M. Y. Adamu Mathematical Sciences Programme, AbubakarTafawaBalewa University, Bauchi, Nigeria
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1 IOSR Journal of Mahemacs (IOSR-JM e-issn: , p-issn: 9-765X. Volume 0, Issue 4 Ver. IV (Jul-Aug. 04, PP Mulple SolonSoluons for a (+-dmensonalhroa-sasuma shallow waer wave equaon UsngPanlevé-Bӓclund Transformaon and he Smplfed Hroa s Mehod M. Y. Adamu Mahemacal Scences Programme, AbubaarTafawaBalewa Unversy, Bauch, Ngera Absrac: The mulple solonsoluons of (+-dmensonal Hroa Sasuma shallow waer wave equaon s suded usngpanlevé- Bӓclundransformaon and he smplfed Hroa s mehod.also he hyperbolc and heexp-rgonomerc funcon mehods are used o oban some more nd of solary wave soluons. Keywords:Hroa Sasuma waer wave equaon, mulplesolon soluon, SmplfedHroa s mehod, hyperbolc and Exp-rgnomerc soluons. I. Inroducon The search for exac soluon o non-lnear dfferenal equaon s sgnfcanly mporan o mahemacal physcs. Exac soluon plays val roles n undersandng varous qualave and quanave feaure of nonlnear phenomenon. There areclasses of neresng exac soluons such as solonand he ravellng wave soluon, bu ofen needs for a specfc mahemacal echnque o consruc exac soluon due o non-lneary propery n dynamcs (Su, e al,007 Varous mehods have been employed o sudy negrable sysem and nonlnear evoluon equaons. Such as:hroa blnear mehod, (Hroa, 004he nverse scaerng mehod,(ablowz, 98he Darboux ransformaon, he generalzed symmery mehod (Adamu and Suleman, 0,Wronsan deermnan and ec (Ma, 00.They all mae possble o creae mulple solon soluons for many negrable sysems and nonlnear evoluon equaons.ye, he Hroa s blnear mehod and he smplfed Hroa s mehod are more appealng o he soluons of he mulple solon soluons for he nonlnear dfferenal equaon.(hroa, 004 We wll n hs woruse he dea proposed by (Hereman and Nuser, 997, Wazwaz, 0 o nvesgae he (+ dmensonalhroa Sasuma shallow waer wave equaon of he form: U UU U V U U, V x xx x x U for mulple solon soluon usng he combne Panlëve-Báclund ransformaon wh smplfed Hroa mehod and also o oban ravellng wave and sngular soluon usng some hyperbolc and exp-rgonomerc funcon mehods. Equaon ( models he undreconal propagaon of shallow waer wave, where V(x, represens he horzonal velocy of waer and V(x,gves he devaon hegh from he equlbrum poson of he lqud.(hearana,,009 II. Mulple solon soluons Accordng o (Wazwaz, 0 we can defnehe Pannlelvë Báclund ransformaon U( x, ( Inf x U0( x, ( V ( x, ( Inf xx V0 where f s an auxlary funcon ofx, ha wll be used o ge he one solon soluons. The runcaedpanlevéexpresson of he sysem ( s used o oban he ransformaon (. For he soluon of ( he funcon U ( x, 0 and V 0 are consdered as arbraryansaz and for smplcy we se U0 V0 0 whch gves V( x, Ux( x, ( Subsung ( no ( gves a sngle nonlnear equaon U UU U xu x Ux 0 (4 In order o ge he dsperson relaon, we se ( 40 Page
2 U( x, ( Inf ( x, xx (5 where he auxlary funcon f ( x, s gven by f ( x, e xc,,, (6 Subsung (5 n (4 gves he dsperson relaon by C,,, (7 Nong ha he dsperson relaon depends only on he coeffcens of he varable x Usng (5 gves he n and he solon soluons U ( x, e x x ( e x x e ( e V ( x, x ( e Respecvely, where we used V( x, Ux( x, as gven n ( For he wo solary wave soluon we se he auxlary funcon as ( x ( x f ( x, e e (9 Now subsung (9 and (5 no our equaon ( we oban he wo n soluonsand he wo solon soluons by ( x ( x ( x( ( ( x ( x e e e U ( x, e e ( x ( ( x x( ( x( e ( x ( x e e ( ( ( e V ( x, e e (0 respecvely, For he hree solary wave soluons, we se he auxlary funcon by x x x f ( x, e e e ( proceedng as before, he hree n soluons and he hree-solon soluons can easly be found o be U x (, e e ( x e ( x( (8 4 Page
3 V x ( (, x e e ( e x ( x( ( ( x x x e e ( e, e ( I s well nown ha he exsence of hree solary wave soluons shows ha he mulple solary waves, ncludng he N-solon soluonssoluons are obanable (Hearana, 987, Wazwaz, 008 III. Oher soluons: he hyperbolc funcons mehods In hs secon we wll apply he hyperbolc funcons approaches n order o deermnng some ravellng wave soluon. The schemes ha wll be used nclude:. The anh mehod In he anh mehodwe employ he use of he expresson as used by Wazwaz, (0 U( x, anh( x s allowed ( as a soluon of he sysem (. To deermne α,β and he wave speed ω we subsue ( no ( collec he coeffcen of anh, 0,,and equae o zero o oban 0 (4 Solvng for we have (5 where α& β are assume o be free parameers. Ths gves he solary waves soluons of he sysem ( as U( x, anh( x V ( x, sec h ( x (6 Subsung anh forcoh n (, and proceed n he same way as before we oban he sngular soluons. U ( x, coh( x (7 V ( x, csc h ( x The an mehod The an mehod adms he used of he expresson U( x, an( x (8 as he soluon of he sysem (. To deermned α,β and he wave speed, we subsue (8 no (, collec he coeffcen of an, 0, and equae o zero we oban 0 Solvng for we oban 4 Page
4 where α s lef as a free parameer. Ths gves he solary waves soluons of he sysem ( by U( x, an( x V sec ( x. The co mehod Replacng an by co n (8 and proceedng as before we oban he sngular soluons U( x, co( x V ( x, cos ec ( x. The exp -rgonomerc mehod In heexp -rgonomerc we use he expresson x U( x, e ( cos( x sn( x (9 as a soluon of he sysem (. To deermne he wave speed, we subsue (9 no (, collec he coeffcens of x x e cos( x, e sn( x and equae o zero o oban ( ( 0 ( ( 0 Solvng for,, and we oban where α s chosen o be a free parameer,hs gves he solary waves soluon o he sysem ( By x U( x, e (cos( x sn( x x x V( x, e (cos( x sn( x e ( sn( x cos( x IV. Dscussons The (+ dmensonal Hroa Sasuma waer- wave equaon s nvesgaed for mulple solon soluons usng dfferen approach. Equaon ( models he undreconal propagaon of shallow waer wave. The exsence of he mulple solon soluons and some ravellng wave soluons ndcaes he varey of he applcaon of he equaon n engneerng and scence. References []. Ablowz, M. J, H. Segur (98,.Solons and he nverse scaerng ransform (Phladelpha: SIAM, []. A-M,.Wazwaz. (0 Mulple solon soluons and raonal soluons for he (+-dmensonal dspersve long waer-wave sysem OceancEeng.60, []. Freeman, N.C and Nmmo, J.J.C.(98The use of Baclund ransformaon n obanng N-solon soluons n Wronsan form. Phys. le. A 95, -. [4]. Hereman, W. Nuser, A. (997, Symbolc mehod o consruc exac soluon of nonlnear paral dfferenal equaon Mah, Comp, Smul., 4, -7 [5]. Hearna, J, (987 A search for blnear equaon passng hree-solon condon I. KdV ype blnear equaon, J. Mah. Phys. 8, 7-74 [6]. M.Y Adamu, E. sulaman, (0, On he generalzed blnear dfferenal equaons,j. Mah (4, 4-0 [7]. Ma.W. X (00,dversy of exac soluon o a resrced Ba- panpnell dspersve long way sysem, Physcs,Leer. A0, 5- [8]. R Hroa, 004 The drec mehod n solon heory, Cambrdge Unversy Press 4 Page
5 [9]. Su, T., Geng, X. G.and Yun-Lng, M.A. (007,Wronsan form of N solon soluons for he (+-dmensonal breang solon equaons. Chn. Phy. Le. 4 ( [0]. Wazwaz, A. (008Mulplesolon soluons for Calogero-Bogoyavlenss-Schff, Jmbo- Mwa and YTSF equaons, App. Mah and comp. 0,, []. Wazwaz, A-M, (007 Mulple solon soluon for he Boussnesq equaon, Appl. Mah.Compu. 9, []. Xu, G. G. and Xu, Z. L. (004,Symbolc compuaon of he Panleve es for nonlnear Dfferenal equaon, usng maple. Comp. phys. Comm., Page
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