On the Pseudo-Spectral Method of Solving Linear Ordinary Differential Equations

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1 Journl of Mhemics nd Sisics 5 ():136-14, 9 ISS Science Publicions On he Pseudo-Specrl Mehod of Solving Liner Ordinry Differenil Equions B.S. Ogundre Deprmen of Pure nd Applied Mhemics, Universiy of For Hre, Alice 57 RSA Absrc: Problem semen: o ll differenil equions cn be solved nlyiclly, o overcome his problem, here is need o serch for n ccure pproime soluion. Approch: The objecive of his sudy ws o find n ccure pproimion echnique (scheme) for solving liner differenil equions. By eploiing he Trigonomeric ideniy propery of he Chebyshev polynomil, we developed numericl scheme referred o s he eudo-eudo-specrl mehod. Resuls: Wih he scheme developed, we were ble o obin pproime soluion for cerin liner differenil equions. Conclusion: The numericl scheme developed in his sudy compees fvorbly wih soluions obined wih sndrd nd well nown specrl mehods. We presened numericl emples o vlide our resuls nd clim. Key words: Chebyshev polynomil, liner ordinry differenil equions, Specrl mehod, Pseudospecrl mehod, eudo-eudo-specrl mehod. ITRODUCTIO The fundmenl problem of pproimion of funcion by inerpolion on n inervl pved wy for he specrl mehods which re found o be successful for he numericl soluion of ordinry nd pril differenil equions. Specrl represenions of nlyic sudies of differenil equions hve been in used since he dys of Fourier. Their pplicion o umericl soluion of ordinry differenil equions refers, les o he ime of Lnczos [1]. Summry of survey of some pplicions is given in [8]. Some presen specrl mehods cn lso be rced bc o he "mehod of weighed residuls" of Finlyson nd Scriven [6]. Specrl mehods cn be viewed s n ereme developmen of he clss of discreizion scheme for differenil equions nown s he Mehod of Weighed Residuls (MWR) [6]. In MWR, he use of pproiming funcions (clled ril funcions) is cenrl. These funcions re used s bsis funcions for runced series epnsion of he soluion. Anoher funcion clled he es funcions (lso nown s he weigh funcions) re used o ensure h he differenil equion is sisfied s close s possible by he runced series epnsion. Among he specrl schemes he hree mos commonly used re he Tu, Glerin nd collocion (lso clled eudo-specrl) mehods. Wh disinguishes beween hese mehods is he choice of he es funcions employed. Glerin nd 136 Tu mehod re implemened in erms of he epnsion coefficiens [5], wheres collocion mehods re implemened in erms of physicl spce vlues of unnown funcion. Over he wo decdes, specrl mehods wih heir curren forms ppered s rcive wys in mos pplicions. Some more deils on specrl mehods could be seen in [9,11-13]. The bsic ide of specrl mehods o solve differenil equions is o epnd he soluion funcion s finie series of very smooh bsis funcions, s given below: (1) = = φ () where, φ represens Chebyshev or Legendre polynomils [14] (for more on Chebyshev polynomils). If y C,b, he error produced by he pproimion pproches zero wih eponenil re [4] s becomes oo lrge (ends o infiniy). This phenomenon is referred o s specrl ccurcy [8]. The ccurcy of he derivive obined by direc erm-by-erm differeniion of such runced epnsion nurlly deeriores [4], bu for low-order derivives nd sufficienly high-order runcions his deeriorion is negligible, compred o he resricions in ccurcy inroduced by ypicl difference pproimions. In [] nd [3], he reserchers focused on differenil equions in which one of he coefficien funcion or

2 J. Mh. & S., 5 ():136-14, 9 soluion funcion is no nlyic on he inervl of definiion. We spec of specrl mehods in solving his ind of problems were sudied in [] nd [3] nd he reserchers cme up wih modificions o he specrl mehod which proved o be more efficien when compred wih eising ones. In his ricle, we presen vriion of he collocion (eudo-specrl) mehod o solve he problems of [] nd [3]. The eudo-eudo-specrl mehod (he mehod inroduced in his ricle) is seen o be efficien nd compees fvorble wih oher wellnown sndrd mehods lie he Tu mehod, Glerin Mehod nd he Pseudo-specrl (collocion) mehod. MATERIALS AD METHODS Pseudo-eudo-specrl mehod: Consider he following differenil equion: Tu nd he eudo-specrl mehods for numericl soluion of second order liner differenil equions o compre he resul wih eudo-eudo-specrl mehod. We need o se here h his discussion cn be eended o he generl problem of he form () nd (3). Consider he following differenil equion: P() + Q() + R() = S(), [ 1,1] y( 1) = α, y(1) = β (6) Wih he eudo-eudo-specrl mehod, we suppose h he pproime soluion of he Eq. 6 is given by: = T () (7) m i Ly f mi()d y f (),,b i= = = () Ty = K (3) where, f i, i =,1, m f, re nown funcions of, D i is he order of differeniion wih respec o he independen vrible, T is liner funcionl of rn m nd K. Here (3) cn eiher be iniil, boundry or mied condiions. To solve he bove clss of equions using he specrl mehod is o epnd he soluion funcion y, in () nd (3) s finie series of very smooh funcions in he form below: = T () (4) where, { T ()} is he sequence of Chebyshev polynomils of he firs ind. Replcing y by y in (3) he residul is defined s: r () Ly f = (5) The min rge nd objecive in specrl mehod is o minimize r s much s possible wih regrd o (3). The implemenion of he specrl mehods leds o sysem of liner equions wih +1 equions in +1 unnowns, 1,, Here we presen vriion (eudo) of one of he hree specrl mehods clled collocion (lso nown s eudo-specrl) mehod. We cll his mehod eudo-eudo-specrl mehod. Also, we use boh he 137 insed of (4) for n rbirry nurl number, where T + 1 = (,,..., ) is he consn coefficiens 1 vecor nd { T () } is he sequence of Chebyshev polynomils of he firs ind. The prime denoes h he firs erm in he epnsion is hlved. In his mehod, s gins he use of funcion V() s in he sndrd Tu mehod nd he Pseudospecrl mehod [,3,7], we insed of using he Chebyshev polynomil s polynomil we eploi he rigonomeric propery of Chebyshev funcion. Le: = T () (8) be he pproime soluion for he Eq. 6, s soluion i mus sisfy he equion. Recll he definiion of Chebyshev polynomil: Le: Then: T () = 1 cos( cos ()) 1 θ = cos, = cosθ T () = T ( θ ) = cosθ Using he ideniy defined bove, (8) becomes: = cos θ (9)

3 J. Mh. & S., 5 ():136-14, 9 The firs nd second derivives of (9) re respecively given s: sin θ (1) y() = ( sin θ ) sin θ cosθ cos θ sin θ 3 = sin θ y() = ( ) (11) Subsiuing (9-11) ino he equion (6) wih he funcions P, Q, R nd S epressed in erms of θ we hve: sin θ cosθ cos θ sin θ P( θ) 3 = sin θ sin θ + Q( θ) sin θ R( ) cos S( ),, + θ θ = θ θ π π y( π ) = α, y( π ) = β (1) Relion (15) form sysem wih wo equions nd +1 unnowns, o consruc he remining -1 equions we Colloce (13) he zeros of T -1 (), which re he inerior poins beween -1 nd 1 nd re ( 1) π given s θ =, = 1,..., 1, which is in gre 1 vrince o he Tu Mehod, he Glerin mehod nd he Pseudo-specrl mehod. The sysem obined here solves for he coefficiens. RESULTS Here we consider some ordinry differenil equions wih Tu Mehod, Pseudo-specrl mehod nd he Pseudo-eudo-specrl mehod nd presen our resuls in Tble-7. As noions, we represen he pproimions wih he Tu mehod, Pseudo-specrl mehod nd he Pseudo-eudo-specrl mehod s y, y, y respecively. φ( θ ) = S( θ) (13) = Simplifying Eq. 11, we hve: sin θ cosθ cos θ sin θ φ( θ ) = P( θ) 3 sin θ sin θ + Q( θ ) + R( θ) cosθ sin θ (14) If we impose he ssocied condiions on (14), we hve: So: = = y( 1) = α T ( 1) = ( 1) = α = = y(1) = β T (1) = (1) = β ( 1) α. = β. (15) 138 umericl eperimens: We shll consider he following problems: Problem 1: y() + y() + y = cos, y( 1) = sin( 1), y(1) = sin(1) wih he ec soluion = sin. Tble 1: Mimum error of pproimion of problem 1 y () Tble : Mimum error of pproimion of problem y () Tble 3: Mimum error of pproimion of problem 3 y ()

4 J. Mh. & S., 5 ():136-14, 9 Tble 4: Mimum error of pproimion of problem 4 y () Tble 5: Mimum error of pproimion of problem 5 y () m Tble 6: Mimum error of pproimion of problem 6 y () Tble 7: Mimum error of pproimion of problem 7 y () Problem : 1 8 y() + y() =, (,1), 8 y(1) =, y() = 7 wih he ec soluion ln 8. Problem 3: = [ 1,1], y( 1) = ep( 1), 3 3 y() y() ( ) 1 ( ) ep(), y(1) = ep(1.) wih he ec soluion = ep(). Problem 4: 3 3 y() + y() + = , [ 1,1], y( 1) = y(1) = 1 wih he ec soluion Problem 5: 3 = ep(). [ 1,1], y( 1) = 1, 3 y() + ep( )y() = ep( ), y(1) = 1 wih he ec soluion = 3. Problem 6: y() y() + =, y( 1) = 1, y(1) = 1 wih he ec soluion =. Problem 7: y() + y() + = +, y( 1) = y(1) = 1 wih he ec soluion =. DISCUSSIO Problem 1 ws en from [3]. This problem ws solved wih Runge-Ku of differen orders nd mimum error of 3. were recorded. I ws lso solved wih Tu mehod nd he mehod described in his sudy wih = 5, 8, 16. The mimum error produced for hese wo mehods for he vrious is shown in he Tble 1. Tble 1 shows he power of specrl mehods over Runge-Ku. Problem ws en from [1]. I ws solved by erpolion mehod wih mimum error of 1 8. I ws solved in [3] by Tu mehod for differen vlues of. Here we solved i by he eudo-eudo-specrl mehod for differen vlues of s in he Tu mehod of [3], he mimum error for he wo mehods is shown in Tble. From Tble i could be seen h s increses, he re of improvemen is very low wih he Tu mehod. Pseudo-specrl mehod which ws he subjec of [3] produced mimum error of 4 9 wih = 18 s gins of he eudo-eudo-specrl mehod. This shows h boh he eudo-specrl nd he eudo-eudo-specrl mehods re more successful hn he Tu mehod. 139

5 Problem 3 ws chosen from []. I ws solved wih he mehod described in his ricle nd he error produced for vrious is shown in he Tble 3 wih he mimum error produced when he problem ws solved by he Tu mehod nd Pseudo-specrl mehod of [3]. Problem 4 ws en from [3]. The problem hs non nlyic soluion funcion which mes ccompny error indispensble. We pplied our mehod o he problem, he error produced by he mehod s well s he error produced when i ws solved wih he Tu mehod nd Pseudospecrl mehod of [3] is shown in Tble 4. Problem 5 ws chosen from [3]. When he problem is solved using he Tu mehod nd he eudo-specrl. mehod in [3], he mehods filed nd modified eudospecrl (m) mehod which ws he subjec of he ricle ws used o solve he problem nd he mimum error produced in [3] for he problem is shown in he Tble 5 wih he error produced by he mehod of his ricle. This mehod performs beer hn he modified mehod of [3]. Problem 6 ws lso from [3]. We esed our mehod on his problem wih differen vlues of $$, he resuls re shown in Tble 6. From he Tble 6 i could be seen h he eudoeudo-specrl mehod is performing considerbly well in he clss of specrl mehods. Problem ws 7 chosen from [3]. The resuls for vrious vlues of re shown in Tble 7. The resuls displyed in he Tble 7 show he power of he eudo-eudo-specrl mehod over he Tu nd he eudo-specrl mehods. COCLUSIO The eudo-eudo-specrl mehod of his ricle is seen o be efficien nd compees fvorble wih oher well-nown sndrd mehods lie he Tu mehod, Glerin Mehod nd he Pseudo-specrl (collocion) mehods. One mjor dvnge wih his mehod is h i does no require edious mens of evluing he unnown coefficiens of he pproiming funcion s in oher specrl mehods. The mehod is esy o progrm nd require moderely less of compuer ime o evlue. I is lso seen o be suible for ny clss of liner differenil equions wih or wihou nlyicl soluions. REFERECES 1. Ascher, U.M., R.M. Mhheij nd R.d. Russell, umericl Soluion of Boundry Vlue Problem for Ordinry Differenil Equions. 1s Edn., Prenice-Hll Inc., USA., ISB: , pp: 619. J. Mh. & S., 5 ():136-14, Bbolin, E. nd M.M. Hosseini,. A modified specrl mehod for numericl soluion of ordinry differenil equions wih non-nlyic soluion. Applied Mh. Compu., 13: DOI: 1.116/S96-33(1) Bbolin, E., M. Bromilow, R. Englnd nd M. Svri, 7. A modificion of eudo-specrl mehod for solving liner ODEs wih singulriy. Applied Mh. Compu., 188: DOI: 1.116/j.mc Cnuo, C., M. Hussini, A. Qureroni nd T. Zng, Specrl Mehods in Fluid Dynmics. Springer, Berlin, ISB: 1: , pp: Delves, L.M. nd J.L. Mohmed, Compuionl Mehods for Inegrl Equions. 1s Edn., Cmbridge Universiy Press, Cmbridge, ISB: 1: , pp: Finlyson, A. nd L.E. Scriven, The mehod of weighed residuls. Applied Mech. Rev., 19: hp:// Weighed-Residuls 7. Fornberg, B., A Prcicl Guide o Pseudo- Specrl Mehods. Illusred Edn., Cmbridge Universiy Press, Cmbridge, ISB: , pp: Golieb, D. nd S. Orszg, umericl Anlysis of Specrl Mehods, Theory nd Applicions. 6h Edn., SIAM, Phildephi, PA., ISB: , pp: E. Gourgoulhon,. Inroducion o Specrl Mehods, 4h EU ewor meeing, Plm de Mllorc, Sep.. 1. Lnczos, C., Trigonomeric inerpolion of empiricl nd nlyic funcions, J. Mh. Phys., 17: Oriz, E.L., The u mehod. SIAM. J. umer. Anl., 6: hp:// 1. Oriz, E.L. nd J.H. Freilich, 198. umericl soluion of sysem of ordinry differenil equions wih u mehod: An error nlysis. Mh. Compu., 39: hp:// 13. Oriz, E.L. nd T. Chves, On he numericl soluion of wo-poin boundry vlue problems for liner differenil equions ZAMM. J. Applied Mh. Mech., 48: DOI: 1.1/zmm Prer, I.B. nd L. Fo, 197. Chebyshev Polynomils in umericl Anlysis. nd Edn., Oford Universiy Press, Oford, ISB: 13: , pp: 16.