The Generalized Laguerre Matrix Method or Solving Linear Differential-Difference Equations with Variable Coefficients

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1 Avalable at Appl. Appl. Math. ISSN: Vol. 9, Issue 1 (Jue 2014), pp Applcatos ad Appled Matheatcs: A Iteratoal Joural (AAM) he Geeralzed Laguerre Matrx Method or Solvg Lear Dfferetal-Dfferece Equatos wth Varable Coeffcets * Correspodg Author Z. Kalateh Bod ad S. Ahad-Asl Departet of Matheatcs Brad Uversty Brad, Ira salaahadasl@gal.co; ahd_kb@yahoo.co A. Aatae * Departet of Appled Matheatcs Faculty of Matheatcs K. N. oos Uversty of echology ehra, Ira atae@ktu.ac.r Receved: Jauary 14, 2013; Accepted: March 11, 2014 Abstract I ths paper, a ew ad effcet approach based o the geeralzed Laguerre atrx ethod for uercal approxato of the lear dfferetal-dfferece equatos (DDEs) wth varable coeffcets s troduced. Explct forulae whch express the geeralzed Laguerre expaso coeffcets for the oets of the dervatves of ay dfferetable fucto ters of the orgal expaso coeffcets of the fucto tself are gve the atrx for. I the schee, by usg ths approach we reduce solvg the lear dfferetal equatos to solvg a syste of lear algebrac equatos, thus greatly splfy the proble. I addto, several uercal experets are gve to deostrate the valdty ad applcablty of the ethod. Keywords: Geeralzed Laguerre atrx ethod; Operatoal atrces; Laguerre polyoals; Lear dfferetal-dfferece equatos wth varable coeffcets AMS-MSC 2010 No.: 33C45, 39A10 272

2 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) Itroducto Orthogoal polyoals play a proet role pure, appled ad coputatoal atheatcs, as well as the appled sceces ad ay other felds of uercal aalyses such as quadratures, approxato theory ad so o [Gautsch (2004), Dukl ad Xu (2001), Marcella ad Assche (2006) ad Askey (1975)]. I partcular, these polyoals play a sgfcat role the spectral ethods, whch have bee successfully appled the approxato of partal, dfferetal ad tegral equatos. hree ost wdely used spectral versos are the Galerk, collocato ad au ethods. her utlty s based o the fact that f the soluto sought s sooth, usually oly a few ters a expaso of global bass fuctos are eeded to represet t to hgh accuracy [Gottleb ad Orszag (1977), Boyd (2000), Cauto et al. (2006) ad (1984), refethe (2000), Hesthave et al. (2009) ad Be-yu (1996)]. We ote at ths pot that uercal ethods for ordary, partal ad tegral dfferetal equatos ca be classfed to the local ad global categores. he fte-dfferece ad fte-eleet ethods are based o local arguets, whereas the spectral ethods are the global class [She et al. (2011) ad Fuaro (1992)]. Spectral ethods, the cotext of uercal schees for dfferetal equatos, belog to the faly of weghted resdual ethods, whch are tradtoally regarded as the foudato of ay uercal ethods such as fte eleet, spectral, fte volue ad boudary eleet ethods. Also the lear DDEs wth varable coeffcets ad ther solutos play a aor role the brach of oder atheatcs ad arse frequetly ay appled areas. herefore, a relable ad effcet techque for ther soluto s extreely portat. he aalytc results o the exstece ad uqueess of solutos to the secod order lear DDEs have bee vestgated by ay authors [Agraval ad Orega (2009) ad Kg et al. (2003)], however the exstece ad uqueess of the soluto for DDEs uder these codtos s beyod the scope of ths paper. We assue that the DDEs whch we cosder ths paper wth ther codtos have solutos. Durg the last decades, several ethods have bee used to solve hgh-order lear DDEs such as Adoa's decoposto ethod [ Wazwaz (2010), Aatae ad Hussa (2007) ad (2010)], aylor collocato ethod [Gulsu et al. (2006), Gulsu ad Sezer (2006), Sezer ad Gulsu (2005) ad Gulsu ad Sezer (2005)], Haar fuctos ethod [Malekead ad Mrzaee (2006), Reha ad Abad (2007)], au ethod [Ortz ad Saara (1981), Vaa ad Aatae (2011) ad Ortz(1978)], Wavelet ethod [Dafu ad Xufeg (2007)], Hybrd fucto ethod [Hsao (2009)], Legedre wavelet ethod [Razzagh ad Yousef (2005)], collocato ethod based o Jacob, Laguerre ad Legedre polyoals [Ia et al. (2011), Vaa ad Aatae (2012) ad Aatae ad Vaa (2013)], aylor polyoal solutos [Sezer ad Dascoglu (2006)], Boubaker polyoal approach [Akkaya ad Yalcbas (2012)], ad Beroull polyoal approach [Erde ad Yalcbas (2012)]. I ths paper, we develop a ew ad effcet approach to obta the uercal soluto of the geeral lear DDEs wth varable coeffcets of the for (1) d d 1 ( ) ( 1) Ak, x y hk, x f k, Ak, 1 x y hk, 1 x f k, 1 k 1 k 1 ( ) ( ) ( ) ( ) d0 (0) +... A y ( h x f ) g, k1 k,0 k,0 k,0 0 x, 0, f, h, d 0, t 0,...,, k, k, t

3 274 Z. Kalateh Bod et al. wth the codtos ( k ) k y ( a ), 0,1,...,. (2) k 0 he a advatage of our work s ts cosderato of the geeral lear DDEs (1) wth respect to (2), whereas the other papers oly cosdered partcular cases of our geeral proble. Also usg the geeralzed Laguerre polyoals as the bass fuctos for uercal approxato whereas the classcal Laguerre polyoals are partcular cases of the, s aother advatage. he reader of our paper s orgazed as follows: I Secto 2, we troduce the propertes of geeralzed Laguerre polyoals ad ther basc forulato requred for our subsequet developet. Secto 3, s devoted to the operatoal atrces of the geeralzed Laguerre polyoals (dervatve ad oet) wth soe useful theores. Secto 4, suarzes the applcato of the geeralzed Laguerre polyoals to the soluto of proble (1) ad (2). hus, a set of lear equatos s fored ad a soluto of the cosdered proble s troduced. Secto 5, s devoted to approxatos by the geeralzed Laguerre polyoals ad a useful theore. I Secto 6, the proposed ethod s appled for three uercal experets. A applcato of the ethod for a hgher order lear dfferetal equato s preseted Secto 7. Fally, we ake a bref cocluso Secto 8. Note that we have coputed the uercal results by Matlab (verso 2013) prograg. 2. he Geeralzed Laguerre Polyoals I ths part, we defe the geeralzed Laguerre polyoals ad ther propertes such as ther Stur-Louvlle ODEs, three-ter recurso forula, etc. Let (0, ), the the Laguerre polyoals are deoted by L ( 1), ad they are the ege-fuctos of the Stur- Louvlle proble ( ) ( ) 0,, x 1 x x e x e L x L x x wth the egevalues [Fuaro (1992)]. 2 x Laguerre polyoals are orthogoal L ( ) space wth the weght fucto w ( x ) x e, satsfyg the followg relato ( 1) ( 1) w L ( ) ( ) ( ) 0 x L x w x dx,,, (3) where, s a Kroecker delta fucto. he explct for of these polyoals s the

4 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 275 for L( x ) E x, 0 where E 1. (4)! hese polyoals are satsfed the followg three-ter recurrece forula ( 1) L ( x ) (2 1 x ) L ( x ) ( ) L ( x ), 1 1 (5) L ( x ) 1, L ( x ) 1 x. 0 1 he case 0 leads to the classcal Laguerre polyoals, whch are used ost frequetly practce ad wll sply be deoted by L x. A portat property of the Laguerre polyoals s the followg dervatve relato [Fuaro (1992)]: 1 L ( x ) L ( x ). (6) 0 Further, ( L ) k ( ) are orthogoal wth respect to the weght fucto w k,.e., ( ) ( ) ( ) k ( )( ) k L ( ) ( ) k 0 x L x w k x dx k,, k where s defed equato (3). k A fucto as y x 2 ( ) Lw [0, ), ca be expressed ters of the geeralzed Laguerre polyoals y ( x ) al ( x ), 0 where the coeffcets a are gve by 1 a L x y x w x dx 0 ( ) ( ) ( ) ( ).

5 276 Z. Kalateh Bod et al. I practce, oly the frst 1ters of the geeralzed Laguerre polyoals are cosdered. he we have 0 y ( x ) a L ( x ) L ( x ) A, where the geeralzed Laguerre polyoals coeffcets vector A ad the geeralzed Laguerre ( polyoals vector L ) are gve by A a0 a1 a,,...,, ( ) ( ), ( ),..., ( ). ( ) L x L0 x L1 x L x Reark 1. Fro equato (1), for hk, 0, we coclude that ( ) k, k, 0 L ( h x ) E ( h ) x ; 1. (7) Now, fro reark 1 ad the followg theore 1, we ca obta the atrx relato betwee the ( ) ( ) ( ) geeralzed Laguerre polyoals space (set) { L ( h x ), L ( h x ),..., L ( h x )} ad stadard polyoal space (set) as the followg 0 k, 1 k, k, ad ( ) ( ) ( ) L0 ( hk, x ), L1 ( hk, x ),..., L ( hk, x ) K 1, x,..., x, (8) ( ) ( ) ( ) L0 ( hk, x ), L1 ( hk, x ),..., L ( hk, x ) 1, x,..., x, (9) where K ad are lower tragular atrces K,, 0,, ( hk ) E,, (10) ad 0,,, E,. (11)

6 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 277 heore 1. he atrces K ad are vertble f ad oly f hk, 0. Proof: For establshg the vertblty of atrx K, t s suffcet to show that Det ( K ) 0, where Det ( K ) s a deterat of the square atrx K. But because K s a lower tragular atrx, the we have Det ( K ) ( h ) E, 0 k, but fro hk, 0, t s suffcet to establsh that Det ( K ) E 0. 0 Now fro equato (4), t s ot dffcult to see that 0 E 0. he vertblty of atrx alog slar les of dscusso of atrx K s obvous. herefore, the proof s copleted. Now fro equatos (8) ad (9), we obta the followg portat atrx relato ( ) ( ) ( ) 1 0 k, 1 k, k, 0 1 L ( h x ), L ( h x ),..., L ( h x ) K L ( x ), L ( x ),..., L ( x ), 0. (12) 3. he Operatoal Matrces of the Geeralzed Laguerre Polyoals (Dervatve ad Moet) I ths secto, we preset the operatoal atrces of the geeralzed Laguerre polyoals (dervatve ad oet). o do ths, frst we troduce the cocept of the operatoal atrx.

7 278 Z. Kalateh Bod et al he Operatoal Matrx Defto 1. Suppose [ 0, 1,..., ], where 0, 1,..., are the bass fuctos o the gve terval [ ab, ]. he atrces E ad F are aed as the operatoal atrces of dervatves ad tegrals, respectvely, f ad oly f d ( t ) E ( t), dt ad x ( t ) dt F( t ). a Further assue g [ g 0, g1,..., g ], aed as the operatoal atrx of the product, f ad oly f ( x ) ( x ) G ( x ). (13) g I other words, to obta the operatoal atrx of a product, t s suffcet to fd g,, k the relato ( x ) ( x ) g ( x ), (14),, k k k 0 whch s called the learzato forula [Eslahch ad Dehgha (2011)]. Operatoal atrces are used several areas of uercal aalyses ad cotue to be portat varous subects such as tegral equatos [Razzagh ad Ordokha (2001)], dfferetal ad partal dfferetal equatos [Khellat ad Yousef (2006)], etc. Also ay textbooks ad papers have eployed the operatoal atrces for spectral ethods. Reark 2. he reaso for usg the equaltes (8), (9), (10) ad (11) s that for soe bases such as polyoal bass, the tegral of a polyoal wth degree, s a polyoal wth degree 1, so t caot be represeted wth a polyoal wth degree. A slar ferece ca be draw for the product of two bases. Reark 3. he reaso for usg three paraeters,, k, for the product of two fuctos ( x ) ad ( x ) s that the coeffcets g,,, k copletely deped o two fuctos ( x ) ad ( x ). Reark 4. I the geeral case, the coeffcet atrx G, ad the coeffcets g,, k of equatos (13) ad (14) respectvely, are dfferet for dfferet bases, ad the followg we obta these

8 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 279 coeffcets for the geeralzed Laguerre polyoals. o ths goal, we use the followg two portat forulas 0 L ( x ) L ( x ) c (,,, ) L ( x ), (15) ad 1 L( x ) L ( x ), 0 (16) where c (,,, ) 1. k 0k k k By cobg the relatos (15) ad (16), we obta k 0 L ( x ) L ( x ) g (,, ) L ( x ), where k 1 g (,, ) d (,, ), k 0 k ad k 1 d (,, ) c(,,, ). k 0 k So the cosdered coeffcets are obtaed. Now we preset the followg theore. heore 2. If we cosder the geeralzed Laguerre approxato the ( ) ( ) ( ) ( ) ( ), 0 y x a L x L x A

9 280 Z. Kalateh Bod et al. where ( ) ( ) ( ) x y ( x ) B L ( x ) G D A L ( x ), D, 1,, 0,, (17) ad G, ( ), 1,,, ( ), 1, 0, otherwse. (18) Proof: Frst, we obta the operatoal atrx wth respect to the dervatve operator. For ths goal, we ust obta a atrx D whch satsfy the followg forula L L L ( ) 0 ( ) 1 ( ) L L D L ( ) 0 ( ) 1 ( ), (19) but by usg equato (16), we ca obta the atrx D as the followg D, 1,, 0,. Now by -tes repeatg the forula (19), we ca obta the operatoal atrx wth respect to ( y ) as the followg L L L ( ) 0 ( ) 1 ( ) D L L L ( ) 0 ( ) 1 ( ). (20)

10 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 281 Also for obtag the operatoal atrx wth respect to the oet operator we ust obta a atrx G, whch satsfy the followg relato ( ) ( ) xl0 ( x ) L0 ( x ) ( ) ( ) xl1 ( x ) L1 ( x ) G, ( ) ( ) xl ( x ) L ( x ) (21) but by usg equato (5), we ca obta the atrx G as the followg G, ( ), 1,,, ( ), 1, 0, otherwse. Now by -tes repeatg the forula (21), we ca obta the operatoal atrx wth respect to x y ( x ), as the followg ( ) ( ) x L0 ( x ) L0 ( x ) ( ) ( ) x L1 ( x ) L1 ( x ) G. ( ) ( ) x L ( x ) L ( x ) (22) Now usg forulae (20) ad (22), yeld ( ) 0 ( ) ( ) ( ) ( ) 1 ( ) k k ( ) k 0 x y x a x L x A x so the proof s coplete. L ( ) ( ) ( ) ( ) ( ) 0 ( ) L1 ( ) A G D G D A L x L ( ) L L L ( ),

11 282 Z. Kalateh Bod et al. heore 3. If cr ad 0, the c 0 1 L ( x c), L ( x c),..., L ( x c) W L ( x ), L ( x ),..., L ( x ), where W s a lower tragular atrx c Wc (, ), 0,, D,, (23) ad D (, ) k k E c, k where E s defed equato (4). Proof: Fro equato (7), we have ( ) L ( x c) E ( x c) E c x, so f we defe D (, ) k k E c, k therefore we have ( ) (, ) 0 L ( x c) D x. (24) Usg obtaed result fro equato (24), we have ( ) ( ) ( ) L0 ( x c), L1 ( x c),..., L ( x c) W c 1, x,..., x, 0, where W s gve equato (23), ad usg forula (12), we obta c ( ) ( ) ( ) 1 ( ) ( ) ( ) 0 1 c 0 1 L ( x c), L ( x c),..., L ( x c) W L ( x ), L ( x ),..., L ( x ), 1 therefore W s the shft operatoal atrx ad the proof of theore s coplete. c

12 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 283 Now by theore 3, we ca obta the odfed verso of theore 2, for geeralzed Laguerre polyoals as kt, ( k ) ( ) 1 ( x y h ) k, tx f k, t B L ( x ) G D A W f K L ( x ), he Method of Soluto I ths secto, we descrbe our ew approach for solvg the lear dfferetal-dfferece equatos wth varable coeffcets (1), wth respect to the codtos (2). Our approach s based o approxatg the exact soluto of equato (1), by trucatg the geeralzed Laguerre expaso as ( ) ( ) ( ) ( ) ( ), 0 y x a L x L x A (25) where A [ a0, a1,..., a ], ad ( ) ( ) ( ) ( ) L ( x ) L0 ( x ), L1 ( x ),..., L ( x ). Also we assue that the coeffcets A, have the aylor seres expaso the followg for k ( ) k, k, 0 A ( x ) e x. (26) Now by substtutg equatos (25) ad (26) to equato (1), we obta s s 1 1 ( ) ( ) ( 1) ( 1) ek, x y hk, x f k, ek, x y hk, 1 x f k, 1 k 1 0 k 1 0 s0 0 k 1 0 ( ) ( )... e x y ( h x f ) f ( x ), (0) ( ) k, k,0 k,0 (27) ( herefore, fro equato (27), we ust splfy ) ( ) x y x as the followg ( ) ( ( ) ( ) ( ) k, k, ) ( k, k, ) ( ) ( ) 0 x y h x f a L h x f L x B fkt, 1 ( ) G D A W K L x ( ), (28) where D ad, G are defed equatos (17) ad (18), respectvely. Also we approxate the rght had sde of equato (1), as

13 284 Z. Kalateh Bod et al. ( ) ( ) ( ) ( ) ( ), 0 f x b L x L x B (29) where,,...,, ad B b0 b1 b ( ) ( ) ( ) ( ) L ( x ) L0 ( x ), L1 ( x ),..., L ( x ). Usg equatos (28) ad (29), to equato (27), we obta L x e B e B e B s s s ( ) ( ) ( ) ( 1) ( ) (0) ( ) ( ), k ( ), k ( 1)..., k (0) k1 0 k1 0 k1 0 ( ) ( ) L F L B. Now, fro lear depedecy of the geeralzed Laguerre polyoals, we coclude that F B, (30) where F [ f 0, f 1,..., f ]. herefore, fro detty (30), we have a syste of 1 algebrac equatos of 1 ukow a 0,..,. Fally, we ust obta the correspodg atrx for of the coeffcets boudary codtos. For ths purpose fro equato (2), the values ( y ) ( a ) ca be wrtte as ( ) ( ) y a L a D A a ( ) ( ), [0, ). (31) Substtutg equato (31), the boudary codtos (2) ad the splfyg t, we obta the followg atrx for ( l) ( ) b, l y ( a ) L ( a ) b, l D A l, a [0, ). (32) 0 0 Now fro equatos (30) ad (32), we have 1 algebrac equatos of 1 ukow coeffcets. hus for obtag the ukow coeffcets, we ust elate arbtrary equatos fro these 1equatos. But because of the ecessty of holdg the boudary codtos, we elate the last equatos fro equalty (30). Fally, replacg the last equatos of equalty (30) by the equatos of equalty (32), we obta a syste of 1 equatos of 1 a 0,,. ukows

14 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) Approxatos by Geeralzed Laguerre Polyoals Now ths secto, we preset a useful theore whch shows the approxatos of fuctos by the geeralzed Laguerre polyoals. For ths purpose, let us defe { x 0 x } ad J spa { L ( x ), L ( x ),..., L ( x )}. ( ) ( ) ( ) ( ) N 0 1 he y x ( ) ( ) 2 ( ) w ( ) orthogoal proecto N : L ( ) JN s a appg a way that for ay 2 L 2 ( ) L ( ), we have ( y ) y, 0, J. ( ) ( ) N N Due to the orthogoally, we ca wrte costats are the followg for N 1 ( ) ( ) N k k k 0 ( y ) c L ( x ), where c ( 0,1,..., N 1) 1 c y ( x ), L. ( ) ( ) k 2 L w ( ) k ( I the lterature of spectral ethods, ) N ( y ) s aed as the geeralzed Laguerre expaso of y ad approxates y o (0, ). I the spectral ethods, by substtutg the ( geeralzed Laguerre expaso ) N ( y ) the DDEs ad ther boudary codtos, we obta a resdual ter whch s sybolcally showed by Res ( x ) as a fucto of x, N, ad. Dfferet strateges for zg a resdual ter Res ( x ), lead to the dfferet versos of spectral ethods such as Galerk, au ad collocato ethods. For stace, the collocato ethods the resdual ter Res(x) s vashed partcular pots aed as collocated pots. Also estatg the dstace betwee y( x ) ad ts geeralzed Laguerre expaso as easured the weghted or. w s a portat proble uercal aalyss. he followg theore provdes the basc approxato results for geeralzed Laguerre expaso. heore 4. We have l d dx l ( ) ( l)/2 d ( N ( y ) y ) ( l ) N y ( x ) w, 0 l, y B w ( )( ), dx where d y B y L L l dx l 2 2 ( )( ) { : w w ( ), 0 l l }.

15 286 Z. Kalateh Bod et al. heore 4 states that f the fucto y ( x ) B( )( ), or other words the fucto y( x ), s suffcetly cotuous, we recover spectral decay of the expaso coeffcets,.e., ck decays faster tha ay algebrac order of 1/ N. hs result s vald depedet of specfc boudary codtos o y x. Proof: See Fuaro (1992). 6. he est Experets I ths secto, soe uercal experets are gve to llustrate the propertes of the ethod ad all of the were perfored o a coputer usg a progra wrtte Matlab Experet 1. Cosder the followg secod-order lear dfferetal equato wth varable coeffcets [Kesa (2003)]: y x xy x xy x x x x 2 ( ) ( ) ( ) 1, 1 1, y (0) 1, y (0) 2 y (1) y ( 1) 1. (33) Now we approxate the exact soluto of equato (33), by ( ) 6 ( ) ( ) ( ) ( ), 0 y x a L x L x A where A = [a 0,a 1,...,a 6]. Also we expad the rght had sde of equato (33) as 1 x x b L ( x ) L ( x ) B, 2 6 ( ) ( ) where B 0 1,2,3/ 2,0,0,0,0. 2 Frst, we reduce equato (33) to the followg atrx ford GD G A B. Also ts boudary codtos as 6 0 a L (0) L (0) A 1. ( ) ( )

16 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 287 Fgure 1. he coparso betwee exact ad approxate solutos of 14 3/ 2 of experet 1 able 1. he coparso betwee preset ethod ( 6 ad 3) approxate solutos of aylor ethod 4 of experet 1. ad ad the x Preset ethod aylor ethod ad 6 0 a L (1) L (1) A 1. ( ) ( ) By pleetato of our ethod whch s preseted secto 4, ad also after the augeted atrces of the syste ad boudary codtos are coputed, we obta the uercal solutos. he coparso betwee our ethod ad aylor ethod s show table 1. Also the approxate ad exact solutos are show Fgure 1.

17 288 Z. Kalateh Bod et al. Experet 2. Cosder the secod-order lear dfferetal equato: (34) 2 ( x 1) y ( x ) y ( x ) 1, wth the boudary codtos y(0)=0, y(1)=1.he exact soluto of equato (34) s y ( x ) x. Now we approxate the exact soluto of equato (34), by ( ) 5 ( ) ( ) ( ) ( ), 0 y x a L x L x A where A = [a 0,a 1,...,a 5]. Also we expad the rght had sde of equato (34) as 5 ( ) ( ) 1 bl ( x ) L ( x ) B, 0 where B 1,0,0,0,0, Now, frst we reduce equato (34) to the followg atrx for G D D D A B. Also ts boudary codtos as ad a L (0) L (0) A 0, ( ) ( ) a L (1) L (1) A 1. ( ) ( ) By pleetato of our ethod whch s preseted secto 4, ad also after the augeted atrces of the syste ad boudary codtos are coputed, we obta the soluto y(x)=x, whch s the exact soluto. Experet 3. Cosder the thrd-order lear dfferetal equato: ( ) ( ) 2, 2 x y x y x

18 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 289 y(0) = 0, y(1) = 1, y(-1)=1. (35) Now we approxate the exact soluto of equato (35) by ( ) 5 ( ) ( ) ( ) ( ). 0 y x a L x L x A Also we expad the rght had sde of equato (35) as 5 ( ) ( ) 2 bl ( x ) L ( x ) B, 0 where B [2,0,0,0,0,0]. Now we ust reduce equato (35) to the followg atrx for G D D A B ad also ts boudary codtos as ad a L (0) L (0) A 0, ( ) ( ) a L (1) L (1) A 1, ( ) ( ) a L ( 1) L ( 1) A 1. ( ) ( ), After the augeted atrces of the syste ad boudary codtos are coputed, we obta 2 the soluto y ( x ) x, whch s the exact soluto. 7. Applcato of the Method for the Hgh-Order Lear Dfferetal Equato I ths secto, we report the uercal results obtaed for a hgh-order lear dfferetal equato by the aforeetoed procedure. hs shows that t s straghtforward to exted the ethod to the hgh-order lear dfferetal equatos as follows. Experet 4.

19 290 Z. Kalateh Bod et al. Let us cosder the eghth-order lear dfferetal equato [Kurt ad Golbaba ad Javd (2007)]: Sezer (2008) ad y (8) ( x ) y ( x ) 8 e x, 0 x 1, wth the tal codtos y (0) 1, y (0) 0, y (0) 1, y y y (4) (5) (0) 2, (0) 3, (0) 4, y (0) 5, y (0) 6. (6) (7) able 2. he coparso betwee the exact ad approxate solutos of HPM, MDM ad preset ethods of experet 4 x Exact Preset ethod HPM ethod MDM ethod

20 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 291

21 292 Z. Kalateh Bod et al. Fgure 2. he coparso betwee the exact ad approxate solutos of 12 2 of experet 4 able 3. he coparso betwee the exact ad approxate solutos of preset ad aylor ethods of experet 4 x Exact Preset ethod aylor ethod x he exact soluto of ths equato s y ( x ) (1 x ) e. By pleetato of our ethod whch s preseted secto 4, ad also after the augeted atrces of the syste ad boudary codtos are coputed, we obta the uercal solutos. he coparso betwee our ethod ad other uercal ethods are show tables 2 ad 3. Also the exact ad approxate solutos are show fgure 2. We see that our ethod, HPM ad MDM ethods obta better results tha the other ethods for ths experet. hese ethods rather tha the aylor polyoal set obta better results ear the corer of terval. I other words, the teror pots betwee 0 ad 1, the aylor ethod gves better results. hs atter s see by [Aatae ad Hussa (2007) ad (2010)] also, whch s due to the affty of aylor seres to the org. ad

22 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) Cocluso I ths paper, we have troduced a ew ad effcet approach for uercal approxato of the lear dfferetal-dfferece equatos. he ethod s based o the approxato of the exact soluto wth the geeralzed Laguerre polyoals approxato wth varable coeffcets by aylor seres expaso. Ipleetato of the ethod reduces the proble to a syste of algebrac equatos. Soe test experets are preseted for showg the accuracy ad effcecy of the preset ethod wth the other ethods such as HPM, MDM ad aylor seres. Applcato of the ethod for uercal soluto of hgh-order lear dfferetal equatos s also cosdered. I addto, we would lke to ephasze that the a portace of the preset schee s cosderg the geeral lear DDEs (1) ad (2), whereas the other auscrpts oly cosdered the partcular cases of our geeral proble. Further, usg the geeralzed Laguerre polyoals as the bass fuctos for uercal approxato whereas the classcal Laguerre polyoals are partcular cases of the s aother advatage of the preset study. Ackowledgets We are grateful to the aoyous revewers ad the edtor Professor Dr. Alakbar Motazer Haghgh for ther helpful coets ad suggestos whch deed proved the qualty of ths auscrpt. REFERENCES Agraval, R.P. ad Orega, D.O. (2009). Ordary ad Partal Dfferetal Equatos, Sprger. Akkaya,. ad Yalcbas, S. (2012). Boubaker polyoal approach for solvg hgh-order lear dfferetal-dfferece equatos, AIP Coferece Proceedgs of 9th teratoal coferece o atheatcal probles egeerg, Vol. 56, pp Aatae, A. ad Hussa, S.S. (2007). he coparso of the stablty of decoposto ethod wth uercal ethods of equato soluto, Appl. Math. Coput., Vol. 186, pp Aatae, A. ad Hussa, S.S. (2010). he barrer of decoposto ethod, It. J. Cotep. Math. Sc., Vol. 5, pp Aatae, A. ad Vaa, S.K. (2013). Nuercal soluto of fractoal Fokker-Plack equato usg the operatoal collocato ethod, Appl. Coput. Math., Vol. 12, pp Askey, R. (1975). Orthogoal Polyoals ad Specal Fuctos. SIAM-CBMS, Phladelpha. Be-yu, G. (1996). he State of Art Spectral Methods. Hog Kog Uversty. Boyd, J.P. (2000). Chebyshev ad Fourer Spectral Methods. Dover Publcatos, Ic, New York. Cauto, C., Hussa, M.Y., Quartero, A. ad Zag,.A. (1984). Spectral Method Flud Dyacs, Pretce Hall, Egelwood Clffs, NJ. Cauto, C., Hussa, M.Y., Quartero, A. ad Zag,.A. (2006). Spectral Methods: Fudaetals Sgle Doas, Sprger-Verlag. Dafu, H. ad Xufeg, S. (2007). Nuercal soluto of tegro-dfferetal equatos by usg

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24 AAM: Iter. J., Vol. 9, Issue 1 (Jue 2014) 295 cal soluto of olear dfferetal equatos, Coputg, Vol. 27, pp Razzagh, M. ad Ordokha, Y. (2001). Soluto of olear Volterra Haerste tegral equatos va ratoalzed Haar fuctos, Math. Prob. Eg., Vol. 7, pp Razzagh, M. ad Yousef, S.A. (2005). Legedre wavelets ethod for the olear Volterra- Fredhol tegral equatos, Math. Coput. Sul., Vol. 70, pp Reha, M. H. ad Abad, Z. (2007). Ratoalzed Haar fuctos ethod for solvg Fredhol ad Volterra tegral equatos, Coput. Appl. Math, Vol. 200, pp Sezer, M. ad Dascoglu, A.A. (2006). aylor polyoal solutos of geeral lear dfferetaldfferece equatos wth varable coeffcets. Appl. Math. Coput., Vol. 174, pp Sezer, M. ad Gulsu, M. (2005). Polyoal soluto of the ost geeral lear Fredhol tegro-dfferetal-dfferece equato by eas of aylor atrx ethod, It. J. Coplex Varables, Vol. 50, pp She, J., ag,. ad Wag, L.L. (2011). Spectral Methods Algorths, Aalyss ad Applcatos, Sprger. refethe, L.N. (2000). Spectral Methods Matlab, SIAM, Phladelpha, PA. Vaa, S. K. ad Aatae, A. (2012). A uercal algorth for the space ad te fractoal Fokker-Plack equato, It. J. of Nuer. Meth. for Heat & Flud Flow, Vol. 22, pp Vaa, S.K. ad Aatae, A. (2011). au approxate soluto of fractoal partal dfferetal equatos, Coput. Math. Appl., Vol. 62, pp Wazwaz, A.M. (2010). he cobed Laplace trasfor-adoa decoposto ethod for hadlg olear Volterra tegro-dfferetal equatos, Appl. Math. Coput., Vol. 216, pp