SOLUTIONS OF THE TELEGRAPH EQUATIONS USING A FRACTIONAL CALCULUS APPROACH

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1 THE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 5, Number /04, pp SOLUTIONS OF THE TELEGRAPH EQUATIONS USING A FRACTIONAL CAULUS APPROACH José Francisco GÓMEZ AGUILAR, Dumiru BALEANU,3,4 Universidad Nacional Auónoma de Méico, Insiuo de Energías Renovables, Dpo. Maeriales Solares, Priv. Xochicalco s/n. Col. Cenro. Temico Morelos, Méico jfga@ier.unam.m King Abdulaziz Universiy, Faculy of Engineering, Deparmen of Chemical and Maerials Engineering, P.O. Bo: 8004, Jeddah, 589, Saudi Arabia 3 Cankaya Universiy, Deparmen of Mahemaics and Compuer Science, 06530, Ankara, Turkey 4 Insiue of Space Sciences, Magurele-Buchares, Romania Corresponding auhor: Dumiru BALEANU, dumiru@cankaya.edu.r In his paper, he fracional differenial equaion for he ransmission line wihou losses in erms of he fracional ime derivaives of he Capuo ype is considered. In order o keep he physical meaning of he governing parameers, new parameers σ and α were inroduced. These parameers characerize he eisence of he fracional componens in he sysem. A relaion beween hese parameers is also repored. Fracional differenial equaions are eamined wih boh emporal and spaial fracional derivaives. We show a few illusraive eamples when he wave periodiciy is broken in eiher emporal or spaial variables. Finally, we presen he oupu of numerical simulaions ha were performed wih boh emporal and spaial fracional derivaives. Key words: ransmission line, fracional differenial equaions, Miag-Leffler funcion.. INTRODUCTION The Fracional Calculus (FC) is he generalizaion of he ordinary inegrals and derivaives of ineger orders o arbirary ones. Mahemaical and physical consideraions in favor of he use of models based on derivaives of non-ineger order are given in [ ] and he references herein. The auhors of Ref. [] presen a numerical mehod for solving a class of fracional differenial equaions based on Bernsein polynomials basis; hese marices are uilized o reduce he muli-erm orders fracional differenial equaion o a sysem of algebraic equaions. In Ref. [3] i was esablished he double Laplace formulas for he parial fracional inegrals and derivaives in he sense of Capuo. The elegraph equaions are a pair of linear parial differenial equaions which describe he evoluion of volage and curren on an elecrical ransmission line wih disance and ime. The elegraph equaions are due o Oliver Heaviside who in he 880s developed he ransmission line model. This model demonsraes ha he elecromagneic waves can be refleced on he wire, and ha appear wave paerns along he ransmission line. Recenly, he fracional elegraph equaions were considered by many auhors [4 30]. Unlike he work of he auhors menioned above, in which he passing from an ordinary derivaive o a fracional one is direc, here we firs analyze he ordinary derivaive operaor and ry o bring i o he fracional form in a consisen manner [3]. This paper is organized as follows: In he second secion we presen he basic definiions of fracional calculus. The hird secion presens he fracional ransmission line and he mehodology proposed. Fracional differenial equaions are eamined separaely, wih emporal and spaial derivaive, respecively. Finally, we show he complee soluion and numerical simulaions by aking simulaneously boh derivaives (ime-space derivaives). In he fourh secion we depic our conclusions.

2 8 José Francisco Gómez Aguilar, Dumiru Baleanu. BASIC DEFINITIONS The definiions of he fracional order derivaive are no unique and here eis several definiions, including he Riemann-Liouville, he Grünwald-Lenikov, and he Capuo represenaion for Fracional Derivaive (CFD). In he Capuo case, he derivaive of a consan is zero and he iniial condiions for he fracional differenial equaions have a known physical inerpreaion; his is imporan from he physical and engineering poin of view. For a funcion f(), he CFD is defined as [3] ( n) C f ( τ) o D f 0 n+ () = d. τ Γ( n ) () ( τ) In his case, 0 <, is he order of he fracional derivaive, n=,,, N and n < n. The Laplace ransform o CFD is given by [3] m C k ( k) o k = 0 L D f() = S F( s) S f (0). The Miag-Leffler funcion has caused eensive ineres among physiciss due o is vas poenial of applicaions describing realisic physical sysems wih memory and delay. The Miag-Leffler funcion is defined by m Ea () =, ( a > 0), (3) Γ ( am + ) m= 0 where Г(.) is he gamma funcion. If a=, from (3), we obain e, he eponenial funcion as a special case of he Miag-Leffler funcion [3]. () 3. THE FRACTIONAL TRANSMISSION LINE The idea is replace he ime derivaive operaor d by a new fracional operaor d ( represens he d d order of he derivaive). The proposed alernaive is inroducing an addiional parameer, which mus have dimension of seconds o be consisen wih he dimension of he ordinary derivaive operaor. Thus, we replace he ordinary ime derivaive operaor by he fracional one as: d d, 0, < d ( σ ) d where 0 <, and, σ is a new parameer represening he fracional ime componens in he sysem (componens ha show an inermediae behavior beween a conservaive and a dissipaive sysem); i has he dimension of ime and can be called cosmic ime ; his is a non-local ime [3]. In he case = he epression (4) becomes an ordinary ime derivaive operaor d d =, 0 <. d ( σ ) d = In he case of he spaial derivaive operaor fracional spaial derivaive as follows: (4) (5) d, we can replace he ordinary derivaive by he d d d, 0 <β, d β β ( α ) d β (6)

3 3 Soluions of he elegraph equaions using a fracional calculus approach 9 in his case, he parameer α has he dimension of lengh and i represens he spaial fracional componen. When β= he epression (6) becomes a classical space derivaive operaor. In he following, we will apply his idea o invesigae he case of he ransmission line. Fracional ime ransmission line equaion. Using (4) he ransmission line equaions [4] can be wrien in erms of he fracional ime derivaives as: V(, ) V(, ) = 0, 0 <, ( σ ) ( ) (7) A paricular soluion of (7) is I(, ) I(, ) = 0, 0 <. ( σ ) ( ) V = V u (9) ik (, ) 0e ( ), where k is he wave vecor in he direcion and V 0 is he iniial volage. Subsiuing (9) in (7) we obain d ( ) u + ω u () = 0, (0) d where k ( ω ) = σ. () The soluion of he equaion (0) reads as u () = E ω. () Subsiuing he epression () in (9) we obain a paricular soluion of he equaion as { } V(, ) = V e E ω. (3) ik 0 Firs case. When =, he soluion of (7) follows from (3) and i is wrien by V = V ω (4) i( k) (, ) Re 0e. The soluion (4) represens a monochromaic wave respec o and (Fig. ). (8) Second case. When Fig. Periodic wave wih respec o and. = he equaion (7) is wrien as

4 30 José Francisco Gómez Aguilar, Dumiru Baleanu 4 V(, ) V(, ) = 0, 0 <. (5) σ The soluion can be found in he form (9), hen we obain he following equaion for u() du u () 0, d + ω = (6) where, in his case ω =σ follows from (). The soluion of he equaion (6) is obained in erms of he ω Miag-Leffler funcion. Then, when =, we have: u () = E { ω } = e. The paricular soluion has he form ik (, ) 0e e. V = V ω (7) We noice ha, for his case he soluion is periodic only regarding, and i is no periodic regarding. The soluion represens a plane wave wih eponenial ime-decaying ampliude. This is a direc consequence of he fracional ime derivaive [33]. On he oher hand, we have ha he velociy is given by k σ V =, k and he angular frequency ω= kυ, hen ω=, if we ake: =σω=. I is imporan o see ha here eiss a direc relaion beween he parameer σ and he period T 0 given by he order of he differenial equaion where, σ =σω=, 0 < σ. (8) T0 T0 Taking ino accoun he relaion (0), he soluion (5) can be wrien as 0 V(, ) = V e E, (9) ik ( ) 0 =, is a dimensionless parameer, and T0 = is he wave's period. The wave's periodiciy is T k broken in he region: 0< σ < < T 0. k Fig. shows he loss of periodiciy of he wave wih respec o. Fig. Periodic wave wih respec o, he periodiciy of he wave is broken in. Fracional space ransmission line equaion. Now will consider he equaion (7) assuming ha he spaial derivaive is fracional (6) and he ime derivaive is ineger. Then, we have he spaial fracional equaion

5 5 Soluions of he elegraph equaions using a fracional calculus approach 3 β V(, ) V(, ) = 0, 0 < β, (0) ( β) β α where he order of he spaial fracional differenial equaion is represened by 0<β, and α has dimension of lengh. A paricular soluion o he equaion (0) may be as follows: iω V(, ) = V0e u( ), () and subsiuing () in (0), we obain β d u ( ) + ku ( ) = 0, β d () ( ) ( ) where k β β =ω α = k α is he wave vecor in he medium in presence of fracional componens and k is he wave vecor wihou is presence. The wave vecors are equal, k = k, only in he case, β =, when do no eis fracional componens. Using (3) he soluion o he equaion () is given in erms of he Miag-Leffler funcion β β ( k ) u ( ) = Eβ k =. (3) n= 0 β( nβ+ ) Therefore, by placing (3) in () we obain he soluion β iω 0 β n V (, ) = V e E k. (4) Firs case. For he fracional space case, when β =, he soluion follows from i( k) V (, ) = ReV 0e ω, (5) wih k = k =ω, where k is he componen of he wave vecor in he direcion and is relaed wih he wavelengh by k =. λ The soluion (5) represens a monochromaic wave wih respec o and, (Fig. 3). Fig. 3 Periodic wave wih respec o and. Second case. We have, β= k = k α =ωα, and k = has dimensions of he inverse of ι k u ( ) = E k = e. he lengh. From (3) we conclude ha The soluion (4) is wrien as V V ω i k (, ) = 0e e. (6)

6 3 José Francisco Gómez Aguilar, Dumiru Baleanu 6 This wave is periodic only wih respec o, bu no wih respec o, so in his case he periodiciy is los. In his case, a direc relaion beween α and he wavelengh λ given by β is described by: α β= kα =, for 0 <α λ. We can use his relaion in order o wrie he equaion (4) as follows λ Here V (, ) = V e E β. (7) i ω ( β ) β 0 β = is a dimensionless parameer. Fig. 4 shows he loss of periodiciy of he wave wih respec o. λ Fig. 4 Periodic wave wih respec o, he periodiciy of he wave is broken in. Fracional ime-space ransmission line equaion. Now we consider he equaions (9) and (0) assuming ha in ime and space he derivaive are fracional, (Eqs. (4) and (6)). Then, we obain he imespace fracional equaion β ( α ) ( σ ) V(, ) V(, ) = 0, 0 < β, 0 <, ( β) β ( ) β β β ( α ) ( σ ) I(, ) I(, ) = 0, 0 < β, 0 <, ( ) ( ) (8) (9) where he order of he ime-space fracional differenial equaion is represened by 0 < β and 0 <, α has dimension of lengh and σ of ime. The full soluion of he equaion (8) is ( β) β ( ) V(, ) = A E β E = 0, 0 < β, 0 <, (30) β where = k, =ω are dimensionless parameers and A is a consan. Figs. 5, 6, 7, and 8 show numerical simulaions where he fracional ime derivaive and he spaial fracional derivaive are aken a he same ime for differen paricular cases of and β. Fig. 5 Fracional wave wih respec o and. Here = 0.99 and β = Fig. 6 Fracional wave wih respec o and.here = 0.98 and β = 0.98.

7 7 Soluions of he elegraph equaions using a fracional calculus approach 33 Fig. 7 Fracional wave wih respec o and. Here = and β = Fig. 8 Fracional wave wih respec o and. Here = 0.95 and β = CONCLUSIONS In his paper we have considered he fracional differenial equaion for he ransmission line wihou losses in erms of he Capuo fracional derivaive (CFD). We have used he idea suggesed in [3] o consruc he corresponding fracional differenial equaions. Two new parameers σ and α were inroduced, hese parameers represening he componens ha show an inermediae behavior beween conservaive and dissipaive sysems. The general soluions of he CFD, depending only on he parameers and β are given in he form of he mulivariae Miag-Leffler funcions ha preserve he physical unis of he sudied sysem. We also show ha he periodiciy wih respec o ime and o space is broken and he wave behaves like a wave wih ime-decaying ampliude or spaial-decaying ampliude for he emporal and spaial case, respecively. Also, he equaions (9) and (7) have a universal characer and hey are relaed wih he σ α condiions =σω= for 0< σ T0 and β= kα = for 0 < α λ. In he case where boh T0 λ ime-space derivaives are simulaneously considered, he equaion (30) shows he complee soluion by applying he separaion of variables mehod. Besides, i was shown ha when he parameers and β are less han, he wave losses is periodiciy wih respec o ime and space. We believe ha wih his approach i will be possible o have a beer undersanding of he ransien effecs in elecrical sysems and ransmission lines. This research was suppored by CONACYT. ACKNOWLEDGMENTS REFERENCES. K. B. OLDHAM, J. SPANIER, The fracional calculus, New York, Academic Press, K. S. MILLER, B. ROSS, An inroducion o he fracional calculus and fracional differenial equaions, New York, Wiley, S. G. SAMKO, A. A. KILBAS, O. I. MARICHEV, Fracional inegrals and derivaives, heory and applicaions, Langhorne, PA, Gordon and Breach Science Publishers, I. PODLUBNY, Fracional differenial equaions, New York, Academic Press, D. BALEANU, K. DIETHELM, E. SCALAS, J. J. TRUJILLO. Fracional calculus models and numerical mehods, Series on Compleiy, Nonlineariy and Chaos, World Scienific, F. GÓMEZ, J. ROSALES, M. GUÍA, R elecrical circui of non-ineger order, Cen. Eur. J. Phys, DOI: 0.478/s , M. GUIA, F. GÓMEZ, J. ROSALES. Analysis on he ime and frequency domain for he RC elecric circui of fracional order, Cen. Eur. J. Phys., DOI: 0.478/s y, R. METZLER, J. KLAFTER, The ramdom walk's guide o anomalous difussion: a fracional dynamics approach, Phys. Rep., 339, pp. 77, R. METZLER, J. KLAFTER, The resauran a he end of random walk: recen developmen in descripion of anomalous ranspor by fracional dynamics, J. Phys., A37, pp. R6 R08, 004.

8 34 José Francisco Gómez Aguilar, Dumiru Baleanu 8 0. O. P. AGRAWAL, J. A. TENREIRO-MACHADO, I. SABATIER (Eds.), Fracional derivaives and heir applicaions: Nonlinear dynamics, 38, Berlin, Springer, R. HILFER, (Ed.) Applicaions of fracional calculus in physics, Singapore, World Scienific, B. J. WEST, M. BOLOGNA, P. GRIGOLINI, Physics of fracional operaors, Berlin, Springer, R. L. MAGIN, Fracional calculus in bioengineering, Roddin: Begell House Publisher, M. CAPUTO, F. MAINARDI, A new dissipaion model based on memory mechanism, Pure and App. Geo., 9, pp , S. WESTERLUND, Causaliy, Universiy of Kalmar, Rep. No , D. BALEANU, Z. B. GÜNVENC, J. A. TENREIRO MACHADO (Eds.), New rends in nanoechnology and fracional calculus applicaions, Springer, F. GÓMEZ, J. BERNAL, J. ROSALES, T. CÓRDOVA, Modeling and simulaion of equivalen circuis in descripion of biological sysems a fracional calculus approach, J. Elecric. Bioimped., 3, pp., A. E. M. HERZALLAH, D. BALEANU, Eisence of a periodic mild soluion for a nonlinear fracional differenial equaion, Compu. Mah. Appl., 64, pp , S. BHALEKAR, V. DAFTARDAR-GEJJI, D. BALEANU, Transien chaos in fracional Bloch equaions, Compu. Mah. Appl., 64, pp , A. RAZMINIA, D. BALEANU, Fracional synchronizaion of chaoic sysems wih differen orders, Proc. Romanian Acad. A., 3, pp. 34 3, 0. J. A. TENREIRO MACHADO e al., Fracional calculus analysis of he cosmic microwave background, Rom. Rep. Phys., 65, pp , 03. J. J. ROSALES GARCIA, M. GUIA CALDERON, J. MARTINEZ ORTIZ, D. BALEANU, Moion of a paricle in a resising medium using fracional calculus approach, Proc. Romanian Acad. A, 4, pp. 4 47, 03. XIAO-JUN YANG e al., Approimae soluions for diffusion equaions on Canor space-ime, Proc. Romanian Acad. A, 4, pp. 7 33, 03.. A. K. GOLMANKHANEH, N. A. PORGHOVEH, D. BALEANU, Mean square soluions of second-order random differenial equaions by using homoopy analysis mehod, Rom. Rep. Phys., 65, pp , 03. D. BALEANU, J. H. ASAD, I. PETRAS, S. ELAGAN, A. BILGEN, Fracional Euler-Lagrange equaion of Caldirola-Kanai oscillaor, Rom. Rep. Phys., 64, pp.7 77, 0. D. BALEANU, J. H. ASAD, I. PETRAS, Fracional-order wo-elecric pendulum, Rom. Rep. Phys., 64, pp , 0. A. K. GOLMANKHANEH e al., Newonian mechanics on fracals subse of real-line, Rom. Rep. Phys., 65, pp , 03.. D. ROSTAMY, M. ALIPOUR, H. JAFARI, D. BALEANU, Solving muli-erm orders fracional differenial equaions by operaional marices of BPs wih convergence analysis, Rom. Rep. Phys., 65, pp , A. M. O. ANWAR, F. JARAD, D. BALEANU, F. AYAZ, Fracional Capuo hea equaion wihin he double Laplace ransform, Rom. J. Phys., 58, pp. 5, R. C. CASCAVAL, E. C. ECKSTEIN, C. L. FROTA, J. A. GOLDSTEIN, Fracional elegraph equaions, J. Mah. Anal. Appl., 76, pp , E. ORSINGHER, L. BEGHIN, Time-fracional elegraph equaions and elegraph processes wih brownian ime, Probab. Theor. Rel., 8, pp. 4 60, J. CHEN, F. LIU, V. ANH, Analyical soluion for he ime-fracional elegraph equaion by he mehod of separaing variables, J. Mah. Anal. Appl., 338, pp , E. ORSINGHER, X. ZHAO, The space-fracional elegraph equaion and he relaed fracional elegraph process, Chin. Ann. Mah. Ser. B, 4, pp , S. MOMANI, Analyic and approimae soluions of he space- and ime-fracional elegraph equaions, Appl. Mah. Com., 70, pp. 6 34, R. F. CAMARGO, A. O. CHIACCHIO, E. CAPELAS DE OLIVEIRA, Differeniaion o fracional orders and he fracional elegraph equaion, J. Mah. Phys., 49, Aricle ID , F. HUANG, Analyical soluion for he ime-fracional elegraph equaion, J. Appl. Mah., 009, Aricle ID 89058, J. F. GÓMEZ-AGUILAR, J. J. ROSALES-GARCÍA, J. J. BERNAL-ALVARADO, T. CÓRDOVA-FRAGA, R. GUZMÁN- CABRERA, Fracional mechanical oscillaors, Rev. Me. Fís, 58, pp , I. PODLUBNY, Geomeric and physical inerpreaion of fracional inegraion and fracional differeniaion, Frac. Calc. Appl. Anal., 5, pp , M. MOSHREFI-TORBATI, J.K. HAMMOND, Physical and geomerical inerpreaion of fracional operaors, J. Franklin Ins., 335 B, pp , 998.