Power Series Solutions for Nonlinear Systems. of Partial Differential Equations

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1 Appled Mhemcl Scences, Vol. 6, 1, no. 14, Power Seres Soluons for Nonlner Sysems of Prl Dfferenl Equons Amen S. Nuser Jordn Unversy of Scence nd Technology P. O. Bo 33, Irbd, 11, Jordn Abeer Al-Hsoon Jordn Unversy of Scence nd Technology P. O. Bo 33, Irbd, 11, Jordn Absrc In hs pper nlycl soluons of nonlner prl dfferenl sysems re ddressed. The soluons re obned usng he echnque of power seres o solve lner ordnry dfferenl equons. Ths mehod ensures he heorecl ecness of he pprome soluon. Severl sysems re solved usng hs mehod nd comprsons of he pprome soluons wh he ec ones re demonsred. Mhemcs Subec Clssfcon: 35-4, 35D99 Keywords: Nonlner PDEs, Power Seres Mehod, Anlycl Soluons 1. Inroducon I s well known h here re severl mehods h cn be used o fnd generl soluons o lner PDEs. On he conrry, for non-lner PDEs s well known h here re no generlly pplcble mehods o solve such nonlner equons. A glnce he lerure shows h here re some known mehods whch hve been ppled o solve specl cses of nonlner PDEs. For emple he spl-sep mehod s compuonl mehod h hs been used o solve specfc equons lke nonlner Schrödnger equon [1, 37].

2 5148 A. S. Nuser nd A. Al-Hsson Neverheless, some echnques cn be used o solve severl ypes of nonlner equons such s he homoopy prncple whch s he mos powerful mehod o solve underdeermned equons [9]. In some cses, PDE cn be solved v perurbon nlyss n whch he soluon s consdered o be correcon o n equon wh known soluon [8, 5]. Alernvely, here re numercl echnques h solve nonlner PDEs such s he fne dfference mehod [,19,3] nd he fne elemen mehods [7,33,34,36]. Mny neresng problems n scence nd engneerng cn be solved n hs wy usng compuers. A generl pproch o solve PDEs uses he symmery propery of dfferenl equons, he connuous nfnesml rnsformons of soluons o soluons (Le heor [14,6,3,35]. The connuous group heory, Le lgebrs nd dfferenl geomery re used o undersnd he srucure of lner nd nonlner prl dfferenl equons. Then generng negrble equons o fnd her L prs, recurson operors, Bäcklund rnsform nd fnlly fndng ec nlyc soluons o PDEs [17,, 3, 8, 31]. Furhermore, oher drec mehods were developed o fnd closed form soluons for nonlner PDEs such s he Tnh mehod [18, 4], eended Tnh mehod [39], Ep-funcon mehod [15,16], ronl eponenl mehod [38], nd ohers [4-4].. Power Seres Mehod for Nonlner Prl Dfferenl Equons Power seres s n old echnque for solvng lner ordnry dfferenl equons [7,]. The effcency of hs sndrd echnque n solvng lner ODE s wh vrble coeffcens s well known. An eenson known s Frobenus mehod llows cklng dfferenl equons wh coeffcens h re no nlyc [3]. Recenly he mehod hs been used o solve nonlner ODEs, [5,1-13]. Furhermore, Kuruly nd Byrm [1] used power seres o solve lner second order PDEs. In hs work we pply he power seres mehod o nonlner PDEs. Anlycl soluons re found by usng lgebrc seres. Mnpulon of he equons leds o very convenen recurrence relons h ensure he ecness of he soluon s well s he compuonl effcency of he mehod. The mehod s srghforwrd nd cn be progrmmed usng ny mhemcl pckge. The effcency of hs mehod s llusred hrough some emples nd obned soluons compred wh ec soluons. The generl lgebr for solvng nonlner ODEs s eplned by consderng n nlycl funcon = (τ ) defned n { I : τ 1}. Assume s epnson n power seres s

3 Power seres soluons for nonlner sysems 5149 nd for ny neger m, (τ ) ( ) = k τ 1 kτ (1) = k of power m s epressed s m = k ( τ ) mkτ () k = The followng relon s n essenl condon o be ssfed n order o revel he desred recurrence relon m m 1 ( τ ) = ( τ ) ( τ ) (3) Afer replcng he seres epressons n ech fcor of equon (3), he followng recurrence relon s obned mk = k p= k m 1) p1( k p) = p= ( (4) 1p ( m 1)( k p) To fnd he seres epnson of produc of wo funcons ssumes h he funcons f ( τ ) nd g( τ ) re nlyc = I : τ 1. Le nd = τ nd re defned n { } f ( τ ) = τ ; g( τ ) = b τ (5) = h ( τ ) = f ( τ ) g( τ ) (6) hen he funcon h s lso nlyc τ = nd defned n I, herefore he seres epnson for h(τ ) s where, = h( τ ) c τ (7) = s= s = sbs = s= c = sb ; 1,,,... (8) And he seres epnson for he n h dervve (τ ), cn be wren s ( n) k = k ( τ ) = φ τ (9) nk ( k + n) where, ϕ = ( + 1)( + )...( k), where k nd re posve negers. k + For more dels on solvng nonlner ODEs, see [31, 3].

4 515 A. S. Nuser nd A. Al-Hsson To generlze hs mehod for solvng nonlner PDEs, le u(, be funcon of wo vrbles, nd suppose h s nlyc n he domn G R nd ssume h he pon (, y ) n G. The funcon u(, s hen represened s u (, = ( ) ( y y ) (1) = = To fnd he represenon seres for ny power of u(, condon s n (3) wll be ppled m m 1 u (, = u (, u(,. (11) If he seres epnson of u m (, s wren s ( m) ( m) u (, = ( ) ( y y ) (1) = = hen usng he relon n (11), he coeffcens of u m epressed s ( m) = s= p= ( m 1) ps (1) ( p)( s) (13) A represenon for ny dervve of u wh respec o or y, for ny order, nd for ny power of hem, cn be found by generlzon he equvlen relon for ODEs. Some emples wll be used o epln he mehod. 3. Numercl Emples To llusre he echnque nd ecness of he pprome soluon, we now nvesge some emples of nonlner PDEs n del. Emple 1. The nonlner dffuson equon s consdered m u = ( u u ) (14) where m s posve neger. Le m=, hen he equon s u u = ( u ) = u u uu + (15) wh nl condon

5 Power seres soluons for nonlner sysems h u(,) = (16) c where c >, nd h s n rbrry consn. The ec soluon of he gven + h equon s u(, ) = [6]. c Assume he soluon u(,) s power seres n nd, u (, ) = (17) = = By dfferenng boh sdes of (17) wh respec o, we wll ge he epnson seres of u nd u The power seres of u, u u nd seres u = ( + 1) (18) = = ( + 1), u = ( + 1)( + ) (19) = = ( + ), uu re obned by pplyng (11) nd he bove u = = = s= = s ( )( s) () q p u u = ( p + 1)( p + ) ( p+ )( q) s( p )( q s) (1) = = q= p= s= = q p uu = ( p)( q) ( + 1)( p + 1) ( + 1) s( p + 1)( q s) () = = q= p= s= = Subsue hese seres no equon (11), o obn he recurrence relon: q p 1 ( + 1) = ( p + 1)( p + ) ( p+ )( q) s( p )( q s) q= p= s= = where q= p= ( p)( q) q p s= = ( + 1)( p + 1) ( + 1) s ( p + 1)( q s) (3)

6 515 A. S. Nuser nd A. Al-Hsson, h c 1 = c f f = = 1 o. w By pplyng he recurrence relons (3) for severl vlues of nd. he polynoml ppromon for u(,) s obned h h 3h 3 5h u %(, ) = (4) c 4c 16c 3c c 4c 16c 3c Tble 1 demonsres he dfference beween he pprome soluon nd he ec one for severl vlues of nd when h=1 nd c=1. Tble 1: The dfference beween ec nd pprome soluons for Emple 1. u(,)-u(,) % u(.,)-u(.,) % u(.4,)-u(.4,) % u(.6,)-u(.6,) % u(.8,)-u(.8,) % u(1,)-u(1,) % Emple. Consder he nonlner PDE 3 u = u u + u + u + u (5) wh nl condon u(,) = sn. The ec soluon o (5) s u(, ) = e sn( + ). Assume h hen u (, ) = (6) = = 1 u = ( + 1)( + ) (7) = = 1( + )

7 Power seres soluons for nonlner sysems 5153 nd Therefore, u = = = s= = 1s 1( )( s) (8) q p u u = ( p + 1)( p + ) 1( p+ )( q) 1 s1( p )( q s) (9) = = q= p= s= = nd, s ( 1qp1( q)( s p) ) 1( )( s) (3) = = s= = p= q= 3 u = Subsung he bove seres no equon (5), he followng recurrence equon s obned 1 ( + 1) = q= p= ( p + 1)( p + ) 1( p+ )( q) s ( 1qp1( q)( s p) ) 1( )( s) + ( + 1) 1( + 1) + 1 s= = p= q= The pprome soluon s compued for =,1,,3, 4 nd =,1,,3, 4 q p s= = 1s 1( p )( q s) + (31) u% (, ) = (3) Tble below llusres he dfference beween he pprome soluon obned by equon (3) nd he ec soluon. Tble : The dfference beween ec nd pprome soluons for Emple. u(,)-u(,) % u(.,)-u(.,) % u(.4,)-u(.4,) % u(.6,)-u(.6,) % u(.8,)-u(.8,) % u(1,)-u(1,) %

8 5154 A. S. Nuser nd A. Al-Hsson Emple 3. Consder he nonlner sysem u + vu + u 1 = v + uv v 1 = (33) Subec o he nl condons u = e v ) = e (,), (,. The ec soluon s u(, ) = e, v(, ) = e [4]. In order o solve he gven sysem usng he power seres mehod, he soluons u nd v re consdered s u(,) = v(,) = N N = = N N = =, (34) b, (35) we use he represenon of he soluons n equons (34) nd (35) o wre he power seres epnson of he producs vu nd uv. Then we obn he recurson formuls where δ,,, + 1 = 1 δ,,, + ( s + 1) b s, s+ 1, + 1 = s= (36) b, + 1 = 1 δ,, + b, + ( s + 1) s, bs+ 1, + 1 = s= (37) 1 f = = = o.w Afer solvng (36), nd (37) for =,, 3 nd polynomls 3 3 u %(, ) = =,,3 we obn he (38) nd, 3 3 v% (, ) = (39)

9 Power seres soluons for nonlner sysems 5155 The dfference beween pprome soluons nd he ec soluons for equon (33) re shown n bles 3 nd 4. Tble 3: The dfference beween ec nd pprome soluons of u(,) for Emple 3. u(,)-u(,) % u(.,)-u(.,) % u(.4,)-u(.4,) % u(.6,)-u(.6,) % u(.8,)-u(.8,) % u(1,)-u(1,) % Tble 4: The dfference beween ec nd pprome soluons of v(,) for Emple 3. v(,)-v(,) % v(.,)-v(.,) % v(.4,)-v(.4,) % v(.6,)-v(.6,) % v(.8,)-v(.8,) % v(1,)-v(1,) % Emple 4. The coupled Burgers equon u u uu + ( uv) v v vv + ( uv) = = (4) wh nl condons u(,) = sn, v(, ) = sn. The ec soluon of he bove sysem s u(, )= v(, )= sn e - [9]. In order o solve he gven sysem usng he PSM, he soluons u nd v s consdered s u(,) = (41) = = = =, v(,) = b (4),

10 5156 A. S. Nuser nd A. Al-Hsson Afer subsuon he ppropre seres n he equon, we ge he followng recurrence relons 1, + 1 = ( + 1)( + ) +, + ( s + 1) s, s+ 1, + 1 = s= (43) + ( s + 1) s, bs+ 1, ( s + 1) b s, s+ 1, = s= = s= 1 b, + 1 = ( + 1)( + ) b +, + ( s + 1) b s, bs+ 1, + 1 = s= (44) + ( s + 1) s, bs+ 1, ( s + 1) b s, s+ 1, = s= = s= Then he desred coeffcens re obned by repeed pplcon of he recurrence relons. Usng mhemc progrm o solve he precedng recurrence sysem, for =,,3 nd =,,3. We obn he followng polynomls for u(,) nd v(,) u %(, ) = v% (, ) = (45) Snce he ec soluons re he sme, hen he pprome soluons u(,) nd v(,) re lso he sme. Tbles 5 nd 6 show he convergence of he pprome soluons o he ec ones for dfferen vlues of nd. Tble 5: The dfference beween ec nd pprome soluons of u(,) for Emple 4. u(,)-u(,) % u(.,)-u(.,) % u(.4,)-u(.4,) % u(.6,)-u(.6,) % u(.8,)-u(.8,) % u(1,)-u(1,) %

11 Power seres soluons for nonlner sysems 5157 Tble 6: The dfference beween ec nd pprome soluons of v(,) for Emple 4. v(,)-v(,) % v(.,)-v(.,) % v(.4,)-v(.4,) % v(.6,)-v(.6,) % v(.8,)-v(.8,) % v(1,)-v(1,) % Concluson The mehod hs been successfully ppled drecly o some emples of nonlner PDEs whou usng lnerzon, perurbon, or resrcve ssumpons. I provdes he soluon n erms of convergen seres wh esly compuble componens nd he resuls hve shown remrkble performnce. The effcency of hs mehod hs been demonsred by solvng some nonlner PDEs nd some sysems of nonlner PDEs. A comprson of hs mehod wh he ec soluons were performed nd presened. References [1] G. P. Agrwl, Nonlner Fber Opcs, Acdemc Press, New York (1). [] W. F. Ames, Numercl Mehods for Prl Dfferenl Equons, Acdemc Press, New York (1977). [3] G. Arfken, H. Weber, Mhemcl Mehods for Physcss, Acdemc Press, New York (1985). [4] A. S. Bneh, M.S. Noorn, nd I. Hshm, Approme nlycl soluons of sysems of PDEs by homoopy nlyss mehod. Compu. Mh. Appl., 55: (8). [5] J. Bzr, M. Ile, A. Khoshkenr, A new pproch o he soluon of he prey nd predor problem nd comprson of he resuls wh he Adomn mehod, Appl. Mh. Compu. 171, (5). [6] C. Chun, H. Jfr, Y. Km, Numercl mehod for he wve nd he nonlner dffuson equons wh he homoopy perurbon mehod. Compu. Mh. Appl., 57: (9). [7] E.A. Coddngon; An Inroducon o Ordnry Dfferenl Equons, Dover Publcons, New York (1989). [8] W. H. Cropper, Gre Physcss: The Lfe nd Tmes of Ledng Physcss from Glleo o Hwkng, Oford Unversy Press (4) [9] Y. Elshberg, N. M. Mshchev, Inroducon o he h-prncple, Amercn Mhemcl Soc. ().

12 5158 A. S. Nuser nd A. Al-Hsson [1] V. Fren, V. Lopez, nd L. Conde; Power seres ppromon o soluons of nonlner sysems of dfferenl equons, Amercn Journl of Physcs; 56, (1988). [11] C.P. Flpch, M.B. Rosles; A Srghforwrd pproch o solve ordnry nonlner dfferenl sysems, Mecnc Compuconl; Vol.XXI; (). [1] C. P. Flpch, M.B. Rosles, nd F. Buezs, Some nonlner mechncl problems solved wh nlycl soluons, J. Ln Amercn Appl. Reserch; 34, (4). [13] N. Guzel, M. Byrm, Power seres soluon of non-lner frs order dfferenl equon sysems, Trky Unv J. Sc, 6(1): (5). [14] T. Hwkns, Emergence of he Theory of Le Groups: n Essy n he Hsory of Mhemcs, Sprnger (). [15] J. H. He, X. H. Wu, Ep-funcon mehod for nonlner wve equons, Chos, Solons Frcls, 3, 7 78 (6). [16] J.H. He, L.N. Zhng, Generlzed solry soluon nd compcon-lke soluon of he Julen Modek equons usng he Ep-funcon mehod, Phys. Le. A, 37, (8). [17] R. Hermnn, The Geomery of Non-Lner Dfferenl Equons, Bcklund Trnsformons, nd Solons, Pr A (Inerdscplnry Mhemcs Seres No. 1): Mh Scence Press (1976). [18] W. Heremn nd W. Mlfle, The nh mehod: ool o solve nonlner prl dfferenl equons wh symbolc sofwre, Proceedngs 9h World Mul-Conference on Sysemcs, Cybernecs nd Informcs, Orlndo, FL, (5). [19] F. B. Hldebrnd, Fne-Dfference Equons nd Smulons, Prence- Hll, New Jersey (1968). [] E. Kreyszg; Advnced Engneerng Mhemcs, Wley & Sons, New York (1999). [1] M. Kuruly nd M. Byrm; A novel power seres mehod for solvng second order prl dfferenl equons, Europen Journl Of Pure And Appled Mhemcs Vol., No., (9). [] P. L, Inegrls of nonlner equons of evoluon nd solry wves, Comm. Pure Appled Mh. 1, (1968). [3] P. L nd R.S. Phllps, Scerng Theory for Auomorphc Funcons, Prnceon Unversy Press (1976). [4] W. Mlfle, Solry wve soluons of nonlner wve equons, Amer. J. Phys (199). [5] A. H. Nyfeh, Perurbon Mehods, Wley & Sons, New York (1973) [6] P.J. Olver, Equvlence, Invrns nd Symmery, Cmbrdge Press (1995). [7] G. Pelos, The fne-elemen mehod, Pr I: R. L. Courn: Hsorcl Corner, Anenns nd Propgon Mgzne, IEEE, (7). [8] A. D. Polynn nd V. F. Zsev, Hndbook of Nonlner Prl Dfferenl Equons, Chpmn & Hll, CRC Press (4).

13 Power seres soluons for nonlner sysems 5159 [9] A.S. Rv Knh, K. Arun, Dfferenl rnsform mehod for solvng lner nd non-lner sysems of prl dfferenl equons. Phys. Le. A., 37, (8). [3] R. D. Rchmyer nd K. W. Moron, Dfference Mehods for Inl Vlue Problems, nd ed., Wley, New York (1967). [31] C. Rogers, W.F. Shdwck, Bäcklund rnsformons nd her pplcons, Acdemc Press, New York (198). [3] D. H. Snger, O.L. Wever, Le Groups And Algebrs Wh Applcons To Physcs, Geomery, And Mechncs, Sprnger-Verlg (1986). [33] P. Soln, K. Segeh, I. Dolezel, Hgher-Order Fne Elemen Mehods, Chpmn & Hll, CRC Press (3). [34] E. Sen, O.C. Zenkewcz, A poneer n he developmen of he fne elemen mehod n engneerng scence, Seel Consrucon, (4), 64-7 (9). [35] H. Sephn, Dfferenl Equons: Ther Soluon Usng Symmeres, Eded by M. McCllum, Cmbrdge Unversy Press (1989). [36] G. Srng, G. F, An Anlyss of he Fne Elemen Mehod, Prence Hll (1973). [37] T. R. Th nd M. J. Ablowz. Anlycl nd numercl specs of cern nonlner evoluon equons. II. Numercl, nonlner Schrödnger equon, J. Compu. Phys. 55 (), 3 3 (1984). [38] A M Wzwz, The nh coh mehod for new compcons nd solons soluons for he K(n,n) nd he K(n + 1,n + 1) equons, Appl. Mh. nd Comp., 188, (7). [39] Wzwz A. M., The eended nh mehod for new solons soluons for mny forms of he ffh-order KdV equons, Appl. Mh. nd Comp., 184, (7). [4] Wzwz A. M., Prl dfferenl equons: mehods nd pplcons. The Neherlnds: Blkem Publshers;. [41] Wzwz A. M., A compuonl pproch o solon soluons of he Kdomsev-Pevshvl equon, Appl. Mh. nd Comp., 13, 5-17(1) [4] Wzwz A. M., The nh nd he sne cosne mehods for relble remen of he modfed equl wdh equon nd s vrns, Commun. Nonlner Sc. Numer. Smul., 11(), (6). Receved: Aprl, 1

THE EXISTENCE OF SOLUTIONS FOR A CLASS OF IMPULSIVE FRACTIONAL Q-DIFFERENCE EQUATIONS

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