Research Article Laplace Transform Collocation Method for Solving Hyperbolic Telegraph Equation

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1 Hidawi Iteratioal Joural of Egieerig Mathematics Volume 7, Article ID 596, 9 pages Research Article Laplace Trasform Collocatio Method for Solvig Hyperbolic Telegraph Equatio Adebayo O. Adewumi, Saheed O. Akideide, Adebayo A. Aderogba, ad Babatude S. Ogudare Research Group i Computatioal Mathematics (RGCM), Departmet of Mathematics, Obafemi Awolowo Uiversity, Ile-Ife 5, Nigeria Correspodece should be addressed to Babatude S. Ogudare; boguda@oauife.edu.g Received July 6; Revised February 7; Accepted 8 February 7; Published April 7 Academic Editor: Bhabai S. Dadapat Copyright 7 Adebayo O. Adewumi et al. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. This article presets a ew umerical scheme to approimate the solutio of oe-dimesioal telegraph equatios. With the use of Laplace trasform techique, a ew form of trial fuctio from the origial equatio is obtaied. The ukow coefficiets i the trial fuctios are determied usig collocatio method. The efficiecy of the ew scheme is demostrated with eamples ad the approimatios are i ecellet agreemet with the aalytical solutios. This method produced better approimatios tha the oes produced with the stadard weighted residual methods.. Itroductio I this paper, we cosider the secod-order oe-dimesioal telegraph equatio t +α u t +βu= u +f(, t), () where α, β are kow costats ad f(, t) is cotiuous i the displayed argumets. Equatio () describes a electrical sigal travelig alog a trasmissio cable; this was first derived i the horse ad buggy days of the telegraph (from where it derived its ame) ad it is still useful for describig log distace power lies ad cable TV systems []. The study of electric sigal i a trasmissio lie, dispersive wave propagatio, pulsatig blood flow i arteries, ad radom motio of bugs alog a hedge is amogst a host of physical ad biological pheomea which ca be described by (). For epository details o the abovemetioed pheomea, readers are advised to see [ ]. Recetly, telegraph equatio is foud to be more suitable tha ordiary diffusio equatio i modellig reactio diffusio for such braches of sciece [5]. Without ay doubt, () ad its solutio are of great importace i may areas of applicatio. Various aalytical ad umerical methods have bee developed ad employed to solve this equatio. These iclude the Method of Weighted Residuals [6], Laplace trasform iversio techique with homotopy perturbatio method [7], radial basis fuctio method [8], Chebyshev tau method [9], Legedre multiwavelet Galerki method [], reciprocity boudary itegral equatio method [], Adomia decompositio method [], ucoditioally stable differece scheme [], ad the Reduced Differetial Trasform Method (RDTM) [] to metio just a few. Other researchers have also proposed differet umerical schemes for solvig telegraph equatio; for eample, Dehgha ad Lakestai [] proposed a method based o Chebyshev cardial fuctios to solve oe-dimesioal hyperbolic telegraph equatio, ad Javidi [5] used Chebyshev spectral collocatio method for computig umerical solutio of telegraph equatio. Borhaifar ad Abazari [6] developed a ucoditioally stable parallel differece scheme for telegraph equatio. Lakestai ad Saray [8] developed a umerical techique for the solutio of secod-order oe-dimesioal liear hyperbolic equatio. The method cosists of epadig the required approimate solutio as the elemets of iterpolatig scalig fuctio. I their techique, by usig operatioal

2 Iteratioal Joural of Egieerig Mathematics matri of derivatives, they reduced the problem to a set of algebraic equatios [7]. Hesameddii ad Asadolahifard [7] applied the Sic-Collocatio Method to approimate the solutio of (). Mittal ad Bhatia [8] ad Rashidiia et al. [9] employed the Cubic B-splie Collocatio Method (CuBSCM) to approimate the solutio of (). I [], the authors employed the Fiboacci Polyomials approach to approimate solutio of telegraph equatios. Motivated by the works of Odejide ad Biuyo [6] where the weighted residual method was applied to the oedimesioal telegraph equatio, i this work, a ew ad efficiet collocatio method based o the Laplace trasform is proposed to approimate the solutio of (). This ew method shall be called Laplace Trasform Collocatio Method (LTCM). The rest of this paper is orgaized as follows: I Sectio, brief descriptio of the method is preseted, ad Sectio is devoted to the error aalysis of the method. Implemetatio of the method usig umerical eamples is preseted i Sectio while the last sectio presets our coclusio.. Laplace Trasform Collocatio Method (LTCM) To put emphasis o the essetial mathematical details of the ew method, we cosider the followig oe-dimesioal hyperbolic telegraph equatio: t +α u t +βu= u +f(, t), with the iitial coditios Ω=[a, b] R, <t T u (, ) =g (), u t (, ) =g (), ad Dirichlet boudary coditio Ω Ω () () u (, t) =h(, t), Ω, <t T, () where α ad β are kow costat coefficiets, f, g, g, ad h are kow cotiuous fuctios i their respective domais, ad the fuctio u is ukow. Takig the Laplace trasform of (), we have s U (, s) su(, ) u t (, ) = L [ u (5) ]+L[f (, t)] αl [ u t ] βl[u]. After simple algebraic simplificatio, we get U (, s) = s [su (, ) +u t (, ) + L [ u ] + L [f (, t)] αl [ u ] βl[u]]. t (6) The fuctio u(, t) ad its derivatives i (6) are thereafter replaced with a trial fuctio of the form u=u + c i u i, (7) where c i are costats to be determied which satisfy the give coditios () ad (). Thus, we have the followig: U (, s) = s [s (u (, ) + + t (u (, t) + i= c i u i (, t)) c i u i (, )) t= + L [ (u (, t) + c i u i (, t))] + L [f (, t)] αl [ t (u (, t) + βl [ t (u (, t) + c i u i (, t))] c i u i (, t))]]. Takig the iverse Laplace trasform of (8), we have u ew (, t) = L [ s (s (u (, ) + + t (u (, t) + i= c i u i (, t)) t= + L [ (u (, t) + c i u i (, t))] + L [f (, t)] αl [ t (u (, t) + βl [ t (u (, t) + Substitutig (9) ito (), we get c i u i (, t))] c i u i (, t))]]. c i u i (, ))) t u ew (, t) +α t u ew (, t) +βu ew (, t) = u ew (, t) +f(, t). Now, collocatig () at poits = j,wehave t u ew ( j,t)+α t u ew ( j,t)+βu ew ( j,t) = j u ew ( j,t)+f( j,t), (8) (9) () ()

3 Iteratioal Joural of Egieerig Mathematics where j =a+ (b a) j, j=,,...,. () + Thus, () costitutes -equatios i -ukows which ca be determied by usig Gaussia elimiatio method. Substitutig these coefficiets ito (9) gives the approimate solutios.. Error Aalysis of Laplace Trasform Collocatio Method (LTCM) Let us defie the error fuctio e (, t) = u(, t) u (, t), where u(, t) ad u (, t) deote, respectively, the eact ad approimatesolutioobtaiedviaourproposedmethod.i lie with [], we defie the residual fuctio where R (, t) =L[u (, t)] f(, t), (). Numerical Eamples I this sectio, we implemet the ew method o some eamples to test its efficiecy ad applicability. Eample. Wecosiderthecaseiwhichα=, β=,ad f(, t) =, ad () becomes (see [6]) subject to t + u t +u= u, (9) u (, ) = si (), u t (, ) = si (), u (, t) =, u (π, t) =, π π, t>. The eact solutio is give by u(, t) = e t si(). () L[u (, t)] = u t It the follows that t +α u t +βu u. () +α u t +βu u =f+r (5) We assume the trial fuctio of the form: u (, t) = ( t) si () +c ( π) t +c ( π) t. Takig the Laplace trasform of (9), we get () subject to iitial coditios s U (, s) su(, ) u t (, ) +L [ u (, t)] t u (, ) =g () (u ) t (, ) =g (), (6) +L [u (, t)] = L [ u (, t)]. () u (, t) =h(), Ω. Rearragig (), we have Now sice L is a liear operator, we obtai for the error fuctio e (, t) e (, t) t = R (, t) +α e (, t) t with the homogeeous coditios +βe (, t) e (, t) (7) U (, s) = [s si () si () s L [c ( π) t+c ( π) t si ()] L [c ( π) t +c ( π) t + ( t) si ()] +L [c ( π) t +c t +c ( π) t +c t ( t) si ()]]. () e (, ) =, (e ) t (, ) =, e (a, t) =, e (b, t) =. (8) Takig the iverse Laplace trasform of (), we have the followig ew trial solutio: u ew (, t) =( t c 6 t c t c 6 t c ) Bysolvig(7)subjecttothehomogeeouscoditiosabove, we obtai the error fuctio e (, t). This allows us to compute u(, t) = u (, t)+e (, t) eve for problems without kow eact solutios. +( 8 t c π+ t c π+ 6 t c π+ t c π) +( t c 6 t c π + t c t c π )

4 Iteratioal Joural of Egieerig Mathematics +(+ t + t t)si () ( c + 6 c )t π. () Substitutig () ito (9), we have the followig residual fuctio: R(,t,c,c )=( 6 t c t c 6 t c t c 8c t 8c t 8c t 8c t) +(6c tπ + t c π+8c tπ + 8t c π+ t c π+ 6 t c π u(, t) t c π+ t c π) +(6c t +t c 8t c π +6c t 8c tπ 6 t c π +t c (5) Approimate solutio Eact solutio Figure : Compariso of approimate ad eact solutio for Eample. t c π +6t c +6t c ) ( t c +t c + t c +t c )π+(t+ t + 5 t ) si () ( 6 t π+ t π) c. 8 Collocatig (5) at equally spaced poits = π/ ad = π/ for t =. ad equatig to zero, we the use the Gaussia elimiatio method to solve the two systems of equatios ad obtaied c = ad c = Substitutig these values ito (), we obtai the followig approimate solutio: u (, t) = (.5 7 t t ) + ( t π.. t.6.8. Figure : The error plot for Eample t π) + (.88 7 t t π t π )+(+ t + t t)si () t π. (6) Table gives the compariso betwee the ew method (LTCM) ad the Method of Weighted Residual (MWR) for Eample.Figureshowsthecomparisoofapproimatead eact solutio for Eample. The error plot for Eample is show i Figure. Eample. We cosider et the case i which α=6, β=, ad f(, t) = e t si(),ad()becomes subject to t +6 u t +u= u e t si (), (7) u (, ) = si (), u t (, ) = si (), u (, t) =, u (π, t) =, π π, t>. The eact solutio is give by u e =e t si() [6]. (8)

5 Iteratioal Joural of Egieerig Mathematics 5 Table:ComparisoofLTCMwithMWRatt =. for Eample. Eact solutio LTCM solutio Error of LTCM Error i [6] E E E E 8.599E E E E.68E 8 We assume a approimate solutio of the form u (, t) = ( t) si () +c ( π) t +c ( π) t. (9) Followig the same procedure as discussed i Eample, we have the followig ew trial fuctio: u ew =( t c 6 t c +t c 6 t c ) +(t c π+t c π+ 6 t c π+ t c π) +( t c 6 t c π+ t c t c π ) +( e t + t + t t)si () ( t c + 6 t c )π. () Substitutig () ito (7) ad collocatig at poits =π/ ad = π/ for t =. ad also equatig to zero ad solvig the resultig equatios by Gaussia elimiatio method, we obtai c = ad c = Similarly, substitutig these values ito (), we determie the followig approimate solutio: u (, t) = (5.5 7 t t ) + ( t π t π t π )+( e t + t + t t)si ().7655t π. () Compariso betwee the approimatio with the ew method (LTCM) ad the covetioal Method of Weighted Residual (MWR) for Eample is give i Table. Figure u(, t) Approimate solutio Eact solutio Figure : Compariso of approimate ad eact solutio for Eample. shows the compariso of approimate ad eact solutio for Eample. The error plot for Eample is show i Figure. Eample. We ow cosider the case i which α=, β=, ad f(, t) = ( t + t )( )e t,adwesolve t + u t +u= u +( t+t )( )e t subject to +te t, u (, ) =, u t (, ) =, u (, t) =, u (, t) =,,, t>. The eact solutio is give by u e =( )t e t [6]. () ()

6 6 Iteratioal Joural of Egieerig Mathematics Table:ComparisoofLTCMwithMWRatt =. for Eample. Eact solutio LTCM solutio Error of LTCM Error i [6] E E 8.75E E 8.88E E 8.7E E.778E Similarly, takig t.6.8. Figure : The error plot for Eample. u (, t) =c ( ) t +c ( ) t, () ad followig the same procedure as i Eample, we have the followig ew trial solutio: u (, t) =( t c t c t c t c ) +(+ t c t t e t + t c e t + t c te t + 6 t c ) +(te t t c +t +t e t + t c +e t + 5 t c ) t c t 6 t c + ( + t +8t)e t. (5) Substitutig (5) ito () ad collocatig at poits = / ad = / ad later solvig the resultig equatios, we obtai the followig values for c ad c : c = ad c = u(, t) Approimate solutio Eact solutio Figure 5: Compariso of approimate ad eact solutio for Eample. Substitutig these values ito (5), we have u (, t) = ( t t t e t t e t te t ) + (te t t +t+t e t t +e t)+(+t +8t)e t +t t. (6) Compariso betwee the approimatio with the ew method (LTCM) ad the covetioal Method of Weighted Residual (MWR) for Eample is give i Table. Figure 5 shows the compariso of approimate ad eact solutio for Eample. TheerrorplotforEampleisshowiFigure6.

7 Iteratioal Joural of Egieerig Mathematics 7 Table:ComparisoofLTCMwithMWRatt =. for Eample. Eact solutio LTCM solutio Error of LTCM Error i [6]....E E.67E E.8988E E.67E 7...E t Figure 6: The error plot for Eample.. u(, t) Approimate solutio Eact solutio Figure 7: Compariso of approimate ad eact solutio for Eample. Eample. We cosider the case i which α=, β=,ad f(, t) = +t,adwehave subject to t + u t +u= u + +t, (7) u (, ) =, u t (, ) =, u (, t) =t, u (, t) =+t,,, t>. The eact solutio is give by u e = +t[6]. We assume a trial solutio of the form: (8) u (, t) = +t+c ( ) t +c ( ) t. (9) Followig the same procedure as i other eamples, we obtai aewformoftrialsolutiowhichisgiveas u (, t) =( c t c t c t c t ) +(+ c t + c t + c t + 6 c t ) +( 5 c t + c t c t )+t ( 6 c c )t. () Substitutig () ito (7) yields a residual fuctio which is the collocated at poits =/ad =/for t =.. Solvig the resultig equatios, we obtai c =c =.. Substitutig these values ito (), we have +twhich is the eact solutio. Table gives the compariso betwee approimatio with the ew method (LTCM) ad the Method of Weighted Residual (MWR) for Eample. Figure 7 shows the compariso of approimate ad eact solutio for Eample. TheabsoluteerrorplotforEampleisshowiFigure8.

8 8 Iteratioal Joural of Egieerig Mathematics Table:ComparisoofLTCMwithMWRatt =. for Eample. Eact solutio LTCM solutio Error of LTCM Error i [6] Refereces Coclusio.... t Figure 8: The error plot for Eample. I this paper, we adopted a combiatio of Laplace trasform scheme ad collocatio method to develop a ew umerical method for solvig oe-dimesioal liear hyperbolic telegraph equatio. Four umerical eamples were cosidered to demostrate the efficiecy ad accuracy of the method. It is observed from the solutio process that the preset method provides a good approimate solutio i compariso to the eact solutio as it ca be see i Figures,, 5, ad7withtheabsoluteerrorplotsdisplayedifigures,, 6, ad 8, respectively. Our approimatios are compared with the results obtaied by Odejide ad Biuyo [6] ad from the tables of results (Tables ) it is evidet that the LTCM produced better results tha the MWR. The method proposed i this paper will be eteded to solve parabolic Volterra itegrodifferetial equatio i future work. Coflicts of Iterest The authors declare that there are o coflicts of iterest regardig the publicatio of this paper. [] H. Pascal, Pressure wave propagatio i a fluid flowig through a porous medium ad problems related to iterpretatio of Stoeley s wave atteuatio i acoustical well loggig, Iteratioal Joural of Egieerig Sciece, vol.,o.9,pp.55 57, 986. [] G. Bohme, No-Newtoia Fluid Mechaics, vol. ofnorth- Hollad Series i Applied Mathematics ad Mechaics, North- Hollad, Amsterdam, The Netherlads, 987. [] M. Dehgha ad A. Ghesmati, Solutio of the secod-order oe-dimesioal hyperbolic telegraph equatio by usig the dual reciprocity boudary itegral equatio (DRBIE) method, Egieerig Aalysis with Boudary Elemets,vol.,o.,pp. 5 59,. [] R.K.MohatyadM.K.Jai, Aucoditioallystablealteratig directio implicit scheme for the two space dimesioal liear hyperbolic equatio, Numerical Methods for Partial Differetial Equatios, vol. 7, o. 6, pp ,. [5] M. Dehgha ad A. Shokri, A umerical method for solvig the hyperbolic telegraph equatio, Numerical Methods for Partial Differetial Equatios,vol.,o.,pp.8 9,8. [6] S. A. Odejide ad A. O. Biuyo, Numerical solutio of hyperbolic telegraph equatio usig method of weighted residuals, Iteratioal Joural of Noliear Sciece,vol.8,o.,pp.65 7,. [7] M. Javidi ad N. Nyamoradi, Numerical solutio of telegraph equatio by usig LT iversio techique, Iteratioal Joural of Advaced Mathematical Scieces,vol.,o.,pp.6 77,. [8] M.LakestaiadB.N.Saray, Numericalsolutiooftelegraph equatio usig iterpolatig scalig fuctios, Computers & Mathematics with Applicatios, vol.6,o.7,pp.96 97,. [9] A. Saadatmadi ad M. Dehgha, Numerical solutio of hyperbolic telegraph equatio usig the Chebyshev Tau method, Numerical Methods for Partial Differetial Equatios, vol.6,o.,pp.9 5,. [] S. A. Yousefi, Legedre multiwavelet Galerki method for solvig the hyperbolic telegraph equatio, Numerical Methods for Partial Differetial Equatios, vol.6,o.,pp.55 5,. [] M. A. Abdou, Adomia decompositio method for solvig the telegraph equatio i charged particle trasport, Joural of Quatitative Spectroscopy ad Radiative Trasfer,vol.95,o., pp.7,5. [] F. Gao ad C. Chi, Ucoditioally stable differece schemes for a oe-space-dimesioal liear hyperbolic equatio,

9 Iteratioal Joural of Egieerig Mathematics 9 Applied Mathematics ad Computatio,vol.87,o.,pp.7 76, 7. [] V. K. Srivastava, M. K. Awasthi, R. K. Chaurasia, ad M. Tamsir, The telegraph equatio ad its solutio by reduced differetial trasform method, Modellig ad Simulatio i Egieerig, vol., Article ID 765, 6 pages,. [] M. Dehgha ad M. Lakestai, The use of Chebyshev cardial fuctios for solutio of the secod-order oe-dimesioal telegraph equatio, Numerical Methods for Partial Differetial Equatios,vol.5,o.,pp.9 98,9. [5] M. Javidi, Chebyshev spectral collocatio method for computig umerical solutio of telegraph equatio, Computatioal Methods for Differetial Equatios,vol.,o.,pp.6 9,. [6] A. Borhaifar ad R. Abazari, A ucoditioally stable parallel differece scheme for telegraph equatio, Mathematical Problems i Egieerig,vol.9,ArticleID9696,7pages, 9. [7] E. Hesameddii ad E. Asadolahifard, The sic-collocatio method for solvig the telegraph equatio, Joural of Computer Egieerig ad Iformatics,vol.,o.,pp. 7,. [8] R. C. Mittal ad R. Bhatia, Numerical Solutio of secod order hyperbolic telegraph equatio via ew cubic trigoometric B- splies approach, Applied Mathematics ad Computatio, vol., o., pp ,. [9] J. Rashidiia, S. Jamalzadeh, ad F. Esfahai, Numerical solutio of oe-dimesioal telegraph equatio usig cubic B-splie collocatio method, Joural of Iterpolatio ad ApproimatioiScietificComputig,vol.,ArticleID jiasc-, 8 pages,. [] A. K. Bahşı ad S. Yalçıbaş, A ew algorithm for the umerical solutio of telegraph equatios by usig fiboacci polyomials, Mathematical ad Computatioal Applicatios,vol.,o.,p. 5, 6.

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