The Matrix Padé Approximation in Systems of Differential Equations and Partial Differential Equations

Transcription

1 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna The Marx Padé Approxmaon n Sysems of Dfferenal Equaons and Paral Dfferenal Equaons C PESTANO-GABINO, C GONZΑLEZ-CONCEPCION, MC GIL-FARIÑA Deparmen of Appled Economcs, Unversy of La Launa, 87 La Launa, SPAIN cpesano@ulles, coonzal@ulles, ml@ulles Absrac: In [] we presened a echnque o sudy he exsence of raonal soluons for sysems of lnear frsorder ordnary dfferenal equaons The mehod s based on a raonaly characerzaon ha nvolves Marx Padé Approxmans Moreover he man deas were only appled n he numercal resoluon of a parcular paral dfferenal equaon Ths paper may be consdered as an exenson of [], n he sense ha we propose fundamenal marces drecly for lnear m-order ordnary dfferenal equaons whou man a ransformaon o an equvalen sysem of frs order In addon, we ncrease s feld of applcaons o parcular soluons of he menoned sysems and o Paral Dfferenal Equaons Key-Words: Sysems of Dfferenal Equaons, Paral Dfferenal Equaons (PDE), Marx Padé Approxmaon (MPA), raonal soluons, mnmum derees (md) AMS subec classfcaon: A, A5, 5A5 Inroducon We wll use he marx noaon n Sysems of Lnear m-order Ordnary Dfferenal Equaons as follows Consdern A :D R C mxn (,m) and Y:D R C n+, le Y (m ()=A () Y (m- ()+A () Y (m- ()+ + A m- () Y()+A m () Y() () be an homoeneous sysem of lnear m-order ordnary dfferenal equaons Normally, o resolve he sysem () we ransform no an equvalen one of frs order, by chanes of varables We wll resolve laer bu no man use of ha ransformaon Defnon : F() s a fundamenal marx of () f any soluon of () can be wren as a lnear combnaon of he columns of F() Obann a fundamenal marx s essenal o solve sysems of dfferenal equaons However here s no procedure o fnd from any marx funcons A (), A ()A m () In hs paper we consder ha he elemens of he marces are analyc funcons I s neresn o noe ha we do no ae no accoun he crcle of converence of power seres, when we consder raonal funcons Wha s mporan s o now her poles ([]) Proposon : If A (), A ()A m () are connuous marx funcons hen () has a fundamenal marx of dmenson n x nm Proof: The se of soluons for a homoeneous sysem of lnear frs-order ordnary dfferenal equaons, X()=A() X(), wh A() a connuous marx funcon of order n, s a lnear space of dmenson n ( has a fundamenal marx of dmenson nxn) ([5]) By chanes of varables (V ()=Y (, =,m-) we ransform () no he equvalen sysem of frs order ha follows: I Y () Y() I V () V () = I V () V () m m A m() A m () A m () A () ) m (), Le G()= ( (), be a fundamenal marx for () such ha G()=I nm x nm and : D R C nxn, e, he nm columns of G() consue a base for he se of soluons of () Noe ha ()= ( (), e, he -h row of marces n G() s he (-)-h dervave of he frs row of marces n G() Consdern ()=, we wre he nal condon = G()=I nmxnm as follows: - =I nxn and = f -, for =,m;, m- () Noe ha n () we consder only coeffcens of (), (,m) Le S be he se of soluons of (), he nm columns of F() ( () () m ()) -wh he condon ()- consue a base of he lnear space S Noe ha F() s a fundamenal marx of dmenson n x nm of () Gven he formal power seres, A ()= A ( ), = A C mxn for,m,, we calculae recursvely he coeffcens of he seres for a fundamenal marx of ISSN: Issue 6, Volume 7, June 8

2 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna (), F()= F ( ), F C nxmn, as follows (whou = consdern he equvalen sysem of frs-order) Subsun F (h ()= = ( + h)( + h )( + )F ( ), h=, m, + h n F (m ()=A ()F (m- ()+A ()F (m- ()+ + A m- ()F()+A m ()F() Laer on, we wll explan more explcly he recurrence relaon for he specfc examples we are on o sudy Marx Padé Approxmans and Raonal Funcons Once he formal power seres for a fundamenal marx (or for a parcular soluon) s obaned, s of praccal neres o e he heorec funcon assocaed o he seres I s evden ha hs am s unaanable n eneral and may only be oban n ceran problems In hs secon we wll consder Marx Padé Approxmaon (MPA) resuls o deermne here s a raonal soluon and, f so, oban mnmum derees - n ceran sense- of he polynomals nvolved We denoe as F any formal power seres, wh marx coeffcens as follows: mxn F() = f f = C C () Suppose ha here exs marx polynomals: Q (z) = h h bz = h N h(z) = n z n = P(z) = D(z) = = = a z dz q C p C C d C mxn mxn nxn mxm para =,,,h para =,,,h para =,,, para =,,, where p =I nxn, d =I mxm, F()-Q h ()P - ()=O( h++ ) and F()-D - ()N h ()=O( h++ ), Q h P - s herefore sad o be a rh Marx Padé Approxman (rh MPA) whch s denoed R [h/] F ; smlarly, D - N h s sad o be a lef Marx Padé Approxman (lef MPA), whch s denoed L [h/] F We shall use { L [h/] F } ({ R [h/] F }) o denoe he se of all possble approxmans L [h/] F ( R [h/] F ) As a consequence of he defnon we can say ha: * R [h/] F exss, e, { R [h/] F }, f and only f, he follown sysem has a soluon: f h-+ p +f h-++ p - ++f h+- p =-f h+, =, RS(h,) * L [h/] F exss, e, { L [h/] F }, f and only f he follown sysem can be solved: d f h-+ +d - f h-++ ++d f h+- =-f h+ =, LS(h,) In boh cases s assumed ha f =, < Resolvn LS(h,) and RS(h,) we oban he coeffcens of he denomnaors of L [h/] F and R [h/] F, respecvely The coeffcens of he numeraors can be obaned by n =c, n =d c +d c n h =d h c ++d c h for he lef MPA and q =c, q =c p +c p q h =c p h ++c h p for he rh MPA Noe ha he MPA do no necessarly have o exs Furhermore, unle he scalar case, f one of hem does exs s no necessarly unque [] If here s no need o dsnush beween he lef and rh MPA because hey are dencal, we wll use [h/] F o denoe he MPA of derees (h,) o F and {[h/] F } o he se of all possble approxmans [h/] F Varous ypes of mnmaly relaed o he derees of he polynomals ha nervene n a raonal funcon have been defned n leraure Tan as our sarn pon he dea of havn a ype of on mnmaly for he wo polynomals ha nervene n a raonal funcon, he concep lef mnmum derees n hs wor s defned as follows Defnon : We say ha he derees p and q of he marx polynomals A p and B q, such ha F()= A ()B () and A p ()=I, are lef mnmum derees p q (lef md), f for any oher wo marx polynomals D and N h of derees and h respecvely whch verfy: D () = I and A p - () B q ()=D - ()N h (), follows ha: h<q mples >p and <p mples h>q In he same way we can defne rh md, wh he nvered polynomal o he rh Some Prevous Resuls ([]) Table: Raonaly and Mnmaly We ve he follown defnon assocaed wh () Defnon : For neers and, and >, le M(, ) = ( f + h+ ) h, =, L L M (, ) = (f ),,s+ r sr + h+ h,= M5 (, ) = (f ), R +,s+ r sr + h+ h,= M (, ) = (f ) s+ r, sr + h+ h,= M5 (, ) = (f ) R s+ r,+ sr + h+ h,= By convenon, f he ran of hese marces s zero for any N Defnon : For any nonneave neers,, le T(,)=ran(M(,)) We wll dsplay hese quanes n an nfne wo-dmensonal able, Table, where and serve o enumerae he columns and he rows respecvely Defnon 5: We defne he sared bloc R o be he follown subse of N : R={(,) N /ran(m(,h))=ran(m(+,h+)) for any N, and h } ISSN: Issue 6, Volume 7, June 8

3 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna Alhouh hs se seems raher absrac, he meann of "sared" becomes clear observn Tables n examples below To ben wh, we can propose a mehod, usn Table, o deermne wheher or no a marx seres sems from a raonal funcon Ths can be se ou as follows Raonaly F s raonal f and only f n he boom rh par of Table we can mar a leas one NW-SE daonal of nfne sze where all s boxes have he same value I s mporan o sae ha, here are several such daonals, wh her unon concdn wh R Noe ha, whn R, boxes of dfferen daonals can have dfferen values Furhermore, s assured ha f (a,b) R hen he formal power seres of F and [a/b] F s he same However, (a,b) are no necessarly lef or rh md In pracce, he Table ha we can consruc s "fne" However, n some applcaons where a rule for he formaon of he coeffcens of he seres F s nown, ceran relaons beween he marces ha defne he elemens of Table can help valdae he properes o an nfne sze Oherwse, we can only say he avalable coeffcens of F concde wh he coeffcens of a raonal funcon of ceran derees Mnmaly The boxes (,) such ha T(,) R, T(-,) R and T(,-) R are called corners of R In parcular suaons we can say ha a corner of R corresponds wh a par of lef or rh md, for nsance: Propery : If (,) R, (-,) R and T(,)=n hen (-u,-v) s no a par of rh md for any u, v such ha u and v Propery : If (,) R, (-u,) R, (,-v) R, for any u, v such ha u and v, and hey are no pars of lef (rh) md, hen (,) s a par of lef (rh) md Table : Mnmaly If F s raonal, here are (and we can fnd) ceran derees r and s assocaed o wo pars of marx polynomals whch represen he funcon n raonal form n wo ways, ha s o say, wh he nvered polynomal of deree r, mulpled o he rh or lef However, may be ha r and s are neher lef nor rh md for represenn F snce here are cases n whch, for nsance, he funcon s dencal o an approxman L [/] F bu he approxman R [/] F does no exs, or vceversa; n hs case, ran(m(,)) ran(m(+,+)) and hus Table would no provde nformaon concernn he lef or rh approxman For hs reason, we presen Table -one for he lef and one for he rh approxman-, whose srucure reflecs he possble combnaons of md (lef and rh, respecvely) for represenn F Frsly we presen he follown defnons Defnon 6: Gven (s,r) R, for neer,, such ha s and r, le L T sr (,)=, f ran( L M sr (,))= ran( L M5 sr (,)), or oherwse L T sr (,)= We wll dsplay hese quanes n a fne wo-dmensonal able, Table for lef approxman, whch has s+ columns and r+ rows Defnon 7: We denoe as a sared bloc L R sr, wh (s,r) R, he follown subse of N : L R sr ={(,) N / s, r and ran( L M sr (,))=ran( L M5 sr (,))} To defne Table for rh approxman we mus consder R M sr (,) and R M5 sr (,) nsead of L M sr (,) and L M5 sr (,), respecvely We only expound he heory for he lef case an no accoun ha for he rh case s smlar Propery : F s a raonal funcon dencal o L [q/p] F where q and p are lef md, q s and p r, f and only f, L T sr (q,p)=, L T sr (q-,p)= and L T sr (q,p-)=, ha s, (q,p) s a corner of L R sr The sysem correspondn o he denomnaor of he elemen of { L [q/p] F } ha concdes wh F s: d p f q-p+ +d p- f q-p++ ++d f q+- =-f q+ =,s+r-q whose assocaed marx s L M sr (q,p) Then we wll consder he resuls of MPA o sudy possble raonal soluons of dfferenal equaons sysems and of paral dfferenal equaons Raonal Soluons for Sysems of Ordnary Dfferenal Equaons In order o oban a clear undersandn, s mporan o read he follown remars relan o possble raonal soluons of he sysem () - Even f A (), A ()A m () are no raonal here may exs raonal soluons For nsance, he sysem e e + Y()= + + Y() sn sn has a raonal soluon, Y()= The fac ha here exss raonal soluon does no mply he fundamenal marx s raonal Noe ha enerally he lnear combnaon of no raonal funcons could be a raonal funcon, for nsance, f x()= cos and y()= cos + 8 he lnear 6 combnaon x()+y() s he raonal funcon - The sysem () may have a fundamenal marx wh ISSN: Issue 6, Volume 7, June 8

4 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna some raonal elemens and ohers whch are no Nex we proceed o llusrae dfferen cases Our am n hs secon s o deec raonal fundamenal marces (whch are recanular when he order of he sysem s reaer han ) In oher case, f hs s no possble, hen o deec parcular raonal columns or parcular raonal soluons If F() s raonal hen any soluon of () s raonal complyn wh he ams However, when F() s no raonal and s n analycal form would be neresn o now f here exs any nal condons, such ha he assocaed parcular soluon o be raonal I s no easy o fnd he answer o hs queson n eneral cases We sar wh an example where he fundamenal marx s no raonal bu he sysem does have raonal soluons Example Le Y()=C() Y() be a dfferenal + ( ) ( ) sysem where C()= Consdern 8 6 ( + ) ( + ) C()= C, we oban recursvely he coeffcens of = he seres for a fundamenal marx F()= F as follows: = = = = = ( )( )F ( C )( F ) (5) Then, an no accoun (), F = and F =, and consdern (5) we calculae he oher coeffcens usn he follown expresson: F + =( C F ) /( + )( + ), (6) F : Table for F() Noe ha F() s no raonal of orders (p,q) such ha p<5, q<5 Then we have calculaed Tables for each column of F() -four ables- and, moreover, Tables for each elemen of F() -eh ables- In all cases, ables do no correspond o a raonal funcon However, he sysem does have raonal soluons They are lnear combnaons of he columns of he fundamenal marx F() For nsance, consdern / K =, we oban he f F: Table of funcon Y()=F() K Ths able ndcaes ha Y can be represened as lef and rh approxmans of he se {[/] Y } Tan no accoun ha T(,)= and Properes and, (,) s a par of rh md The represenaon of R [/] Y s Y()= Noe ha we have solved RS(,) o oban he coeffcens of he denomnaor Alhouh we have a represenaon of he soluon, ou of curosy we have calculaed f F : Table (for lef approxman) Table ndcaes ha (,) s a par of lef md The represenaon of L [/] Y s: Y()= 5 5 Noe ha s he parcular soluon correspondn o he nal condons Y = ( 5 5 ), Y = ( 5 ) I s obvous ha any soluon wh nal condons Y = c c, Y = 5c, c C, s raonal wh derees (,) The follown example could be solved wh wo ndependen equaons However, we consder as a sysem o llusrae a fundamenal marx wh raonal columns and non-raonal columns Example Le Y()=C() Y() be a dfferenal sysem where C()= In he same way as he las example, consdern (6) we oban he coeffcens of a fundamenal marx Is Table s n f ISSN: Issue 6, Volume 7, June 8

5 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna F : Table We can see ha he fundamenal marx s no raonal wh orders conaned n he able However, Tables for each column of he fundamenal marx are n f5 F 5: Tables for each column Column Column Column Column These ables ndcae ha he frs column of he fundamenal marx s no raonal; he hrd and fourh columns are raonal (polynomal) Specfcally + and + Reardn Table for column, could be he able of a raonal funcon wh deree n he numeraor and n he denomnaor, bu maybe he bloc wh dsconnuous lne s a fne sared bloc ([6]) To now f he bloc wh dsconnuous lne s R we ncreased he able as f 6 F Noe ha, he menoned bloc s a fne sared bloc Therefore, he funcon s no raonal for derees consdered n hs able We have ncreased ables for columns, and and we reaffrm wha we have commened abou hese columns Raonal Soluons for Paral Dfferenal Equaons In hs secon we am o solve paral dfferenal equaons combnn MPA wh oher nown mehods, n parcular, Galern and Fne Dfferences Mehod of Galern and MPA In eneral, he mehods of Galern [] are used n problems where here s an unnown funcon o be deermned Of course, paral dfferenal equaons are of hs ype In hs mehod we ae a se of basc funcons and ry o solve he equaon, presumn ha he soluon s a suable combnaon of hese basc funcons Somemes hs soluon s no conssen Dependn on he approach consdered we oban a dfferen approxmaed soluon In he follown examples we combne he mehod of Galern and MPA o solve he equaons Example Le us consder he follown hyperbolc equaon: u = u x (7) ( + ) u(x,)=sn πx x u(,) = u(,) = > We propose as soluon he sum: u(x,)= n v ()w (x) (8) We wll choose some basc funcons of x, ha s, w, w w n Noe ha for =, (8) only depends on w (n) because v () (n) are consans Due o he fac ha u(x,) = sn πx, we preend ha sn πx o be a lnear combnaon of w (n) Moreover, f w ()=w ()= (n) he condons u(,)=u(,)= (>) are verfed wh he enave soluon (8) Subsun (8) n (7), he resul s: n (v ()w (x) v ()w (x)) = (9) ( + ) As we have commened, somemes (9) s no conssen Then we loo for approxmaed soluons Applyn he neror produc f, = f (x)(x)dx n equaon (9), we oban: n (v () w, w v () w, w ) =, n () ( + ) The expresson () s a sysem of n homoeneous dfferenal equaons wh n unnown v, n I s very advanaeous ha {w, w w n } o be an oronormal sysem n he nerval [,] A suable sysem such ha w ()=w ()= (n) s ISSN: Issue 6, Volume 7, June 8

6 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna w (x)= sen πx In hs case, he equaon () n marx form s: V()=A() V() () where V() = (v () v () v n ()) and A() s an nxn marx whch elemens are: a ()= w, w ( + ) Consdern he nal condons, we have: n v ()w (x) = sn πx Due o he fac ha {w,w w n } s an oronormal sysem, hen: v ()= sn π x, w (n) () The expresson () provdes nal condons for sysem () To llusrae he procedure, suppose ha n= n (8) Obvously, wh reaer n we oban beer approxmaon If n=, hen A()= Consdern + A()= A ( ) A =, we have ha A + =, + A fundamenal marx F()= such ha F =I, verfes ha F + = A F + for,, F 7: Table for F() F of he sysem () I ndcaes ha F()=[/] F Solvn he sysem LS(,), and calculan he numeraor, F() can be represened as follows: / / F() = + + / / or equvalenly F()= + Gven nal condons, we calculae he parcular soluon V()=F()K, where K R Noe ha V()=K Tan no accoun (), K= / Therefore, V()= and ( + ) sn π x + ( )sn πx u(x,)= + In he case ha we are neresed only n a parcular soluon ha verfes ceran nal condons, s no necessary o calculae a fundamenal marx To now V(), we suppose ha V()= V, wh V = / Then se V + = A V, + F 8: Table for V() Ths able ndcaes ha V can be represened as lef and rh approxmans of he se {[/] V } Consdern rh approxman, Table s no necessary because n Table we can see ah (,) s a par of rh md (an no accoun Properes and and T(,)=) We solve he sysem RS(,) o / oban he nown soluon V()= ( + ) Ou of curosy we have calculaed he Table for lef approxman n f 9 F 9: Table for lef approxman I ndcaes ha (,) and (,) are wo pars of lef md Le us consder he follown example In hs case we need o solve a sysem of frs order ordnary dfferenal equaons, V()=A() V() where A() s a daonal marx, hen each equaon can be solve ndvdually Therefore, nsead of MPA we wll use scalar Padé approxmaon ISSN: Issue 6, Volume 7, June 8

7 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna Example Le be he follown ellpc equaon: u = u xx, >,, x π ( ) u(,)=u(,)=; u(x,)=u (x,)=snπx We propose a soluon of he form u(x,)= v ()w (x) consdern he same oronormal sysem of Example Subsun n he paral dfferenal equaon and dvdn by ), we oban: v ()sn π x+ v ()( π sn π x) + π ( ) v ()snπ x + v ()( π snπ x) = π ( ) 8 Then, v () = v (), v () = v () ( ) ( ) We have wo ndependen equaons Noe ha A() s a daonal marx The nal condons can be deduced by: u(x,)=v ()w (x)+v ()w (x)=sn πx, herefore, v ()= and v ()=, u (x,)= v ()w (x) + v ()w (x) = sn π x, herefore, v () = and v () = Consdern he frs equaon v () = ( ) v () and s condons v ()= and v () =, we denoe he eneral soluon as s()= s Tan no accoun = ha = ( ) + -n he crcle of converence ( ) of he seres- Subsun n he dfferenal equaon we oban: = + = = = ( )( )s ( ( ) )( s ) Therefore, o calculae he coeffcens of he seres s() we use he follown recurrence relaon: s + = ( (+ )s ) /(( + )( + )) F : Table for v () I shows ha v ()=[/] s Solvn Padé equaons for [/] s, v ()= / In a smlar way, solvn 8 v () = v (), v ()= v () =, we oban ( ) v ()= Therefore, he soluon of he paral dfferenal equaon s: u(x,)= sn πx Mehod of Fne Dfferences and MPA We ve he heory for he follown hyperbolc paral dfferenal equaon (eneralzed wave equaon): u =α (x,) u xx x, > u(,)=f (), u(,)=f (); u(x,)= (x), u (x,)= (x) In hs mehod we choose n N, calculae h=/n and consder he pons (x,) where x =h for =,,n Noe ha we do no dscrese varable Subsun hese pons, for =n-, we oban u (x,) = α (x,) u xx (x,) If we replace second paral dervave u xx wh second dfferences we oban: u (x,)=α u(x +, ) u(x, ) + u(x, ) (x,) u xx (x,), h for =n- Denon α (x, )f () u(x, ) h u(x, ) Y()=, b()= and u(x n, ) α (x n, )f () h α (x,) α (x,) (x,) (x,) (x,) α α α A()= α (x,) α(x,), h α (x n,) α (x n,) we oban he sysem of lnear second-order ordnary dfferenal equaons: Y()=A() Y()+b() () Noe ha he soluon vecor Y() s of dmenson n- The nal condons for hs sysem are: (x ) (x ) (x ) Y()=, Y()= (x ) (x n ) (x n ) Le us suppose Y()= = Y ( ), A()= = A ( ) and b()= b ( ) where he coeffcens of hese = seres have suable dmensons Subsun n (), he expresson o calculae he coeffcens of a ISSN: Issue 6, Volume 7, June 8

8 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna parcular soluon, ven he nal condons Y and Y, s Y + = ( A Y + b ) /(( + )( + )) I s neresn o commen ha, from compuaonal pon of vew, f n s lare h s very small and Y ncreases consderably when ncreases To avod errors, we consder he formal power seres * Y, = * wh Y = h Y Noe ha Y() s raonal f and only f Y * () s raonal Le us sudy some parcular examples Example 5 x u = u ( ) xx x, >,, u(,)= u(,)= ; u(x,5)=x u (x,5)=x Consdern, for nsance, n= and =5 (he equaon has no sense n =) he nal condons are: x x x Y(5)=, Y(5)= x x 9 x 9 x x a = x x x A x x h x x where a = + (+), a + = + (+) b = + ( + )( + ), h x9 9 9 F : Table for Y() s: I ndcaes ha Y()=[/] Y Solvn he sysem RS(,), we oban + ( 5) Y()= R 8( 5) [/] Y = + ( 5) 8+ 6( 5) whch can be smplfed as Y()= 8 x Noe ha u(x,)= s he soluon and ha Y() concdes wh u(x,) dscresed n x Example 6 Le us see now an example where lef approxman s more suable (lef md are smaller han rh md) x u = u xx >, x ( + ) u(,)=, u(,)= + ; u(x,)= + x, u x (x,)= ( + x) Consdern dfferen values of n (n=5, n=, n=5) n hs example Table s hhly sensve o he hreshold ha we choose o decde wheher and elemen s o be consdered zero or no (Due o he lmaons of fne arhmec, ceran heorecally zero elemens wll no be exacly zeros, whch s why we have chosen a hreshold so ha any number wh an absolue value of below wll be consdered as ben zero) However Table for lef approxman, n mos cases, shows ha (,) s a par of lef md Noe ha u(x,)= s he soluon and ha + x( + ) Table ndcaes ha Y() { L [/] Y } For nsance, f n=5: Y() = / / / / + 6/6 /6 8/8 /8 e; Y()= ( ) ( + ) + 6( + ) + 8( + ) s u(x,) dscresed n x 5 Conclusons In hs paper we have exended resuls and pracce of [] Our am has been o hhlh mehods of loon for raonal fundamenal marces for sysems of lnear m-order dfferenal equaon and parcular raonal soluons for hese sysems and for paral dfferenal equaons ISSN: Issue 6, Volume 7, June 8

9 C Pesano-Gabno, C Gonz_Lez-Concepcon, MC Gl-Farna The balanced use of numercal mehods, oeher wh approxmaon heory s a very neresn alernave o mprove he calculaon of hese soluons (or, alernavely, approxmaed soluons) In hs paper we combne Galern and Fne Dfference mehods wh MPA Oher applcaons were consdered n [7] n connecon wh me seres analyss and economc models References [] Burden, RL and Fares, JD Análss Numérco Grupo Edoral Iberoamιrca Méxco, 996 [] A Draux, On he Non-Normal Padé Table n a Non-Conmuave Alebra, Journal of Compuaonal and Appled Mahemacs, 998, pp 7-88 [] González-Concepcón, C and Pesano-Gabno, C (999), Approxmaed soluons n raonal form for sysems of dfferenal equaons, Numercal Alorhms, 85- [] Kncad, D and Cheney, W Análss Numérco Addson-Wesley Iberoamercana Arenna, 99 [5] Marcellαn, F, Casasϊs, L and Zarzo, A Ecuacones Dferencales Problemas lneales y aplcacones McGraw-Hll Madrd, 99 [6] Pesano C and González C Marx Padé Approxmaon of Raonal Funcons, Numercal Alorhms, 5 (997), -6 [7] Pesano-Gabno, C and González-Concepcón, C, Raonaly, mnmaly and unqueness of represenaon of marx formal power seres, Journal of Compuaonal an Appled Mahemacs, 9 (998), -8 ISSN: Issue 6, Volume 7, June 8