Mahemaical and Compuaional Applicaions, Vol., No. 4, pp. 99-978,. Associaion for Scienific Research APPLICATION OF CHEBYSHEV APPROXIMATION IN THE PROCESS OF VARIATIONAL ITERATION METHOD FOR SOLVING DIFFERENTIAL- ALGEBRAIC EQUATIONS M. Ghovamand, M.M. Hosseini, M. Nilli Faculy of Mahemaics, Yazd Universiy, P.O. Bo 8995-74,Yazd, Iran. mghova@su.yazduni.ac.ir, hosse_m@yazduni.ac.ir Absrac- In his paper, we use Chebyshev approimaions in he process of He s variaional ieraion mehod for finding he soluion of differenial-algebraic equaions. This allows us o make inegraion a each of he ieraions possible and a he same ime, obain a good accuracy in a reasonable number of ieraions. Numerical resuls show ha using Chebyshev approimaion is much more efficien han using Taylor approimaion which is more popular. Firs, an inde reducion echnique is implemened for semi-eplici differenialalgebraic equaions, hen he obained problem is solved by He s variaional ieraion mehod. The scheme is esed for some high inde differenial-algebraic equaions and he resuls demonsrae reliabiliy and efficiency of he proposed mehod. Key words- Chebyshev approimaion, Variaional ieraion mehod, Differenial algebraic equaions. INTRODUCTION Many physical problems are governed by a sysem of differenial-algebraic equaions (DAEs), and finding he soluion of hese equaions has been he subjec of many invesigaions in recen years. In 999, he variaional ieraion mehod (VIM) was proposed by He [-]. This mehod is now widely used by many researchers o sudy linear and nonlinear problems. The mehod inroduces a reliable and efficien process for a wide variey of scienific and engineering applicaions. I is based on Lagrange muliplier and i has he meris of simpliciy and easy eecuion. Unlike he radiional numerical mehods, VIM needs no discreizaion, linearizaion, ransformaion or perurbaion. The mehod gives rapidly convergen successive approimaions of he eac soluion if such a soluion eiss; oherwise a few approimaions can be used for numerical purposes. The VIM was successfully applied o auonomous ordinary and parial differenial equaions [ 4]. Recenly in [5] he VIM has been implemened for finding he soluion of differenialalgebraic equaions. To avoid edious inegraion, in he laer paper he Taylor approimaion is used. In his paper, we are going o use he Chebyshev approimaion insead of Taylor approimaion. Eamples show ha his procedure is much more efficien. I is well known ha he eigenfuncions of cerain singular Surm-Liouville problems allow he approimaion of funcions [a, b] where runcaion error approaches zero faser han any negaive power of he number of basic funcions used in he approimaion, as ha number (order of runcaion ) ends o infiniy []. This phenomenon is usually referred o as '' specral accuracy ''. In his work, we are using C
97 M. Ghovamand, M.M. Hosseini and M. Nilli firs kind orhogonal Chebyshev polynomials T k k which are eigenfuncions of singular Surm-Liouville problem: k T () Tk (). DAEs AND REDUCING INDEX I is well known ha he inde of DAEs is a measure of he degree of singulariy of he sysem and also widely regarded as an indicaion of cerain difficulies for numerical mehods. So, DAEs can be difficul o solve when hey have a higher inde, i.e., an inde greaer han [7]. In his case, an alernaive reamen is he use of inde reducion mehods [7-9]. In his secion, we briefly review he reducing inde mehod for DAEs which is menioned in [8]. Consider he linear (or linearized) semi-eplici DAEs: X (m) m A X BY q, (a) j j ( j) CX r, (b) where A, B and C are smooh funcions of, j nk kn f, A j nn () R, j,, m, B() R, C() R, n, k n and CB is nonsingular (DAE has inde m ) ecep possibly a a finie number of isolaed poins of. The n inhomogeniies are q() R and r() R. Now suppose ha CBis non-singular, from (a), we can wrie m (m) ( j) Y (CB) CX A jx q, j [, f ] () Subsiuing () ino (a) implies ha m ( m) ( j) I B( CB) CX A j X q j So, problem () ransforms o he over-deermined sysem: m ( m) ( j) I B( CB) C X A j X q, j () CX r, [, f ] Now sysem () can be ransformed o a full-rank DAE sysem wih n equaions and n unknowns wih inde m [8]. Here, for simpliciy, we consider problem () when m (problem has inde ). Theorem : Consider problem () wih inde, n and k. This problem is equivalen o he following inde -DAE sysem:
Applicaion of Chebyshev Approimaion 97 EX EX qˆ (4) such ha ba b a ba b a b b b q bq E, E, c c qˆ r and y (CB) C[X AX q]. (5) Proof: presened in [8]. In his paper, firs we implemen his proposed inde reducion mehod o linear semieplici DAEs. Then we employ he VIM using Chebyshev approimaion o solve he obained problem. Furhermore, we use some eamples o demonsrae he efficiency and effeciveness of he proposed mehod.. HE S VARIATIONAL ITERATION METHOD In his secion, we briefly review he main poins of he powerful mehod, known as he He s variaional ieraion mehod. This mehod is a modificaion of a general Lagrange muliplier mehod proposed by Inokui []. In he VIM, he differenial equaion L[u()] N[u()] g(), () is considered, where L and N are linear and nonlinear operaors, respecively, and g() is an inhomogenous erm. Using he mehod, he correcion funcional u N[u ~ n () u n () [L(u n (s)) n (s)] g(s)]ds (7) is considered, where is a general Lagrange muliplier, is he n -h approimae soluion and u~ n is a resriced variaion which means u~ n. In his mehod, firs we deermine he Lagrange muliplier ha can be idenified via variaional heory, i.e., he muliplier should chosen such ha he correcion funcional is saionary, i.e., u~ n (u n (), ). Then he successive approimaion u n, n of he soluion u will be obained by using any selecive iniial funcion u and he calculaed Lagrange muliplier. Consequenly, u lim u. I means ha, by he correcion n funcional (7) several approimaions will be obained and herefore, he eac soluion emerges as he limi of he resuling successive approimaions. To perform he VIM, in general, for an arbirary naural number, g () epress in Taylor series, g () gi (). (8) i In his paper, we sugges ha g () be epressed in Chebyshev series, n u n
97 M. Ghovamand, M.M. Hosseini and M. Nilli i g() a T (), (9) i i where T i () is he firs kind of orhogonal Chebyshev polynomial, T (), () T, T (), and in general, Tk Tk Tk, k. In he ne secion, his mehod is successfully applied for solving differenial-algebraic equaions. 4. TEST PROBLEMS In his secion, o show he abiliy and efficiency of he proposed mehod, some eamples are presened. In all he eamples, o simplify he compuaions, for an arbirary naural number, every coefficien funcion g() is epressed in Chebyshev series (9). The resuls are compared wih he case in which Taylor epansion is used wih he same number of erms. The algorihms are performed by Maple wih digis precision. Eample : Consider inde- problem: X AX By q, (a) CX r, (b) where and sin() A, B, q, C T, r() (e sin()), wih (), () and he eac soluions () e, () sin() and cos() y(). From Theorem, problem () can be convered o he inde- DAE: e sin() (a) () sin() (b) wih (), and (). To solve he new problem, we ransform he algebraic equaion (a) in he ieraive form wih respec o and by he He s variaional ieraion mehod and using (7), we consruc he correcion funcional in -direcion for he differenial equaion (b). Therefore, we obain he following (n) syse () e sin() () (a) () (n ) () () (s)[ (s) (s) ~ (s) sin(s)]ds (b)
Applicaion of Chebyshev Approimaion 97 where ~ is considered as a resriced variaion,i.e., ~. By aking he variaion from boh sides of he correcion funcional (b), we have (n) ~ () ~ (s)[ (s) (s) ~ (s) sin(s)]ds, or (n) ~ () ~ (s) (s) s [ (s) (s)] (s)ds. (n) By imposing ~, we obain he saionary condiions (s) s (s) (s). Therefore s (s) e. (4) By subsiuing he opimal value (4) ino funcional (b), we obain he following ieraion formula: (n) (n) e sin() e s [ () (s) () (s) ~ () (s) sin(s)]ds, wih n,,,, () and (). Now, we epand he coefficien funcions e and sin() a and e a Taylor series epansion wih v as follow: e 4 4 5 7 9 54 7 4 8 (a) (b) s 88 s by 9, () (5) (a) sin(), (b) e s 5 7 54 88 ( s) ( s) s 7 ( s) 9 54 So afer ieraions, (5) and () yield: 4 ( s) ( s) 7 4 4 ( s) ( s) 8 5 88 ( s) 9 (c). () () () () () () () () 4 4 4 5 7 54 4 4 88 5 9 88 99584 88 99584 In he alernaive mehod, we epanded he coefficien funcions e, sin() and Chebyshev series wih he same number of erms. Using (5) and (), afer ieraions, we obain s e by
974 M. Ghovamand, M.M. Hosseini and M. Nilli () () () () ().79( 4) 4.( ).7( 8) ().. 7.8 ( ) ().79( 4)..( 5) ()...4( 5) 9 9 For he sake of comparison, we have illusraed he absolue errors for () () and using boh Taylor and Chebyshev cases in figures and. These figures, obviously, show he superioriy of using Chebyshev approimaion insead of Taylor approimaion in his eample. I should be noed ha in order o increase he rae of convergence, we have used (n ) insead of in (5b) for compuing (n ). Fig.. Absolue errors of using Taylor polynomials for compuaion of (n) and... in E.. (n )
Applicaion of Chebyshev Approimaion 975 Fig.. Absolue errors of using Chebyshev polynomials for compuaion of (n ) (n) and... in E.. Eample : Consider inde- problem: X AX By q, (7a) CX r, (7b) where and A, C T e ( sin() ) B, q sin(), e (sin() cos() ) r() e ( sin()) wih () () and he eac soluions () e, () e sin() e and y(). By Theorem, he inde - DAEs (7) ransforms o he following inde- DAEs: g() (a) (8) g () g(), (b) wih () (), when g() e ( sin(), g () and g() e (cos() sin()). Similar o eample, we ransform he algebraic equaion (8a) in he ieraive form wih respec o and consruc he suiable correcion funcional in -direcion by using (7) for he differenial equaion (8a). Therefore, sysem (8) is epressed as
97 M. Ghovamand, M.M. Hosseini and M. Nilli (n) () g() (a) (n) s ~ (s) s ~ (s) g (s) ~ (9) () (s) (s) (s) g(s) ds, (b) where ~ and ~ denoe resriced variaions, i.e., ~ ~.To find he opimal value of in (9b), we have (n) () () (s) (s) s ~ (s) s ~ (s) g (s) ~ (s) g (s) ds. Therefore (n) () () (s) (s) s (s) (s)ds. (n) Imposing ~ yields he following saionary condiions: (s) s () (s). Therefore he opimal value of Lagrange muliplier is ( s). () By subsiuing his value ino correcion funcional (9b), he following ieraion formula is obained: (n) () g() (a ) (n) () (s) s ~ (s) s ~ (s) g (s) ~ () (s) g(s) ds (b) () wih n,,,..., () and (). We epand he funcions g (), g () and g () by Taylor series epansion a wih v 4.So afer 8 ieraions, () yields () () (8) (8) () () () () 4 4 5 4 5 () 884 884 5 4 799877 8858 799877 5 8858...... In he alernaive mehod, we epanded he coefficien funcions e, sin( ) and by Chebyshev series wih he same number of erms. Using (7) and (8), afer 8 ieraions, we obain () 4 ().74( 8)..8( 8) () () 9.99( ) 5.( ).97( 9) (8) 9 ().74( 8)..( 4) (8) 9 () 9.99( )..44( )... s e
Applicaion of Chebyshev Approimaion 977 For he sake of comparison, we have illusraed he absolue errors for (8) (8) and using boh Taylor and Chebyshev cases in able. This Table, obviously, shows he superioriy of using Chebyshev approimaion insead of Taylor (n ) approimaion in his eample. In his eample, we have used insead of in (b) o compue (n ). Absolue errors of using Taylor polynomials for compuaion of ~ and ~ in E.. ~ () () ~ () ()..5( 8).5( 7)..9( ).57( )..4( ).7( ).4.55( 8).88( 8).5 5.9( 7).8( ) Table ) ) 9 ) 4 ) ) ) The comparison beween he resuls menioned in Figures and and Table show he power of he proposed mehod of his paper, for hese eamples. 5. CONCLUSION Absolue errors of using Chebyshev polynomials for compuaion of ~ and ~ in E.. ~ () ( ~ () ().8( 5.8( 4).5( 4.79( 4).9( 4 9.7( 4) 4.( 4.8( 4).79( 4.57( ) The variaional ieraion mehod (VIM) has been successful for solving many applicaion problems. However, difficulies may arise in dealing wih deermining he componens u m, (7). To overcome hese difficulies he modified VIM is proposed using Chebyshev polynomials and is applied for solving differenial algebraic equaions in his paper. The resuls are compared wih he VIM using Taylor series. Numerical resuls show ha using Chebyshev approimaion is much more efficien han using Taylor approimaion which is more popular. The proposed mehod can be easily generalized for more funcional equaions.. REFERENCES [] J. H. He, Variaional ieraion mehod- a kind of non-linear analyical echnique: Some eamples, Inernaional Journal of Nonlinear Mechanics 4 99 78, 999. [] J. H. He, Variaional ieraion mehod for auonomous ordinary differenial sysems, Applied Mahemaics and Compuaion 4 5,. [] J. H. He, Variaional ieraion mehod _ some recen resuls and new inerpreaions, Journal of Compuaional and Applied Mahemaics 7-7, 7. [4] T. A. Abassy, Modified variaional ieraion mehod (nonlinear homogeneous iniial value problem), Compuers & Mahemaics wih Applicaions 59 9-98,.
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