ( <) rsn ( ) cos ( <) cos( ) - rsn ( <) 0 d r d whch s the same as the usuall exteror transformaton between the sphercal coordnate and Cartesan coordnate n three- dmensonal space. Exa mple 2 Cylndrcal coordnate Consder the coordnate transformaton of cylndrnal coordnate x 1 = rcos( <), x 2 = rsn ( <), x 3 = z. From (12), t s easy to see that the coordnate transformatons for the fractonal dfferentals are that s, 1 = ( v - 2 m + 1) r 2 v - m z v - m (5 <) v cos v ( <) sn m - v ( <) r 2 v - m ( v - m + 1) z m (1 - m) 2 = ( v - 2 m + 1) r 2 v - m z v - m (5 <) v sn v ( <) r 2 v - m ( v - m + 1) z m (1 - m) 3 = ( v - 2 m + 1) r 2 v - 2 m z v r 2 v - 2 m cos v - m ( <) sn m - v ( <) cos v - m ( <) (5 <) v sn m - v ( <) z v - m cos v ( <) sn m - v ( <) d < r v - m d r d < (17) (18) (19) cos v ( <) sn m - v ( <) dzv, (20) z v - m sn v ( <) d sn v ( <) z v - m cos v - m ( <) sn m - v ( <) d < For v = m = 1, we from (20) - (22) have = cos( <) d r - rsn ( <) d <, r v - m d r dz v, (21) r v - 2 m d r dz v. (22) = sn ( <) d r + rcos( <) d <, = dz, (23) Tsnghua Tongfang Optcal Dsc Co., Ltd. All rghts reserved.

5 260 CHEN Yong, YAN Zhen-ya and ZHANG Hong-qng = cos( <) - rsn( <) 0 sn ( <) rcos( <) whch s the same as the usual exteror transformaton between the sphercal coordnate and Cartesan coordnate n three- dmensonal space. In summary, we have found the two coordnate transformatons for the fractonal dfferentals for three- dmensonal Cartesan coordnates to sphercal and cylndrcal coordnates. In partcular, for m = v = 1, the two above- mentoned coordnate transformatons are the same as the standard results obtaned from the exteror calculus. Ref e re nces : [ 1 ] Dold A, Eckmann B. Fractonal Calculus and Its Aapplcatons[ M ]. Berln :Sprnger- Verlag,1975. [ 2 ] Kerner R. Z 3 - graded exteror dfferental calculus and gauge theores of hgher order[j ]. Lett Math Phys,1996,36(5) : [3] Dubos- Volette M. Generalzed homologes for d N = 0 and graded q- dfferental algebras [ J ]. Contemp Math,1998,219 (1) : [4] Coquereaux R. Dfferentals of hgher order n noncommutatve dfferental geometry[j ]. Lett Math Phys,1997,42(3) : [ 5 ] Madore J. An Introducton to Noncommutatve Dfferental Geometry and Its Applcatons[ M ]. Cam2 brdge : Cambrdge Unversty Press,1995. [ 6 ] Cottrll- Shepherd K, Naber M. Fractonal dfferental forms [ J ]. J Math Phys, 2001, 42 ( 5) : d r d < dz (24) Tsnghua Tongfang Optcal Dsc Co., Ltd. All rghts reserved.