Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol.2, No.4, 2012

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1 Soluion of Telegraph quaion by Modified of Double Sumudu Transform "lzaki Transform" Tarig. M. lzaki * man M. A. Hilal. Mahemaics Deparmen, Faculy of Sciences and Ars-Alkamil, King Abdulaziz Uniersiy, Jeddah-Saudi Arabia. Mahemaics Deparmen, Faculy of Sciences, Sudan Uniersiy of Sciences and Technology-Sudan.. Mahemaics Deparmen, Faculy of Sciences for Girles King Abdulaziz Uniersiy Jeddah-Saudi Arabia * -mail of he corresponding auhor: Tarig.alzaki@gmail.com and farah@kau.edu.sa The research is financed by Asian Deelopmen Bank. No. 006-A7(Sponsoring informaion) Absrac In his paper, we apply modified ersion of double Sumudu ransform which is called double lzaki ransform o sole he general linear elegraph equaion. The applicabiliy of his new ransform is demonsraed using some funcions, which arise in he soluion of general linear elegraph equaion. Keywords: Double lzaki Transform, modified of double Sumudu ransforms, Double Laplace ransform, Telegraph quaion.. Inroducion: Parial differenial equaions are ery imporan in mahemaical physic [7], he wae equaion is known as one of he fundamenal equaions in mahemaical physics is occur in many branches of physics, for eample, in applied mahemaics and engineering. A lo of problems hae been soled by inegral ransforms such as Laplace [7], Fourier, Mellin, and Sumudu [9, 0]. Also hese problems hae been soled by differenial ransform mehod [3-0] and homoopy perurbaion [-5] an ingenious soluion o isualizing he lzaki ransform was proposed originally by Tarig M. lzaki [-4], his new ransform rials Sumudu ransform in problem soling. In his paper we derie, we beliee for he firs ime and sole elegraph and wae equaions by using modified of double Sumudu ransform [8] "double lzaki ransform". We wrie ha Laplace ransform is defined by: [ ] s L f ( ) = e f ( ) d, s > 0 (-) 0 Where ha lzaki ransform is defined oer he se of funcions: k j j A = f ( ) : M, k, k > 0, f ( ) > Me, ( ) Χ[0, ) 95

2 By u [ ] = f ( ) u f ( ) e d, u ( k, k ). (-) 0 By analogy wih he double Laplace ransform, we shall denoe he double lzaki ransform.. Double lzaki Transform: The double Laplace ransform of a funcion of wo ariables is gien by: ( p+ s ) [ (, )] = (, ) = (, ) (-3) L f F p s f e dd Where p, s are he ransform ariables for, respeciely. Definiion: 0 0 Le f (, ),, R +, be a funcion which can be epressed as a conergen infinie series, hen, is double lzaki ransform, gien by [ ] 0 0 ( + ) u f (, ), u, = T ( u, ) = u f (, ) e dd,, > 0 (-4) Where u, are comple alues. To find he soluion of elegraph and wae equaions by double lzaki ransform, firs we mus find double lzaki ransform of parial deriaies as follows: Double Laplace ransform of he firs and second order parial deriaies are gien by: f f F (0, s ) L = pf ( p, s ) F (0, s ) L p F ( p, s ) pf (0, s ) = f f F ( p,0) L = sf ( p, s ) F ( p,0) L s F ( p, s ) sf ( p,0) = f L = psf ( p, s ) pf ( p,0) sf (0, s ) F (0,0) Similarly double lzaki ransform for firs and second parial deriaies are gien by: T u ut T u T u u u f f T (0, ) = (, ) (0, ) (, ) (0, ) = T u T u T u T u f f T ( u,0) = (, ) (,0) (, ) (,0) = f u = T ( u, ) T ( u,0) T (0, ) + ut (0,0) 96

3 Proof: u e f dd e u e f d d f ( + ) u u = (, ) = (, ) The inner inegral gies: T ( u, ) uf (0, ), and hen: u f u = e T ( u, ) d u e f (0, ) d = T ( u, ) ut (0, ) u 0 0 f Also = T ( u, ) T ( u,0) We can proe anoher deriaie easily by using he same mehod.. Applicaions: In his secion we esablish he alidiy of he double lzaki ransform by applying i o sole he general linear elegraph equaions. To sole parial differenial equaions by double lzaki ransform, we need he following seps. (i) Take he double lzaki ransform of parial differenial equaions. (ii) Take he single lzaki ransform of he condiions. (iii) Subsiue (ii) in (i) and sole he algebraic equaion. (i) Take he double inerse of lzaki ransform o ge he soluion Here we need he main equaion: a + b u e = ( au )( b ) Consider he general linear elegraph equaion in he form: Wih he boundary condiions: And he iniial condiions: Soluion: U + au + bu = c U (-) U (0, ) = f ( ), U (0, ) = g ( ) U (,0) = f ( ), U (,0) = g ( ) Take he double lzaki ransform of equaion (-) and single lzaki ransform of condiions, and hen we hae: 97

4 T ( u,0) a T ( u, ) T ( u,0) + T ( u, ) at ( u,0) + bt ( u, ) c T T u c T c u (0, ) + (, ) (0, ) = 0 u And: (-) T (0, ) T (0, ) = F ( ), = G( ) T ( u,0) T ( u,0) = F ( u ), = G ( u ) (-3) Subsiuing (-3) in (-), we obain: u F ( u ) + u G ( u ) + a u F ( u ) + c u F ( ) + c u G( ) T ( u, ) = = H ( u, ) + au + b u + c Take double inerse lzaki ransform o obain he soluion of general linear elegraph equaion (-) in he form: U (, ) = H ( u, ) = K (, ) [ ] Assumed ha he double inerse lzaki ransform is eiss. ample.: Consider he elegraph equaion Wih he boundary condiions: And he iniial condiions: The eac soluion is U (, ) = Soluion U = U + U + U (-4) U (0, ) = e, U (0, ) = e (-5) U (,0) = e, U (,0) = e (-6) e Take he double lzaki ransform of equaion (-4), and single lzaki ransform of condiions (-5), (-6), and hen we hae: 98

5 T (0, ) T ( u,0) T ( u, ) T (0, ) u = T ( u, ) T ( u,0) u + T ( u, ) T ( u,0) + T ( u, ) (-7) And, T (0, ) T (0, ) =, = + + (-8) u T ( u,0) u T ( u,0) =, = u u (-9) Subsiuing (-8) and (-9) in (-7), we obain: u u u T u = T T T + + = u + + u u u Or And u u u u u u u u T = + + u u u + + u ( u u + u ) u T ( u, ) = = + u u u + u + u ) ( ) Inersion o find he soluion of equaion (-4) in he form: ample.: ( )( )( ) ( )( U (, ) = e. e = e Consider he elegraph equaion Wih he boundary condiions: And he iniial condiions: U = U + U U (-0) U (0, ) = e, U (0, ) = e (-) U (,0) = e, U (,0) = e (-) The eac soluion is U (, ) = e Soluion Take he double lzaki ransform of eq (-0), and single lzaki ransform of condiions (-), (-), and 99

6 hen we hae: T (0, ) T ( u,0) T ( u, ) T (0, ) u = T ( u, ) T ( u,0) u + T ( u, ) T ( u,0) T ( u, ) (-3) And T (0, ) T (0, ) =, = + + (-4) u T ( u,0) u T ( u,0) =, = u u (-5) Subsiuing (-4) and (-5) in (-3), o find: u u u u u ( u u + u ) T = + + u u u + + u ( u u + + u ) u And T ( u, ) = = (+ )( u )( u u + + u ) (+ )( u ) The inerse of he las equaion gies he soluion of equaion (-0) in he form: U (, ) = e ample.3: Le us he elegraph equaion Wih he boundary condiions: And he iniial condiions: U = U + 4U + 4U (-6) U = + e U = (-7) (0, ), (0, ) U = + e U = (-8) (,0), (,0) The eac soluion is U (, ) = e + e Soluion Applying double lzaki ransform o eq (-6), and single lzaki ransform o condiions (-7), (-8), we ge: 00

7 T (0, ) T ( u,0) T ( u, ) T (0, ) u = T ( u, ) T ( u,0) u 4 + T ( u, ) 4 T ( u,0) + 4 T ( u, ) (-9) And he ransform of condiions are, 3 + T (0, ) T (0, ) =, = + (-0) 3 u u T ( u,0) T ( u,0) =, = u u (-) By he same mehod in eamples (-4) and (-5), subsiuing (-0) and (-) in (-9) o find: u + u + u ) u T ( u, ) = = + (+ )( u ) ( u ) + Take he double inerse of lzaki ransform o ge he soluion of equaion (-6) in he form: U (, ) = e + e 3. Conclusion: In his work, double lzaki ransform is applied o obain he soluion of general linear elegraph. I may be concluded ha double lzaki ransform is ery powerful and efficien in finding he analyical soluion for a wide class of parial differenial equaions. Acknowledgmen: Auhors graefully acknowledge ha his research paper parially suppored by Faculy of Sciences and Ars-Alkamil, King Abdulaziz Uniersiy, Jeddah-Saudi Arabia, also he firs auhor hanks Sudan Uniersiy of Sciences and Technology-Sudan. References [] Tarig M. lzaki, The New Inegral Transform lzaki Transform Global Journal of Pure and Applied Mahemaics, ISSN ,Number (0), pp [] Tarig M. lzaki & Salih M. lzaki, Applicaion of New Transform lzaki Transform o Parial Differenial quaions, Global Journal of Pure and Applied Mahemaics, ISSN ,Number (0), pp [3] Tarig M. lzaki & Salih M. lzaki, On he Connecions Beween Laplace and lzaki ransforms, Adances in Theoreical and Applied Mahemaics, ISSN Volume 6, Number (0),pp. -. [4] Tarig M. lzaki & Salih M. lzaki, On he lzaki Transform and Ordinary Differenial quaion Wih Variable Coefficiens, Adances in Theoreical and Applied Mahemaics. ISSN Volume 6, 0

8 Number (0),pp [5] Tarig M. lzaki, Adem Kilicman, Hassan layeb. On isence and Uniqueness of Generalized Soluions for a Mied-Type Differenial quaion, Journal of Mahemaics Research, Vol., No. 4 (00) pp [6] Tarig M. lzaki, isence and Uniqueness of Soluions for Composie Type quaion, Journal of Science and Technology, (009). pp [7] Lokenah Debnah and D. Bhaa. Inegral ransform and heir Applicaion second diion, Chapman & Hall /CRC (006). [8] A.Kilicman and H..Gadain. An applicaion of double Laplace ransform and Sumudu ransform, Lobacheskii J. Mah.30 (3) (009), pp.4-3. [9] J. Zhang, A Sumudu based algorihm m for soling differenial equaions, Comp. Sci. J. Moldoa 5(3) (007), pp [0] Hassan layeb and Adem kilicman, A Noe on he Sumudu Transforms and differenial quaions, Applied Mahemaical Sciences, VOL, 4,00, no., [] Kilicman A. & H. Layeb. A noe on Inegral ransform and Parial Differenial quaion, Applied Mahemaical Sciences, 4(3) (00), PP [] Hassan Layeh and Adem kilicman, on Some Applicaions of a new Inegral Transform, In. Journal of Mah. Analysis, Vol, 4, 00, no.3, 3-3. [3] Abdel Hassan, I.H, 004 Differenial ransformaion echnique for soling higher-order iniial alue problem. Applied mah.compu, [4] Ayaz.F.004 soluion of he sysem of differenial equaions by differenial ransforms mehod.applied mah.compu, 47: [5] Bongsoo Jang: Soling linear and nonlinear iniial alue proplems by he projeced differenial ransform mehod. Ulsan Naional Insiue of Science and Technology (UNIST), Banyeon-ri-00, Ulsan korea. Compu. Phys. Communicaion(009) [6] C. Hchen, S. H. Ho. Soling Parial differenial by wo dimensional differenial ransform mehod, APPL. Mah.Compu.06 (999)7-79. [7] Fama Ayaz-Soluion of he sysem of differenial equaions by differenial ransform mehod.applied.mah. Compu. 47(004) [8] F. Kanglgil. F. Ayaz. Soliary wae Soluion for kd and M kd equaions by differenial ransform mehod, chaos soluions and fracions do:0.06/j. Chaos [9]Hashim,IM.SM.Noorani,R.Ahmed.S.A.Bakar..S.I.Ismailand A.M.Zakaria,006.Accuracy of he Adomian decomposiion mehod applied o he Lorenz sysem chaos [0] J. K. Zhou, Differenial Transformaion and is Applicaion for lecrical erucs.hunzhong uniersiy press, wuhan, china, 986. [] Monri Thong moon. Sasiornpusjuso.The numerical Soluions of differenial ransform mehod and he Laplace ransform mehod for a sysem of differenial equaion. Nonlinear Analysis. Hybrid sysems (009) d0i:0.06/j.nahs [] N.H. Sweilam, M.M. Khader. ac Soluions of some capled nonlinear parial differenial equaions using he homoopy perurbaion mehod. Compuers and Mahemaics wih Applicaions 58 (009) 34-0

9 4. [3] P.R. Sharma and Giriraj Mehi. Applicaions of Homoopy Perurbaion mehod o Parial differenial equaions. Asian Journal of Mahemaics and Saisics 4 (3): 40-50, 0. [4] M.A. Jafari, A. Aminaaei. Improed Homoopy Perurbaion Mehod. Inernaional Mahemaical Forum, 5, 00, no, 3, [5] Jagde Singh, Deendra, Sushila. Homoopy Perurbaion Sumudu Transform Mehod for Nonlinear quaions. Ad. Theor. Appl. Mech., Vol. 4, 0, no. 4, Tarig M. lzaki Deparmen of Mahemaics, Faculy of Sciences and Ars-Alkamil, King Abdulaziz Uniersiy, Jeddah-Saudi Arabia -mail: arig.alzaki@gmail.com and farah@kau.edu.sa man M. A. Hilal Deparmen of Mahemaics, Faculy of Sciences for Girls King Abdulaziz Uniersiy, Jeddah- Saudi Arabia -mail: ehilal@kau.edu.sa 03