Z. Ramadan

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224 Nicolae Popovici 8 REvUE D'ANALYSE NUI\{ÉRIQUE ET DE THÉORIE DE L'APPROXII\{ATIONTome 25, No" 1-2' 1996' pP.225-233B IBLIOGRRAPHIE1. Borwein, J,M., Zhuang, D., Super Effcíency irt vector optimísation. Transaptions of A.M.S., 338(1993), tos-122.Z. 1".,'ó. i.,'Theory of Vector Optímízation. Lecture Notes in Econ. and Matlr. Systems, vol. 319'Springer Verlag, Berlin, (1989)¡. Geoftion,-e .M., Proper Effìiency anil rhe Theory of vector Maxùníza\íon. J. Math. Anal Appl'22 (1968),618-630.¿, popoviù N .,"The Degree of Effrcienc¡, in the Multiobjective Optbnization Problents. "Babe¡-Bolyai" Univ., Preprinr'7 (1990)' 143-154.Received 25.10.1994 Uníversité de ClujFaculté de Mathëntalíquesl, rue M. Kogãlníceanuj 4 0 0 Cluj-NaP o ca, Rount att ieSPLINE APPROXIMATION FOR SYSTEM OF TWO THIRDORDER ORDINARY DIFFERENTIAL EQUATIONS, IIZ. RAMADAN(Cairo)DESCRIPTION OF THE I\{ETHODconsider the system ofnonlinear ordinary differential equations:(1) y"' = fr(r,y,r), "y(r0 ) = ys, !' (xo ) = yó, y"(xo)= y(t',(2) z"' = -f2(x,y,z), z(x6) -- zs, z' (xo) = 2f,, z"(xs) = z'dwherefr, fre C ([0,1] x R2).Let Â be the partitionA:0=ïo<Ít<.,.<where xk+r-xk=h <land k=Q(l)n-1.. . Let L, and Lrbe thelipschitz constants satisfied by the functions fr@) undf f ) respectively, i.e,,(3 ) lfÐ r*, vi, zl) - .f{ù @, vz, ,)l < h{l vt - vzl+l zt - zzl}and(4) lt[Ð t*, v1, zt) - .flÐ G, vr, ,)l < h{l )'t - vzl+l zt - zzl} 'Ii:;,,iirll1ii:!ìfor all (x, yt, z) and (x, yz, zr) in the domain of definition of/, and frandall q:0(1) r,The functions fø) ar'ñ ¡[t) , q: 1(l)r are functions of ¿ y and z only andthey are given from the following algorithm:setfr(o) = -ft(x, y, t), -f;o) = fz(x, Y, z)226 Z. Ramadan 2 3 Spline Approximation 22',|and if ¡(ø-t) and f;s-t) are defined, then The following natation will be used along the discussion of the convergenceof those spine approximants:and¡(ø) = *ufr:-t' y,+ü)r,-OVozf;u) =+.#y,+qt)',e(x) = l,rr(x) - Sa (x)1,er =14 -S¡(.,r¿)1,e(x)=lz(x)- S¡(r,,)1,er =lzr -S¡(¡¡,)1,(e)Then, rve define the spline functiorrs approximating y(x) and z(:$ by So(r)and Sol¡¡ where fr\¡l = fr(i ) l* o,t o-r ("¿ ),+-, (.t¡.)landf;t = fli) [*o,s¿-r (.r¡, ), so-, (."0 )]wlrereT : 0(l)r and k= 0(1)n-1.Along this work, we will deal with the general subinterval1,, = fx¡, xt -t], k = 0(l)rz - 1.Now, we are going to estinate l-r'¡ - S*1;r:;¡ .Using (5), (7) the Lipschitz condition (3) and the notation (9), we getso(¡) = s¿(.r) = sr-r(¡¡,) + si-t(-r¿) (x - xr) + si'-r(x¡)(5) r+l ¡u)l{r¡, s¡-1(x¡), so-, (r¿ )li=0ands"(r) = s¿(r) = sr-r(x¡,) + si-r("r) G - xt) + q'-r¡xo, (:'; -r't )2 *2l(6) *f rlÐu*¡,s¡-1(x¡,), s*-r("¿)lj=owlrere-rr-r I x 3 x¡, k = 0(I)n - 1, s-1(x6) = !s, s\(xo)=y'0, s11(.x¡) = yó,5-r(ro) = zs, Í-1(xo) = tó and s-1(xs) = z'å.By construction it is clear that s¡(.r), "¿ (") . C2¡0,t1 .ly(') - sr(")l <lyt - s¡-1(-r¡)l+l yi - s--Jxr)ll; - x¿l++t y'í - sü-t'"Ðt-#* i lrl'.'r - rt:tlll#f .(l 0) +lr('*r)(Er) - fr:,ìl <1 ek + h.L * *. r .nlrÍi*, - f,(,'ìl#,-*l:,(,*r){€o ) - fr::;rlj.'.-ERROR ESTIMATIONS AND CON\/ERGENCEFor all x e lx¡r, xk*t), k = O(l)n - 1, the exact solutions of (l) and (2) can bewritten, by Taylor's expansion, in the following fonns:(i) vG)=i*r-xr)i *v('*3)(1ù (x-x,\''*37uit \ ^' f+3)t \ n'and(8) z(x)='ä+r-xt)i.l#o-,r)'*3wltere €r, ry e(xt, xr*), k = 0(1)r - lNow,letu =lty*tt - fr:!ìlthen using the Lipschitz condition (3), we get:228(1 1)Also letthen, using (3), we get:(12)Z. RarnadanU3L1@¡+e¡), = lr<,*tt çEù - f{]ll, is a constant independ ent of h,Spline Approximation, is a constant independent of /2.229are constants i nde¡relr-54Similarly, we estimate l"'{u.) - SÅ(")1. Thus using (6), (8), the Lipschitzcondition (4), the notation (9), it can be easily shown that:(17) "'(x) < C3he¡ + Crher + ei + ha{ + +r,tçz?*3),lt¡" (r +2)lv < llt'*3) (€ o ) - .rf .') | +lfrØ {h, t o,' ù - f{,lìl<< açy(r+3) ,lt)+ I"(eo + e¡). /- _fwhere v3 - L2Ie+_(r + 2)lwhere o1y('+3), å) is the modulus of continuity of the function ,('+3).From (10-12) and noting thatr-,1 t il.3(13) Ð"r.r'-h'("-r)<h2ewe can see that:(14) e(x)< (r+coh2)e1,+coh2eo +n"¡+(.T *Ær(y('*3),h)2l " (r+3)!( t Iwlrere Co = Ll e * - | is a constarrt inclependent of /¡.u ¡[ (r + 3)l)Similarly, using (6), (8), the Lipschitz condition (4) and the notation (9), wecan see that:(1s) ¿6)<C1h2e¡ +(;+Cú2)ek +na¡+(a¡'* !!-^ç2Q+3),lt)Zt ^ (r+3)!where c, = t2(".."å,) , is a coustant iudependent of /z anclot7z0+3),/z) is the modulus of continuity of the functi on ,?+3) .We are going to estimate ly'@) - SÅ(x)1. For this purpose we use equa-tions (5), (7), the Lipschitz conclition (3), the notation (9) and the irrequalities (l 1),(12) and (13) attd we get:( I 6) e' (x) < C2lte¡ + C2hex + e' ¡,+|rc'i * 9^^ (r('+3) ,h)LrLLQ+2)lWe now estinate ly"(x) - Sí(r)l and lz"(x) - ¡ü(r)1, Thus, using equa-tions (5-8), the Lipschitz conditions (3-4) and the notation (9), we get:e"(x) < Cahe¡ + Col- h'*rrc¡ + e,i. Ë. ,), a(y(,+3), h)""(x) < C5he¡ + Crheo + e[ +# rr7z?*3) , h¡(18)and(t e)where C+ = Ltdent ofå.Ie+_(r+1)! and ¿, = ¡,Ie+_(r + l)!where Cz = LtIe+-(r + 2)lTo cornplete the colrvergence proof, we use the matrix inequality rvhicli isgiven in the following definition:DennqtrloN L Let A:lo,¡1, B : [b¡¡J be ñuo matrices of tlrc same orrÌer, rhenwesaythatA<B iff(i) l:oth a,, and b,,are norutegative,(ii) a, < b,, for all i, j.Accordi'g to this definition. and if we use the matrix notation:E(x) = le(x) "(x) e,(x) .,(x) e,,(x) ",,(x)lrandEr = (.* ¿k e'¡ ei el ¿í!)'then, we can write the estimations (14-19) in the foilowing fomr:(20) E(.r:) I (l + hA)E,, + lt'+ta(Ì)BI is the identity matrix of order 6 anda(h) = max{o (Y('*t', h),' P('*3)' h¡¡'Then, we give the following definition of the matrix nornr'DEFINITIoN 2' Let T: lt,r) be at7 m x n matrix' lhen v'e definellrll = ",e.it,ul.According to this definition, we get:(2r) ll'ttlll=max{e(¡)''(x)'e'(x)'ë'(x)'e"(x)'""(x))'Since (20) is valid for all x e|x¡,xkotf , &: 0(l)n_I, then the followinginequalities hold true:lltt'lll= (t * øll,all)llr.ll ¡ ¡r+t co(ø)llrll(r + ø ll,rll) llø*ll = (r + nllell)'lløo-,ll + h'+ta(h)llnll (t + tr ll'rll)(r + øllzll)'?lløo-,ll . (r + øll,ell)'ll¿o-,ll+ ø'*rro(ø)llnll(r + nll{l)'(r + {l,nll)ellø,ll< (r + nll,ell)&.'lltoll + h'+'\a(h)llall(r + nll{ÙoAdding L.H.S. antl R.H,S. of these inequalities and noting that llfoll = O,Splile Approximationll"t.)l < cuh'a(h)where c. = ]l;ll ("il'rtt - 1), i, u constanr independent or/r,Then applying (21), we get:"(*) < c 6h'a (\ = o(n'*"),¿(r) < c 6h'' ot(\ = o(n'*"),(22) "'(') < c6ta(\ = o(n'*"),ã'(r) < c6h''açt) = o(tt'*"),u"(r) . C6ilo1n¡ = O(n'.")and"" (t) < cuh' o(h) = o(n''*")Now, we estimate lrøl lù- 'Í!)(t)l *r'.r" q : 3(1)r + 2Using (3), (5), (7), (9), (11), (12), and (22), we get:Z. Ramadan 6 231'7230C,,crc2c3c4cscocrc2c3c4cs012l0I2l0l0B_(r+3)!I(r+3)!I(r + 2)lI(r + 2)lI(r + l)!where I I000I000000000I00 I(r+l)!lrtu)i,)- "Í')(,)l =r-lIj=q-,lrf*'- rl,lì(23) *þt*') {E o ) - r{,:ll;:# =< C, hr +3 - ø ^ (tt) = o(t**,*z- ø )where C, is a constant independent of å.For the case q: r * 3, we use (5), (7), (12) and (22) we get:lr('*,){,)- "f .')1,¡l = lr('.'){€) - f{,:)1,< cr'(/,) = o(0")wlrere Cr: l*2 LrCu, is a constant independent of /r.we get:9Spline Approximatiou 233232 Z. Ramaclan IIn a similar mallner, using (4), (6), (8), (9) and (22), it can be shown thatl rr', (r)- ilo) (") l. rr,r,*'- nro (rr) = o(h"*'*t- ø7The t¿bulated results, appearing in the following table, are evaluated at thepoint.x :0.25.Tableandabsolute er¡orl .054 I 7E-041 .0 I 94007E-040.00t44266670.0012666294240.0124354170.01,02t'728310.06262s4t6680.0399292169nurnerical value1.033921.0286988430.282582'7 50 21993258'7sr.2't 1590.768s835t22140-0.8 I 87 3analytical value1.034031.028 80078 30.28402541670.22r19921691,28402541',10.778800783 11.28402s4r'l-0.778800783 1vzvzy"zty"'zntl,t'*,)1"¡ - sf'*3)1"¡ | = r,o'1r¡ = o(n")where Q = 3(l)r+2 arLd Cs, C to are constants independent of å.Thus, we have proved the following theorem:THEoREM, Let S/:.) and S¡(x) be the approxi.mate solutiotts to probletn(1)-(2) given bv equaÍiotts (5-6), and let ft,fre C' ([0,1] x R2)'Then for all x e lxo, xo*1), k:0(l)n-1, v'e have:1,,,l"('lr")1.,,¡- 'f)1' )ch''6(h) ,¡ = o(7)2,ch'o\h) ,¡ = o(l)2,(x -t,s¿ (')l(')lr1J )(¡)J¿' < kh'*3-ja(h)and REFERENC E S1r,,,(r) - rÍ')(")l < k* t{*3- i ot(h) l. A. Ayad, F.S. Holail and Z. Rarnadan, A splíne approxinntion of an arb.ilrary orderfor thesolutío, of slts¡¿¡1 of second order tliferential equatiotts, Studia Univ. Babeç-Bolyai,Mathematic4 XXXV, (1990) l,49-59.2. M jcula G., Approxítnate integration of syslem of dffirentíal equations by spline funcfiolrr, StudiaUniv. B abeg-Bolyai Cluj, Mathematica" 2(197 l)' 27 -39.3. Micul4 Gh., Spiinefuutctions of higlter degree of approxíntatíon þr solutions olsystent of dtf.fer-entíal eqiatior¡s, Stuclia Univ. Babeç-Bolyai, Math-Iylech. 17 (1972) 1,21'32'4. Gh. Micula, Th. Fawzy, Z. Ramadan, A. polynonial spline approxinatíott method .for solvirtgÐ,stem of orclíttary clills,'.r¡¡rl equøtions, Studia Univ. Babeç-Bolyai, Mathernatica, XXXII(1987) 4,55-60.5. Schuìnakei, Larry L., Optímal spline solur.ions of sysfem of ordina4, difþrential equation't,DitÍe-rential equations (são Paulô, 1981), 272-283, Lecrure noles in Math., 957, Springer, Berlin-New York, 1982.6.Th.Fawzy and Z. Rarnadan , Spline approxinralionsþr s¡'sÍem of ordinatl,rli[erential eqttalions,Ill'Mather¡atica-Revue d'analyse numérique et de Théorie de I'approximatiott, 15 (1986) 2.'7 . Z. Ru¡adan, Spline approximatíon þr rystem of ñuo second order ordinaty differentíal equations,Joumal of the Faculty of Education, 16 (1991), 359-369.Received 10,03.94 Depl. of Mathenatics, Fac of EducalionAin Shants Univ., Caíro, EgttPlwhere j: 3(1)r+3, c, kandk* areconslanls independentof lt'NUMERICAL EXAMPLEConsider the following systen of differential equations:y"'= y- z +2x*e-',10) = t,y'(0) - 0,1"(O) = t,z"' = | - z + 2x + e',2(0) = l, t' (0) = 0, z"(O) = t.The method is tested using this example in the interval [0, l] with stepsize h = 0.I where r: 0,The analytical solution is:Y(')=ê'-x'z(x)=e-'+x'

Spline approximation for system of two third order ordinary differential equations, II

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Spline approximation for system of two third order ordinary differential equations, II

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