Solving the generalized regularized long wave equation on the basis of a reproducing kernel space

On the basis of a reproducing kernel space, an iterative algorithm for solving the generalized regularized long wave equation is presented. The analytical solution in the reproducing kernel space is shown in a series form and the approximate solution u"n is constructed by truncating the series to n terms. The convergence of u"n to the analytical solution is also proved. Results obtained by the proposed method imply that it can be considered as a simple and accurate method for solving such evolution equations.

[1]  Juan I. Ramos.  Explicit finite difference methods for the EW and RLW equations , 2006, Appl. Math. Comput..

[2]  Yingzhen Lin,et al.  New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition , 2011 .

[3]  Fazhan Geng,et al.  A computational method for solving one-dimensional variable-coefficient Burgers equation , 2007, Appl. Math. Comput..

[4]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[5]  Abdulkadir Dogan,et al.  Numerical solution of regularized long wave equation using Petrov–Galerkin method , 2001 .

[6]  Minggen Cui,et al.  Numerical algorithm for parabolic problems with non-classical conditions , 2009 .

[7]  D. Peregrine.  Calculations of the development of an undular bore , 1966, Journal of Fluid Mechanics.

[8]  Zhong Chen,et al.  An efficient algorithm for solving Hilbert type singular integral equations of the second kind , 2009, Comput. Math. Appl..

[9]  I Dağ.  Corrigendum to: “Least-squares quadratic B-spline finite element method for the regularized long wave equation” [Comput. Methods Appl. Mech. Engrg. 182 (2000) 205–215] , 2001 .

[10]  Fazhan Geng,et al.  Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space , 2007, Appl. Math. Comput..

[11]  K. R. Raslan,et al.  A computational method for the regularized long wave (RLW) equation , 2005, Appl. Math. Comput..

[12]  Fazhan Geng,et al.  Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method , 2009, Appl. Math. Comput..

[13]  Fazhan Geng,et al.  SOLVING SINGULAR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS IN THE REPRODUCING KERNEL SPACE , 2008 .

[14]  Hong Du,et al.  The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space , 2008, Appl. Math. Comput..

[15]  Zhihua Xie,et al.  High‐order compact difference scheme for the regularized long wave equation , 2006 .

[16]  İdris Daǧ.  Least-squares quadratic B-spline finite element method for the regularised long wave equation , 2000 .

[17]  Chun-li Li,et al.  The exact solution for solving a class nonlinear operator equations in the reproducing kernel space , 2003, Appl. Math. Comput..

[18]  A. A. Soliman,et al.  Collocation solution for RLW equation with septic spline , 2005, Appl. Math. Comput..

[19]  Edward H. Twizell,et al.  A linearized implicit pseudo‐spectral method for some model equations: the regularized long wave equations , 2003 .

[20]  Fazhan Geng,et al.  Solving singular two-point boundary value problem in reproducing kernel space , 2007 .

[21]  I. Dag,et al.  Approximation of the RLW equation by the least square cubic B-spline finite element method , 2001 .

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

[23]  Seokchan Kim,et al.  Numerical solution of one-dimensional Burgers' equation using reproducing kernel function , 2008 .

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

[25]  Selçuk Kutluay,et al.  Application of a lumped Galerkin method to the regularized long wave equation , 2006, Appl. Math. Comput..

[26]  G. R. McGuire,et al.  Numerical Study of the Regularized Long-Wave Equation. II: Interaction of Solitary Waves , 1977 .

[27]  Mehdi Dehghan,et al.  A meshless method using the radial basis functions for numerical solution of the regularized long wave equation , 2010 .

[28]  Hong Du,et al.  Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space , 2008, Appl. Math. Lett..

[29]  Bülent Saka.  A COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF THE RLW EQUATION USING CUBIC B-SPLINE BASIS , 2005 .

[30]  Luming Zhang,et al.  A finite difference scheme for generalized regularized long-wave equation , 2005, Appl. Math. Comput..

[31]  Hong Du,et al.  Reproducing kernel method of solving singular integral equation with cosecant kernel , 2008 .

[32]  M. A. López-Marcos,et al.  Numerical behaviour of stable and unstable solitary waves , 2002 .

[33]  Idris Dag,et al.  B-Spline Collocation Methods For Numerical Solutions Of The Rlw Equation , 2003, Int. J. Comput. Math..

[34]  G. Ben-yu,et al.  A Spectral Method for Solving the RLW Equation , 1985 .

[35]  Abdulkadir Dogan,et al.  Numerical solution of RLW equation using linear finite elements within Galerkin's method , 2002 .

[36]  Labib Iskandar,et al.  Numerical solutions of the regularized long-wave equation , 1979 .

[37]  Hasan Alzubaidi,et al.  A collocation method with cubic B-splines for solving the MRLW equation , 2008 .

[38]  L. Gardner,et al.  A B-spline finite element method for the regularized long wave equation , 1995 .

[39]  J. C. Eilbeck,et al.  Numerical study of the regularized long-wave equation I: Numerical methods , 1975 .

[40]  Dheeraj Bhardwaj,et al.  A computational method for regularized long wave equation , 2000 .

[41]  Minggen Cui,et al.  An efficient computational method for second order boundary value problems of nonlinear differential equations , 2007, Appl. Math. Comput..

[42]  Juan M. Restrepo,et al.  Stable and Unstable Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation , 2000, J. Nonlinear Sci..

[43]  Fazhan Geng,et al.  Solving a nonlinear system of second order boundary value problems , 2007 .

[44]  M. Cui,et al.  THE EXACT SOLUTION OF NONLINEAR AGE-STRUCTURED POPULATION MODEL , 2007 .

[45]  Idris Dag,et al.  Application of cubic B-splines for numerical solution of the RLW equation , 2004, Appl. Math. Comput..

[46]  Siraj-ul-Islam,et al.  A meshfree method for the numerical solution of the RLW equation , 2009 .

[47]  Maryam Mohammadi,et al.  Numerical solution of GRLW equation using Sinc-collocation method , 2010, Comput. Phys. Commun..

[48]  Abdulkadir Dogan,et al.  A least‐squares finite element scheme for the RLW equation , 1996 .

[49]  Ming Gen Cui,et al.  New algorithm for a class of nonlinear integro-differential equations in the reproducing kernel space , 2006, Appl. Math. Comput..