Three different methods for numerical solution of the EW equation

Abstract Numerical solutions of the equal width wave (EW) equation are obtained by using a Galerkin method with quartic B-spline finite elements, a differential quadrature method with cosine expansion basis and a meshless method with radial-basis functions. Solitary wave motion, interaction of two solitary waves and wave undulation are studied to validate the accuracy and efficiency of the proposed methods. Comparisons are made with analytical solutions and those of some earlier papers. The accuracy and efficiency are discussed by computing the numerical conserved laws and L 2 , L ∞ error norms.

[1]  Bülent Saka,et al.  Galerkin method for the numerical solution of the RLW equation using quintic B-splines , 2006 .

[2]  Abdulkadir Dogan,et al.  Application of Galerkin's method to equal width wave equation , 2005, Appl. Math. Comput..

[3]  Scott A. Sarra,et al.  Adaptive radial basis function methods for time dependent partial differential equations , 2005 .

[4]  Solitary waves of the EW and RLW equations , 2007 .

[5]  R. Bellman,et al.  DIFFERENTIAL QUADRATURE: A TECHNIQUE FOR THE RAPID SOLUTION OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS , 1972 .

[6]  C. Shu.  Differential Quadrature and Its Application in Engineering , 2000 .

[7]  F. J. Seabra-Santos,et al.  A Petrov–Galerkin finite element scheme for the regularized long wave equation , 2004 .

[8]  G. A. Gardner,et al.  Simulations of the EW undular bore , 1997 .

[9]  P. M. Prenter.  Splines and variational methods , 1975 .

[10]  Y. C. Hon,et al.  Numerical comparisons of two meshless methods using radial basis functions , 2002 .

[11]  B. García-Archilla.  A Spectral Method for the Equal Width Equation , 1996 .

[12]  Peter J. Olver,et al.  Euler operators and conservation laws of the BBM equation , 1979, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  Zhi Zong,et al.  A localized differential quadrature (LDQ) method and its application to the 2D wave equation , 2002 .

[14]  L. Gardner,et al.  Solitary waves of the equal width wave equation , 1992 .

[15]  K. R. Raslan,et al.  Collocation method using quartic B-spline for the equal width (EW) equation , 2005, Appl. Math. Comput..

[16]  Chang Shu,et al.  Fourier expansion‐based differential quadrature and its application to Helmholtz eigenvalue problems , 1997 .

[17]  İdris Dağ,et al.  A CUBIC B-SPLINE COLLOCATION METHOD FOR THE EW EQUATION , 2004 .

[18]  Bülent Saka,et al.  A finite element method for equal width equation , 2006, Appl. Math. Comput..

[19]  Chang Shu,et al.  Integrated radial basis functions‐based differential quadrature method and its performance , 2007 .

[20]  Alaattin Esen,et al.  A numerical solution of the equal width wave equation by a lumped Galerkin method , 2005, Appl. Math. Comput..

[21]  Idris Dag,et al.  Galerkin method for the numerical solution of the RLW equation using quadratic B-splines , 2004, Int. J. Comput. Math..

[22]  S. G. Rubin,et al.  A cubic spline approximation for problems in fluid mechanics , 1975 .

[23]  I. Dag,et al.  Numerical solutions of KdV equation using radial basis functions , 2008 .

[24]  S. Zaki,et al.  A least-squares finite element scheme for the EW equation , 2000 .