EXACT SOLUTION OF WHITHAM-BROER-KAUP SHALLOW WATER WAVE EQUATIONS

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1 Joural of Sciece ad Arts Year 5, No. (3), pp. 5-, 5 ORIGINAL PAPER EXACT SOLUTION OF WHITHAM-BROER-KAUP SHALLOW WATER WAVE EQUATIONS JAMSHAD AHMAD, MARIYAM MUSHTAQ, NADEEM SAJJAD 3 Mauscript received: 9..5; Accepted paper: 5.3.5; Published olie: Abstract.I this study we preset the aalysis of Adomia Decompositio Method usig He s Polyomials for oliear WhithamBroer-Kaup equatios dealig with propagatio of shallow water waves with differet dispersio relatios. The exact solutios of two variats of WhithamBroer-Kaup equatios are studied. The suggested method is used without discretizatio, liearizatio or restrictive assumptios. Numerical results show that the proposed method was efficiet ad capable to obtai the exact solutio of this set of wave equatios. The obtaied solutios of these equatios could straightforwardly show some facts of the described process deeply such as the propagatio. It is clear that this method ca be easily exteded to other oliear wave equatios arisig i mathematical physics. Keywords: Noliear PDEs, Adomia decompositio (ADM) usig He s polyomials, Coupled WhithamBroer-Kaup equatios.. INTRODUCTION Noliearity exists everywhere ad ature is oliear i geeral. Noliear physical pheomea that appear i may areas of scietific fields such as solid state physics, plasma physics, fluid dyamics, mathematical biology ad chemical kietics ca be modeled by partial differetial equatio. Most of physical systems ca be described by appropriate sets of differetial equatios, which are well suited as models for systems. Hece, uderstadig of differetial equatios ad fidig its solutios are of primary importace for researchers, mathematicias as for physicists. A broad class of aalytical solutios methods ad umerical solutios methods were used i hadle these mathematical problems [-6]. I this paper, we cosider the coupled Whitham Broer Kaup (WBK) equatios which have bee studied by Whitham [], Broer [] ad Kaup [3]. The equatios describe the propagatio of shallow water waves, with differet dispersio relatios. The WBK equatios are as follows, u + uu + v + βu = t x x xx v + vu + uv βv + αu = () t x x xx xxx Uiversity of Gujrat, Faculty of Scieces, Departmet of Mathematics, Gujrat, Pakista. jamshadahmadm@gmail.com. Natioal College of Busiess Admiistratio & Ecoomics (NCBA&E), Departmet of Mathematics, Gujrat (Campus), Pakista. 3 Lahore Leads Uiversity, Departmet of Mathematics, Lahore, Pakista. ISSN:

2 6 where u = u ( x, t ) is the horizotal velocity, v v( x, t) = is the height that deviates from equilibrium positio of the liquid, ad α, β are costats which are represeted i differet diffusio powers [4]. The basic motivatio of this work is to apply the Adomia Decompositio Method (ADM) coupled with He s polyomials to fid travellig wave solutios of Whitham-Broer-Kaup (WBK) equatios which arise quite frequetly i mathematical physics, oliear scieces. It is show that the proposed ADM provides the solutio i a rapid coverget series with easily computable compoets. Numerical results explicitly reveal the complete reliability of the proposed algorithms.. ANALYSIS ADOMIAN DECOMPOSITION METHOD USING HE S POLYNOMIALS To illustrate the basic cocept of He s Adomia decompositio, we cosider the followig geeral differetial equatio L( u) + N ( u) = g ( x) () where L is the liear operator ad N is the oliear operator ad g(x) is the homogeeous term. Accordig to the ADM we costruct the where { } t t = dt. The embeddig parameter (, ] L u = u + Lt N u + g x (3) p ca be cosidered as a Expadig parameters. The homotopy perturbatio method uses the homotopy parameter p as a expadig parameter to obtai (4) = u = p u = u + pu + p u + p u + If p,the approximate solutio of the form, f = lim p u = u (5) = It is well kow that series (3) is coverget for most of the cases ad also the rate of covergece is depedet o L(u). We assume that (3) has a uique solutio. The comparisos of like powers of p give solutios of various orders. I sum, accordig to, He s cosiders the solutio u(x), of the homotopy equatio i a series of p as follows: u x = p u = u + pu + p u + p u +... = ad the method cosider the oliear term N(u), as 3 3

3 7 N u = p H = H + ph + p H + p H +... = 3 3 where H are so called He s polyomials which ca be calculated by usig the formula H u, u, u,... =,,,,3,...! N p u = i ( ) ( i= i ) p (6) p = The successive approximatio u+, of the solutio of u will be obtaied by selective fuctio u. Cosequetly the solutio is give by u = lim u. 3. NUMERICAL APPLICATIONS I this sectio, we apply the Adomia decompositio method usig He s polyomials for solvig coupled Whitham-Broer-Kaupequatios. Example 3. Cosider the equatios () withα = ad β =, we have u = uu v u t x x xx vt = uvx vux + vxx (7) with subject to iitial coditios (,) = ω cot ( + ) u x k h k x x (,) = csc ( + ) v x k h k x x (8) Accordig to the above procedure u ( x, t) = ω k cot h k ( x + x ) plt p ( u ) ux + p v x + p u = = = xx v x, t k csch k x x pl t p ( u ) v x p vu = + + x p v = = = xx (9) u( x, t) = ω k cot h k ( x + x ) plt p ( H ) u + p v x + p u = = = xx ISSN:

4 8 v x, t k csch k x x pl t p ( A ) p B = + + p v = = = xx () where A, B ad H are oliear terms, comparig like powers compoets of p, we get p : u = ω k cot h k x + x p : v = k csc h k x + x p : u = Lt H + v x + u xx = ωk csc h k ( x + x ) t 3 p : v = Lt A + B v xx = ωk csc h k ( x + x ) cot h k ( x + x ) t Ad so o, summig all compoets of u(x,t) ad v(x,t ), we get the series solutio ω ω u x, t = k coth k x + x k csc h k x + x t ω v x, t = k csch k x + x k csch k x + x cot h k x + x t +... The closed form solutio is (, ) = ω cot ( + ω ) u x t k h k x x t (, ) = csc ( + ω ) v x t k h k x x t. Fig.. (a) Exact solutio ad (b) Approximate solutio of (, ) (a) ad ω =.5, k =., x =. (b) u x t with x ad t

5 9 Fig.. (a) Exact solutio ad (b) Approximate solutio of (, ) (a) ω =.5, k =., x =. (b) v x t with x ad t ad Example. Cosider the equatios () withα = 3ad β =, we have ut = uux vx uxx ' v = uv vu + v 3u () t x x xx xxx ' with subject to iitial coditios u x h x (,) = 8ta ( ) Accordig to the above procedure (,) 6 6 ta ( ) v x = h x () { = = = } u x t h x pl p u u p v p u (3) (, ) = 8ta ( ) + + t x x xx (, ) = 6 6 ta ( ) 3 t x + x xx + ( xxx ) v x t h x pl p u v p vu p v p u { = = = = } { = = = } u x t h x pl p H u p v p u (, ) = 8ta ( ) t ( ) + ( x ) + ( xx ) (4) t { = = = xx = xxx } (, ) = 6 6 ta ( ) v x t h x pl p A p B p v p u where A, B ad H are oliear terms, comparig like powers compoets of p, we get ISSN:

6 p : u = 8ta h x p : v = 6 6 ta h x p : u = Lt H + v x + uxx = 8sec h ( x) t { } p : v = L A + B v + 3u = 3sech x tah x t t xx xxx Ad so o, summig all compoets of u(x,t) ad v(x,t ), we get the series solutio u x, t = u + u +... u( x, t) = 8ta h( x) 8sech ( x) t +... v x, y = v + v +... v x, y = 6 6 ta h x 3sech x ta h x t +... The closed form solutio is u( x, t ) 8ta h = x t v( x, y) 6 6 ta h = x t. Fig. 3. (a) Exact solutio ad (b) Approximate solutio of (, ) (a) (b) u x t with x ad t.

7 (a) (b) Fig.. (a) Exact solutio ad (b) Approximate solutio of (, ) v x t with x ad t. 4. CONCLUSIONS I this study, the Adomia Decompositio method usig He s polyomials has bee used for fidig the exact ad approximate travelig wave solutios of the Whitham Broer Kaup (WBK) equatios i shallow water. This method is used without i a direct way without usig liearizatio, trasformatio, discretizatio or restrictive assumptios. From the obtaied results, it may be cocluded that the Adomia decompositio method usig He s polyomials is a very powerful ad efficiet techique to fid exact or approximate solutios for a wide classes of problems. It is also worth poitig out that the advatage of ADM is the fast covergece to the solutios. REFERENCES [] Whitham, G.B., Proceedigs of the Royal Society of Lodo, Series A, 99, 6, 967. [] Broer, L.J.F., Applied Scietific Research,3, 377,975. [3] Kaup, D.J., Progress of Theoretical Physics, 54, 396,975. [4] Kupershmidt, B.A., Commuicatios i Mathematical Physics, 99, 5, 985. [5] Xu, T., Li, J., Zhag, H.Q., Physics Letters A, 369, 458, 7. [6] Momai, S., Odibat, Z., Chaos, Solitos & Fractals, 7(5), 9, 6. [7] Mohyud-Di, S.T., Noor,M.A., Noor, K.I., It.J.Noli.Sci.Num.Sim,(), 3, 9. [8] Ahmad, J., Hassa, Q.M., Mohyud-Di, S.T., J.Fract.Calc.&Appl, 4(), 349, 3. [9] Ahmad, J., Mohyud-Di, S.T., Life Sci J,(4),, 3. [] Ahmad, J., Siddique, H.I., Naeem, M.,It.J.BasicSci.&Appl.Res., 3(3), 73, 4. [] Jafari, H. et all, Romaia Rep. Phys.,65, 9, 3. [] Li,Y.W., Tag,H.W., Che,C.K., J. Applied Mathematics, Art.3887, 4. [3] Ahmed, B.S., Zerrad,E., Biswas,A., Proc. Romaia Acad. A,4, 8, 3. ISSN:

8 [4] AhmadJ., BibiZ., NoorK., Joural of Sciece ad Arts, (7), 3, 4. [5] ShakeelM., Ul-HassaQ. M., Ahmad J. NaqviT., Adv.Math.Phys., Art.8594, 4. [6] Li,Y.W., Lu,T.T., Che,C.K., Commu. Theor. Phys.,6, 59, 3.