Application of Numerical Iterative Methods for Solving Benjamin-Bona-Mahony Equation

Document Type : Original Research (Full Papers)

Author

Department of Mathematics Islamic Azad University Qazvin Branch, Qazvin, Iran

Abstract

In this paper, a generalized Benjamin-Bona-Mahony equation ( BBM)
is solved by using the Adomian's decomposition method (ADM) ,
modified Adomian's decomposition method (MADM), variational
iteration method (VIM), modified variational iteration method (MVIM)
and homotopy analysis method (HAM). The approximate solution of this
equation is calculated in the form of series which its components
are computed by applying a recursive relation. The existence and
uniqueness of the solution and the convergence of the proposed
methods are proved. A numerical example is studied to demonstrate
the accuracy of the presented methods.The MVIM has been shown to solve effectively, easily and accurately
a large class of nonlinear problems with the approximations which
convergent are rapidly to exact solutions.

Keywords


[1 ]. S.Abbasbandy , A .Shirzadi, The first integral method for modified Benjamin-Bona-Mahony equation . Commun Nonlinear Sci Numer Simulat , In press , 2009 .
[2 ]. S.Abbasbandy , Homptopy analysis method for generalized Benjamin-Bona-Mahony
equation . Zeitschriff fur angewandte Mathematik und Physik (ZAMP) , 59(2008) 51-62 .
[3 ]. T.A.Abassy , El-Tawil,H.El.Zoheiry , Toward a modified variational iteration method
(MVIM) , J.Comput.Apll.Math , 207(2007) 137-147 .
[4 ]. T.A.Abassy , El-Tawil,H.El.Zoheiry , Modified variational iteration method for Boussinesq equation , Comput.Math.Appl , 54(2007) 955-956 .
[5 ]. T.B.Benjamin , J.L.Bona , J.J.Mahony , Model equations for long waves in nonlinear dispersive
system . Philos Trans R Soc London , Sera , 27(1972) 47-78 .
[6 ]. S.H.Behriy,H.Hashish , I.L.E-Kalla , A.Elsaid , A new algorithm for the decomposition solution of nonlinear
differential equations.54(2007) 459-466 .
[7 ]. A.O.Celebi , V.H.Kalantarov , M.Polat , Attractors for the generalized Benjamin-Bona-Mahony
equation . Journal of Differential Equations , 157(1999) 439-451 .
[8 ]. I.L.El-Kalla,Convergence of the Adomian method applied to a class of nonlinear integralequations . Appl.Math.Comput , 21(2008) 372-376 .
[9 ]. Sh.Fang , B.Guo , The decay rates of solutions of generalized Benjamin-Bona-Mahony equations in
multi-dimensions . Nonlinear Analysis , 69(2008) 2230-2235 .
[10 ]. M.A.Fariborzi Araghi , Sh . Sadigh Behzadi , Solving nonlinear Volterra-Fredholm integral differential
equations using the modified Adomian decomposition method , Comput . Methods in Appl . Math , 9( 2009) 1-11 .
[11 ]. D.D.Ganji , H.Babazadeh , M.H.Jalaei , H.Tashakkorian , Application of He's variational iteration method for solving nonlinear Benjamin-Bona-Mahony-Burgers equations and free vibration of systems . Acta
Applicanda Mathematica : An International Survey Journal on Applying Mathematics and Mathematical
Applications , 106(2008) 359-367 .
[12 ]. J.H.He , X.H.Wu , Exp-function method for nonlinear wave equations . Chaos , Solitons and
Fractals , 30(2006) 700-708 .[13 ]. J.H.He , Variational principle for some nonlinear partial differential equations with variable cofficients , Chaos , Solitons , Fractals , 19 (2004) 847-851 .
[14 ]. J.H . He , Variational iteration method for autonomous ordinary differential system , Appl . Math . Comput. , 114 (2000) 115-123 .
[15 ]. J.H . He , Wang.Shu-Qiang , Variational iteration method for solving integro-differential
equations , Physics Letters A . 367 (2007) 188-191 .
[16 ]. J.H.He , Variational iteration method some recent results and new interpretations , J . Comp . and
Appl . Math. , 207 (2007) 3-17 .
[17 ]. N.Khiari , K.Omrani , On the convergence of difference scheme for the Benjamin-Bona-Mahony
equation . Appl.Math.Comput , 182(2006) 999-1005 .
[18 ]. Sh.Lai , X.LV , M.Shuai , The Jacobi elliptic function solutions to a generalized Benjamin-Bona-Mahony
equation . Mathematical and computer Modelling , 49(2009) 369-378 .
[19 ]. J.Limaco , H.R.Clark , L.A.Medeiros , Remarks on equations of Benjamin-Bona-Mahony
type . J.Math.Anal.Appl , 328(2007) 1117-1140 .
[20 ]. J.Limaco , H.R.Clark , L.A.Medeiros , On equations of Benjamin-Bona-Mahony type . Nonlinear
Analysis , 59(2004) 1243-1265 .
[21 ]. S.J.Liao , Beyond Perturbation : Introduction to the Homotopy Analysis Method.Chapman and Hall/CRC
Press,Boca Raton,2003 .
[22 ]. S.J.Liao , Notes on the homotopy analysis method : some definitions and theorems , Communication in Nonlinear Science and Numerical Simulation 14(2009)983-997 .
[23 ]. S.T.Mohyud-Din,M.A . Noor , Modified variational iteration method for solving Fisher's
equations , J.Comput.Apll.Math , 2008 .
[24 ]. S.Mica , On the controllability of the linearized Benjamin-Bona-Mahony equation . SIAM.Control
Optim , 39(2001) 1667-1696 .
[25 ]. K.Omrani , The convergence of fully discrete Galerkin approximations for the Benjamin-Bona-Mahony
equation . Appl.Math.Comput , 180(2006) 614-621 .
[26 ]. M.A.Raupp , Galerkin methods applied to the Benjamin-Bona-Mahony equation . Bol.Soc.Brasil.Mat , 6(1975) 65-77 .
[27 ]. M.Song , Ch.Yang , B.Zhang , Exact solitary wave
solutions of the Kudomtsov-Petviashvili-Benjamin-
Bona-Mahony equation . Appl.Math.Comput , In
press , 2009 .
[28 ]. H.Tari,D.D.Ganji , Approximate explicit solutions of
nonlinear Benjamin-Bona-Mahony-Burgers equations
by He's methods and cpmparison with the exact
solution.Phys.Lett.A , 367(2007) 95-101 .
[29 ]. L.Wahlbin , Error estimates for a Galerkin method for
a class of model equations for long
waves . Numer.Math , 23(1975) 289-303 .
[30 ]. A.M.Wazwaz , Exact solutions with compact and
noncompact structures for the one-dimensional
generalized Benjamin-Bona-Mahony equation .
Communications in Nonlinear Science and Numerical
Simulation , 10 (2005) 855-867 .
[31 ]. B.Wang , Random attractors for the stochastic
Benjamin-Bona-Mahony equation on unbounded
domains . J.Diffrential Equations , 246(2009) 2506-
2537 .
[32 ]. A.M.Wazwaz , Construction of solitary wave solution
and rational solutions for the KdV equation by
ADM.Chaos,Solution and fractals . 12(2001) 2283-
2293 .
[33 ]. A.M.Wazwaz , A first course in integral
equations , WSPC , New Jersey; 1997 .
[34 ]. E.Yusufoglu , A.Bekri , the variational iteration method
for solitary patterns solutions of generalized
Benjamin-Bona-Mahony
equation.Phys.Lett.A , 367(2007) 461-464 .
[35 ]. E.Yusufoglu , New solitonary solutions for the
modified Benjamin-Bona-Mahony equations using
Exp-function method . Phys.Lett.A , 372(2008) 442-
464 .
[36 ]. Y.H.Ye , L.F.Mo , He's variational method for the
Benjamin-Bona-Mahony equation and Kawahara
equation . Comput and Math with Appl , In press , 2009 .
[37 ]. H.Yin , H.Zhao , J.Kim , Convergence rates of solutions
forward boundary layer solutions of generalized
Benjamin-Bona-Mahony equations in the halfspace
. J : Differential Equations , 245(2008) 3144-
3216 .
[38 ]. X.Zhao , W.Xu , Travelling wave solutions for a class
of the generalized Benjamin-Bona-Mahony
equations . Appl.Math.Comput , 192(2007) 507-519 .
[39 ]. X.Zhao , W.Xu , Sh.Li , J.Shen , Bifurcations of
traveling wave solutions for a class of the generalized
Benjamin-Bona-Mahony
equation . Appl.Math.Comput , 175(2006) 1760-1774 .
[40 ]. L.Zeng , Existence and stability of solitary-wave
solutions of quations of Benjamin-Bona-Mahony
type , Differential Equations , 188(2003) 1-32 .
[41 ]. M.Ikram, A.Muhammad, Muhammmad, Analytic
solution to Benjam-Bona-Mahony by using laplace
Adomian decomposition method ,Matrix, Science
Mathematic ,3 (2019) 1-4.
[42 ]. A. Fakharian, M.T. Hamidi Beheshti and A. Davari,
“Solving the Hamilton-Jacobi-Bellman equation Using
Adomian Decomposition Method”, International
Journal of Comp. Mathematics, 87(2010), 2769-2785.