Comparison of the solutions obtained by B-spline FEM and ADM of KdV equation

A numerical solution to a generalized Korteweg-de Vries (KdV) equation is obtained using the Galerkin method with quadratic B-spline finite element method (FEM) over which the nonlinear term is locally linearized and using the Adomian's Decomposition Method (ADM). Test problems concerning the motion and interaction of soliton solutions are used to compare the FEM with the ADM. The present methods extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.

[1]  Yves Cherruault,et al.  Adomian's polynomials for nonlinear operators , 1996 .

[2]  Dogan Kaya,et al.  Solitary-wave solutions for compound KdV-type and compound KdV-Burgers-type equations with nonlinear terms of any order , 2004, Appl. Math. Comput..

[3]  Y. Cherruault,et al.  New ideas for proving convergence of decomposition methods , 1995 .

[4]  Dogan Kaya,et al.  Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation , 2004, Appl. Math. Comput..

[5]  Y. Cherruault,et al.  Convergence of Adomian's method applied to differential equations , 1994 .

[6]  Salah M. El-Sayed,et al.  On a generalized fifth order KdV equations , 2003 .

[7]  Salah M. El-Sayed,et al.  An application of the decomposition method for the generalized KdV and RLW equations , 2003 .

[8]  Salah M. El-Sayed,et al.  On the solution of the coupled Schrödinger–KdV equation by the decomposition method , 2003 .

[9]  Dogan Kaya,et al.  A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations , 2002, Math. Comput. Simul..

[10]  B. M. Fulk.  MATH , 1992 .

[11]  D. Kaya.  On the solution of a korteweg-de vries like equation by the decomposition method , 1999, Int. J. Comput. Math..

[12]  Abdul-Majid Wazwaz,et al.  A reliable modification of Adomian decomposition method , 1999, Appl. Math. Comput..

[13]  G. Adomian,et al.  On KdV type equations , 1997 .

[14]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[15]  Andrew G. Glen,et al.  APPL , 2001 .

[16]  A. Rèpaci,et al.  Nonlinear dynamical systems: On the accuracy of adomian's decomposition method , 1990 .

[17]  Y. Cherruault.  Convergence of Adomian's method , 1989 .

[18]  G. Adomian,et al.  Noise terms in decomposition solution series , 1992 .

[19]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

[20]  Dogan Kaya.  A numerical solution of the sine-Gordon equation using the modified decomposition method , 2003, Appl. Math. Comput..

[21]  L. Gardner,et al.  Solitary wave solutions of the MKdV− equation , 1995 .

[22]  Dogan Kaya,et al.  Solitary wave solutions for a generalized Hirota-Satsuma coupled KdV equation , 2004, Appl. Math. Comput..

[23]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

[24]  M. J. Pujol,et al.  A new formulation of Adomian method: Convergence result , 2001 .

[25]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[26]  J. M. Sanz-Serna,et al.  Petrov-Galerkin methods for nonlinear dispersive waves , 1981 .

[27]  Bengt Fornberg,et al.  A numerical and theoretical study of certain nonlinear wave phenomena , 1978, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[28]  C. S. Gardner,et al.  The Korteweg-de Vries equation as a Hamiltonian System , 1971 .

[29]  G. Adomian.  Nonlinear Stochastic Operator Equations , 1986 .

[30]  Dogan Kaya,et al.  An application for a generalized KdV equation by the decomposition method , 2002 .

[31]  Y. Cherruault,et al.  Decomposition methods: A new proof of convergence , 1993 .

[32]  L. Gardner,et al.  Simulations of solitons using quadratic spline finite elements , 1991 .

[33]  M. C. Shen,et al.  Asymptotic Theory of Unsteady Three-Dimensional Waves in a Channel of Arbitrary Cross Section , 1969 .

[34]  P ? ? ? ? ? ? ? % ? ? ? ? , 1991 .

[35]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[36]  P. Drazin,et al.  Solitons: An Introduction , 1989 .

[37]  A. Wouwer,et al.  An adaptive method of lines solution of the Korteweg-de Vries equation , 1998 .

[38]  J. Rogers.  Chaos , 1876 .