D Alembert s solution of fractional wave equations using complex fractional transformation

Transcription

1 D Alember s soluion of fracional wave equaions using comple fracional ransformaion Absrac Uam Ghosh a, Md Ramjan Ali b, Sananu Rau, Susmia Sarkar c and Shananu Das 3 Deparmen of Applied Mahemaics, Universiy of Calcua, Kolkaa, India a uam_mah@yahoo.co.in b ramjan.azad@gmail.com c susmia6@yahoo.co.in Mahabhanga College, Cooch Behar, Wes Bengal, India rau_sananu@yahoo.com 3 Reacor Conrol Sysems Design Secion E & I Group BARC Mumbai India shananu@barc.gov.in Fracional wave equaion arises in differen ype of physical problems such as he vibraing srings, propagaion of elecro-magneic waves, and for many oher sysems. The eac analyical soluion of he fracional differenial equaion is difficul o find. Usually Laplace-Fourier ransformaion mehod, along wih mehods where soluions are represened in series form is used o find he soluion of he fracional wave equaion. In his paper we describe he D Alember s soluion of he fracional wave equaion wih he help of comple fracional ransform mehod. We demonsrae ha using his fracional comple ransformaion mehod, we obain he soluions easily as compared o fracional mehod of characerisics; and ge he soluion in analyical form. We show ha he soluion o he fracional wave equaion manifess as ravelling waves wih scaled coordinaes, depending on he considered fracional order value. Key-words Fracional wave equaion, Comple fracional ransformaion, D Alember s soluion, modified fracional derivaive (Jumarie ype, fracional mehod of characerisics.. Inroducion Fracional calculus is one of he oldes invenions of modern mahemaics. I originaed from he leer of Leibniz o L Hospial in 695 [-, 3]. Fracional differenial models are used o describe he physical phenomena hose have memory or herediary effec [] such as viscoelasiciy, anomalous diffusion, fluid mechanics, biology, acousics, di-elecric relaaions, dynamics of super-capaciors, conrol heory ec [,4]. The mos commonly used definiion of fracional derivaive is he Riemann-Liouville (R-L fracional derivaive [, 5-]. The differenial equaion formed wih R-L ype fracional derivaive has iniial/ boundary condiions defined via fracional order derivaive []. In real life problem i is difficul o formulae he iniial/ boundary condiions wih fracional derivaive. Therefore, he researchers are using he Capuo [, ] or Jumarie [] ype fracional derivaive o formulae he problems. To solve he fracional differenial equaion usually mehods used are

2 he Variaion Ieraion Mehod [3-4], ep-mehod [5], fracional sub-equaion mehod [6-8], soluion using Miag-Leffler funcions as eigen-funcion [, 9, and 3] for fracional differenial operaor, Comple fracional ransform mehod [-5]. Among all hese mehods he comple fracional ransformaion mehod is one of he simples mehods o solve he Fracional Parial Differenial Equaion (FPDE. This ransform ransforms he fracional parial differenial equaion o a classical parial differenial equaion and hus he soluion procedure become simple. The D Alember s soluion of Cauchy Problem of he wave equaion plays an imporan role in differen ype of vibraion and wave propagaion problems. Usually Laplace-Fourier ransform mehod gives he eac soluion of he wave equaion, in closed form by use of Miag-Leffler and M-Wrigh funcions, where obaining inverse Laplace-Fourier ransforms are difficul using conour inegraion or Berberan-Sano s mehod [3]. The oher echniques o ge he soluion of fracional differenial equaion are by he mehods such as Variaion Ieraion Mehod [3-4], Adomian Decomposiion Mehod [] ec; where he soluions obained are in form of series soluion; (which is analyical approimae of closed form soluion. In his paper we find he D Alember s soluion of fracional wave equaion using he comple ransformaion mehod as eac analyic soluion in closed form. One of he moivaions for using fracional calculus in physical sysems is ha he space and ime variables which we ofen deal ehibi coarse-grained phenomena. This means infiniesimal quaniies canno be arbirarily aken o zero-raher hey are non-zero wih a minimum spread. This ype of non-zero spread arises in he microscopic o mesoscopic levels of sysem dynamics. This means ha if we denoe as he poin in space and as he poin in ime, hen limi of he differenials d (and d canno be aken zero. To ake his concep of coarse graining ino accoun, use he infiniesimal quaniies as ( (and ( wih ; called as fracional differenials. For arbirarily small differenials are greaer han and (ending owards zero, hese fracional (and i.e. ( and (. This way of defining he fracional differenials helps us o use fracional derivaives in he sudy of dynamic sysems. The coordinae in -space, originaes from he differenial d, as is inegraion i.e.. Now wih a differenial ( d, wih, we have ( d d, where d ( d [3, 3]. Tha is he space is ransformed o a fracal space where he coordinae is ransformed o. This is coarse graining phenomena in paricular scale of observaion. Here we come across he fracal space ime, where he normal classical differenials d and d, canno be aken arbirarily o zero. Thus in hese cases he concep of classical differeniabiliy is los [3]. The fracional order is relaed o roughness characer of he space-ime i.e. he fracal dimension [3, 3]. We will show ha soluion of fracional wave equaions have componen of ravelling wave in he scaled coordinaes and.

3 Organizaion of he paper is as follows: in secion. we describe some review of he fracional derivaive and comple fracional ransformaion. The secion 3. describes he mehod o find soluion of h and h order wave equaion using comple fracional ransformaion. In secion 4. he eisence and uniqueness of he soluion is described. Finally conclusion is given in secion 5.; followed by references.. Review of fracional calculus and Comple ransform mehod In his secion we describe some basic definiions of he fracional derivaives and he heory of comple fracional ransformaion. a Fracional derivaive The mos commonly used fracional derivaive is he Riemann Liouville (R-L [, 4, 3], ha is defined in inegral form as d D f f d (. n RL n ( ( ( n ( n d, n n In erms of his definiion fracional derivaive of he consan K is RL K D K ( [, 4, and 3]. In classical calculus he fracional derivaive of he ( consan equal o zero. To overcome his difference M. Capuo [] proposed he fracional order derivaive. Le be a posiive real number and nbe a posiive ineger saisfying n n. ( n Le f ( eiss (ha is n h order ordinary derivaive eiss. Then -h order fracional derivaive in Capuo sense is defined by C RL n ( n n ( n D f ( I f ( ( f ( d, ( n (., n n where RL I is he R-L inegral operaor[, 4, 3], defined as follows RL I f f d ( ( ( (, C In erms of he Capuo derivaive fracional derivaive of consan (K is zero i.e. D K. Bu his Capuo definiion is applicable only for differeniable funcions [, 4, 3]. Jumarie [, a 3

4 3] proposed he modified R-L derivaive of coninuous (bu no necessarily differeniable funcion f ( in he range a in he following form, (wih assumpion ha f ( is finie J ( f ( d, ( ( d D f ( f ( ( f ( f ( d, ( d (.3 ( n ( n f (, n n, n On he oher hand he general form of fracional differenial equaion of wo independen variables, u, is defined as funcion as follows where and one dependen variable D u,,,...,, u D u D u D u D u, (.4 u(, denoes he Jumarie fracional parial derivaive[] of he form, u(, d ( u(, u(, d, ( d (.5 The funcion u(, in (.5 is coninuous (bu no necessarily differeniable funcion. The Jumarie fracional derivaive of a consan is zero which overcomes he discomfor of RL derivaive. b The fracional Comple Transformaion Le he differenial equaion be of he form, wih D as Jumarie fracional derivaive operaor d D u f (, D (.6 d The comple fracional ransformaion for his differenial equaion (.6 is defined as, X ( p ( where p is consan []. From Fracional Taylor series (of Jumarrie ype [3] we have a ( conversion formula ha is! ( The relaion d ( differenial d f d d. From his have following conversion formula d! d d d ;! ( d d ( d d, as we ge from above, is he conversion formula for fracional o d f. Using his Jumarie conversion formula d u! du ( du and hen doing obvious manipulaions as depiced in following seps, and hereafer using he 4

5 d formula a a ( ge d (comes from Euler formula [, 3] i.e. d u ( du d d du ( dx dx d du d X dx d p du d ( dx d du p dx 5 d d ( ( Wih his above derivaion he equaion (.6 i.e. D u f ( reduces o he ineger order ordinary differenial equaion, ha is du ( p p F( X, X. dx ( For he fracional parial differenial of following form we D u(, D u(, f (, (.7 u u where and D u(,, D u(, ; by using he comple fracional ransformaions i.e. X ( p ( q (, T ( fracional parial derivaive operaors we ge he following ransformed epressions for u ( u u ( X u p ; u q u X X T Using he above ransformaion equaion (.7 i.e. D u(, D u(, f (, ; reduces o he ineger order (classical parial differenial equaion ha is following u u q p F( X, T T X 3. Applicaion of Comple fracional ransformaion In he ne wo sub-secions we shall describe he soluion of h and h order wave equaion using he comple fracional ransformaion. 3. Soluion of h order wave equaion Now consider he h order fracional differenial equaion D u(, c D u(, ha is of he following form

6 u(, u(, c, The iniial condiion of (3. is specified as; u, f (. Firs we will use fracional mehod of characerisic [8, 9]. Using his we have for FPDE (3. he characerisic ha are following du ds ( d ( ds ( d c ( ds (3. (3. Now we use he formula of inegraion w.r.. d inegraion of (3. [8, 9, and 3] as indicaed below for Where ( def f ( d ( f ( d, ( ( ( f ( d ( I f ( I f ( ( f ( d is Riemann-Liouville fracional inegraion of order. On making ( f we ge d,. Solving he firs equaion we ge u(, c (he reason ha Jumarie Fracional Derivaive of consan is zero and from he las wo equaions we ge [8, 9] ( d c ( d (3.3 Inegraing he epression (3.3 wih he formula of inegraion w.r.. d [8, 9, 3] we ge c c ( ( Hence he soluion of he equaion (3. can be wrien as ( ( (3.4 u(, c (3.5 6

7 where ( is an arbirary funcion of single variable [8, 9]. Using he iniial condiion we ge ( f (. Hence he soluion of (3. is given by, u(, f c (3.6 ( ( p q Now use he fracional comple ransformaion, wih ( X and ( T developed by He [-3]; and wrie he following epressions for fracional differenial operaors (ha ge changed o classical differenial operaor, as follows u u q, u p u T X (3.7 Wih above change by use of fracional comple ransformaion, we obain following equivalence in operaor form c q c p T X Using he ransformaion (3.7 equaion (3. reduces o he firs order parial differenial equaion in he following form u(, u(, p q c, X, T ( ( u( X, T u( X, T q c p T X Wih choice of p q (as hese consans are scale facors we ge following (3.8 u( X, T u( X, T X T c ( ( T X wih he iniial condiion u X, f ( X. The characerisics of (3.8 wih p q are following ha is given by, du dx dt ; c ; ds ds ds Soluion of he firs equaion is u ( X, T c and from he las wo equaions afer inegraion we ge X c T c, where c and c are consans. Hence soluion of he parial differenial equaion (3.8 is u ( X, T X c T. Using he iniial condiions we ge, X f ( X Hence he soluion of (3.8 is given by. 7

8 u ( X, T f X c T (3.9 Recalling he original variable in (3.9 i.e. ( X and ( T we ge soluion of he h order parial differenial equaion (3. is u ( X, T u (, f c (3. ( ( The soluion (3.6 and soluion (3. are he same epressions herefore he soluion obained by boh he mehods is same. We observe he soluion obained by using comple ransform mehod is easy comparaive o he earlier mehod i.e. fracional mehod of characerisics. The original iniial funcion f (, ravels wih consan velociy in his case isc, as i ravels in scaled coordinaes ( and (. These are fracional order ravelling waves [3]. 3. D Alember s Soluion of h order fracional Wave Equaion Here we consider he one dimensional Fracional wave equaion in he form, u(, u(, c, (3. wih he iniial condiions epressed as, ( ; (, ( u f u g (3.a where c in (3. is a posiive consan. This c is physically inerpreed as he wave speed. For eample, if displacemen of infinie sring is represened by u(, hen c becomes he quaniy / where is he ension of he sring and is he densiy. Now, he equaion (3. can be wrien (in facored form as following c c u(, (3. Le us consider he fracional comple ransformaion developed by He [-3], ( p ( q X, T ( ( (3.3 Using his ransformaion we ge, u u q, u p u T X. 8

9 Implying he following operaor equivalence c q c p ; c q c p T X T X Using he ransformaion (3.3 he equaion (3. reduces o he following form, q c p q c p u X, T T X T X (3.4 Under his ransformaion he boundary condiions reduces o following ses u( X, f ( X ; u( X, T g( X q (3.5 Le us consider he one facored erm of (3.4 ha is he following q c p u X T v X T T X,, (3.6 hen equaion (3.4 reduces o q c p v X, T T X (3.7 Using he mehod of characerisic [6] he soluion of he equaion (3.7 can be wrien in he following form,, Therefore from (3.6 u X, T saisfies he following equaion, v X T f q X c p T (3.8 q c p u X T f q X c p T T X, (3.9 On he oher hand he equaion (3.4 can be rewrien as, q c p q c p u X, T T X T X (3. T X Again consider q c p u X, T w X, T following hen equaion (3. reduces o 9

10 q c p w X, T T X (3. Thus (on similar lines as described above soluion of he equaion (3. can be wrien in he following form Therefore u X, T saisfies he equaion,, w X T g q X c p T (3. q c p u X T g q X c p T T X, (3.3 where f and g are funcions of single variable. Now adding he equaions (3.9 and (3.3 we ge following epression q u X, T f q X c p T g q X c p T (3.4 Inegraing (3.4 we can wrie he soluion in he following form, u X T F q X c p T G q X c p T (3.5 where F and G arbirary funcions of he single variable. Differeniaing (3.5 wih respec o T we ge T, u X T c p F q X c p T c p G q X c p T (3.6 Then from (3.5 and (3.6 and using iniial condiions we ge he following se of epressions g( X u X, c p F X G X T q u( X, F( X G( X f ( X F( X G( X f ( X (3.7 Solving he equaions in (3.7 we ge, c p q c p q F ( X f ( X g ( X, G ( X f ( X g ( X (3.8 Inegraing (3.8 on boh side we ge following epressions

11 X c p q X F ( X f ( X g ( d X c p q X G ( X f ( X g ( d Thus from (3.5 we ge he general soluion of (3. which is (3.9 u X, T f ( q X c p T f ( q X c p T g( d c p q. (3.3 q X c p T q X c p T p Now ransforming o he original variables i.e. ( q ( X and ( ( T he soluion (3.3 reduces o he following form, ( ( ( ( u(, f p q c f p q c p q c ( ( g( d c p q p q c ( ( (3.3 Wih p q (as hese consans are scale facors and he iniial condiion of (3. is u(, f (. The equaion (3. is hus a classical wave equaion i.e. u(, c u(,. Assume he second iniial condiion as (, ( u g. Placing p q and g( in (3.3, we ge he following u(, f c f c The above epression represens he iniial condiion (funcion geing spli ino wo wih ampliude half each and ravelling in and direcions respecively wih consan velociy c in opposie direcions. This is sandard classical Cauchy problem of a classical wave equaion. The soluion (3.3 is D Alember soluion of he Cauchy problem for one dimensional fracional wave equaion given by (3.3. Now wih g( in (3.3, and wih p q for soluion o fracional wave equaion (3.3, we ge he wo waves (defined by iniial condiion ravelling in opposie direcions wih velociy c in he scaled coordinaes 4. Eisence and uniqueness ( and (. Theorem: Show ha he problem (3. and (3.a in he domain T for a fied T well-posed for f C ( and g C (. Proof: The eisence and uniqueness of he soluions of (3. wih (3.a is direced from D Alember s soluions. Now from smoohness condiions f C ( and g C ( implies

12 ha u(, C ( (, C ( (,. Thus like he classical ineger order soluion he soluion (3.3 is generalized soluion. Now we shall prove he sabiliy of he soluion i.e. for a small change of he iniial condiions give rise o small change of he soluion. Le ui (, be wo soluion of he problem (3. wih wo iniial condiions f, g for i,. Now since fi i i C ( and g C ( for i, wih, and for all i f ( f ( ; g ( g (. Then for all and T we have he following from (3.3 u (, u (, if ( T heorem is proved. ( ( ( ( f p q c f p q c ( ( ( ( f p q c f p q c p q c ( ( g ( g( d p q c c p q ( ( ( T. Therefore for all and T we have u(, u(,. Hence he Eample: consider he fracional wave equaion in he form u(, u(, c, (3.3 wih he iniial condiions u, f ( and ( D u(, g( sin( for firs case represened; he soluion is represened by figure-; and wih iniial condiions ( u he soluion represened by figure-, f ( and D u(, g( sin

13 Graphical presenaion of he soluion u(,for =.7 Graphical presenaion of he soluion u(,for = Graphical presenaion of he soluion u(,for =.9 Graphical presenaion of he soluion u(,for = Fig-: Graphical represenaion of he soluion (3.33 for.7,.8,.9,.. Soluion: Considering p q (as hese consans are scale facors in (3.3 we ge he D Alember s soluion of (3.3 in he following form ( ( ( ( ( ( c c ( ( u (, ( f c f c c g d Puing he iniial condiions f ( and g( sin 3 we ge he following u(, c c ( ( ( ( c ( ( sin( d c c ( ( (3.33 c ( ( cos ( cos c ( c

14 Thus we obain soluion of D'Alember's of he fracional wave equaion. If we consider he iniial condiion as f ( and g( sin hen we ge he soluion as u ( ( (, c sin( cos ( cos ( c d c c (3.34 ( ( c Graphical presenaion of he soluion u(,for =.7 Graphical presenaion of he soluion u(,for = Graphical presenaion of he soluion u(,for =.9 Graphical presenaion of he soluion u(,for = Fig-: Graphical represenaion of he soluion (3.34 for.7,.8,.9, From he graphical presenaions in figure- and i is clear ha he soluion depends on he order of fracional derivaive, wih he increase of order of fracional derivaive he soluion paern changes. 5. Conclusions In his paper we obained he soluion of and order fracional wave equaion; composed via Jumarie ype fracional derivaive using comple fracional ransformaion. We have demonsraed ha using his fracional comple ransformaion mehod, we obained he soluions easily as compared o fracional mehod of characerisics. The soluion of order fracional wave equaion is like he D'Alember's soluion of classical second order wave equaion. We show ha i is having ravelling wave componens bu are ransformed ino scaled 4

15 coordinaes and. When he fracional order is uniy, we ge he classical soluion of he wave equaion. 6. References. Kilbas, AA, Srivasava, HM, Trujillo, JJ: Theory and Applicaions of Fracional Differenial Equaions. Norh-Holland Mahemaics Sudies, vol. 4. Elsevier, Amserdam (6. S. Das. Funcional Fracional Calculus nd Ediion, Springer-Verlag and Funcional Fracional Calculus for sysem idenificaion & conrols s Ediion Springer-Verlag D. Qian, C. Li, R. P. Agarwal and P. J.Y Wong. Sabiliy analysis of fracional differenial sysem wih Riemann-Liouville derivaive. Mahemaical and Compuer Modelling Podlubny, I: Fracional Differenial Equaions. Mahemaics in Science and Engineering, vol. 98. Academic Press, San Diego ( Miller, KS, Ross, B: An Inroducion o he Fracional Calculus and Fracional Differenial Equaions. Wiley, New York ( Hilfer, R: Applicaions of Fracional Calculus in Physics. World Scienific Publishing, River Edge ( 7. Wes, BJ, Bologna, M, Grigolini, P: Physics of Fracal Operaors. Insiue for Nonlinear Science. Springer, New York (3 8. Sabaier, J, Agrawal, OP, Machado, JA: Advance in Fracional Calculus: Theoreical Developmens and Applicaions in Physics and Engineering. Springer, Dordrech (7 9. Baleanu, D, Guvenc, B, Tenreiro, JA: New Trends in Nanoechnology and Fracional Calculus Applicaions. Springer, New York (. Lakshmikanham, V, Leela, S, Vasundhara, J: Theory of Fracional Dynamic Sysems. Cambridge Academic Publishers, Cambridge (9. M. Capuo, Linear models of dissipaion whose q is almos frequency independen, Geophysical Journal of he Royal Asronomical Sociey, ( Jumarie G. Modified Riemann-Liouville derivaive and fracional Taylor series of non differeniable funcions furher resuls. Compu. Mah. Appl. 6; 5(9-: G. C. Wu, E. W. M. Lee. Fracional variaional ieraion mehod. Physics Leers A He J.H., Analyical mehods for hermal science-an elemenary inroducion, Thermal Science, 5(, s, pp. S-S3 5. S. Zhang, Q. A. Zong, D. Liu and Q. Gao, A generalized Ep-funcion mehod for fracional Riccai differenial equaions, Commun. Frac. Calc, ( U. Ghosh, S. Sengupa, S. Sarkar, S. Das. Analyical soluion wih anh-mehod and fracional sub-equaion mehod for non-linear parial differenial equaions and corresponding fracional differenial equaion composed wih Jumarie fracional derivaive. In. J. Appl. Mah. Sa.; ( S. Zhang and H. Q. Zhang, Fracional sub-equaion mehod and is applicaions o nonlinear fracional PDEs, Phys. Le. A U. Ghosh, S. Sarkar, S. Das. Analyical Soluions of Classical and Fracional KP-Burger Equaion and Coupled KdV Equaion. 6. CMST. 6. (

16 9. U. Ghosh, S. Sengupa, S. Sarkar and S. Das. Analyic soluion of linear fracional differenial equaion wih Jumarie derivaive in erm of Miag-Leffler funcion. American Journal of Mahemaical Analysis. 5; 3( El-Sayed AMA, Gaber M. The Adomian decomposiion mehod for solving parial differenial equaions of fracal order in finie domains. Phys. Le. A. 6; 359(:75-8. R. W. Ibrahim, Fracional comple ransforms for fracional differenial equaions, Advances in Difference Equaions, vol., aricle 9,.. He, J.H., Elagan, S.K.: Li ZB. Geomerical eplanaion of he fracional comple ransform and derivaive chain rule for fracional calculus. Phys. Le. A 376, ( 3. Li, Z.B., He, J.H.: Fracional comple ransform for fracional differenial equaions. Mah. Compu. Appl. 5, ( 4. Wang, Q.L., He, J.H., Li, Z.B.: Fracional model for hea conducion in polar bear hairs. Therm. Sci. 5, 5 ( 5. Li, Z.B., Zhu, W.H., He, J.H.: Eac soluions of ime-fracional hea conducion equaion by he fracional comple ransform. Therm. Sci. 6, ( 6. Myin-U, T. and Debnah, L. (7. Linear Parial Differenial Equaions for Scieniss and Engineers, 4h ediion, Birkhäuser, Boson. 7. Ian N. Sneddon. Elemens Of Parial Differenial Equaions. Dover Publicaions G. C. Wu. A fracional characerisic mehod for solving fracional parial differenial equaions. Applied Mahemaics Leers Jumarie G, Tables of some basic fracional calculus formulae derived from modified Riemann- Liouville derivaive for non-differeniable funcions; Applied Mahemaics Leers ( S. Das, Kindergaren of Fracional Calculus, (Lecure noes on fracional calculus-in use as limied prins a Dep. of Phys, Jadavpur Universiy-JU; under publicaion a JU 3. J. Banerjee, U. Ghosh, S. Sarkar and S. Das. A Sudy of Fracional Schrödinger Equaioncomposed via Jumarie fracional derivaive (in-press, arxiv: