Communication in Numerical Analysis
Volume 2012 (2012), Article ID cna-00059, 28 Pages
Numerical simulation of GEW equation using RBF collocation method
Hamid Panahipour *
Safahan Institute of Higher Education, Isfahan 81747-43196, Iran
* Corresponding author. firstname.lastname@example.org Tel: +989131706401
Received: 03 August 2011; Accepted: 19 June 2012
Copyright © 2012 Hamid Panahipour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The generalized equal width (GEW) equation is solved numerically by a meshless method based on a global collocation with standard types of radial basis functions (RBFs). Test problems including propagation of single solitons, interaction of two and three solitons, development of the Maxwellian initial condition pulses, wave undulation and wave generation are used to indicate the efficiency and accuracy of the method. Comparisons are made between the results of the proposed method and some other published numerical methods.
Keywords: Equal width equation, modified equal width equation, RBF collocation method
Hamid Panahipour, Application of Extended Tanh Method to Generalized Burgers-type Equations, Communications in Numerical Analysis, Volume 2012 (2012), 1-14.
P. Donald Ariel, On a second parameter in the solution of the flow near a rotating disk by homotopy analysis method, Communication in Numerical Analysis, Volume 2012 (2012) , 1-13.
A. H. A. Ali, Spectral method for solving the equal width equation based on Chebyshev polynomials, Nonlinear Dyn. 51 (2008), 59-70.
T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for waves in nonlinear dispersive systems, Phil. Trans. Roy. Soc. (London). A 227 (1972), 47-78.
J.L. Bona, W.G. Pritchard, L.R. Scott, A comparison of solutions of two model equations for long waves, In: Norman R. Lebovitz (Ed.), Fluid Dynamics in Astrophysics and Geophysics, Lectures in Applied Mathematics, (1983), 235-267.
J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation for water waves, Phil. Trans. Roy. Soc. (London). A 302 (1981), 457-510.
A.H.D. Cheng, M.A. Golberg, E.J. Kansa, G. Zammito, Exponential convergence and H-c multiquadric collocation method for partial differential equations, Numer. Methods Partial Differential Equations. 19 (2003), 571-5-94.
İ. Daǧ, Y. Dereli, Numericl Solution of the KDV equation using Radial basis Functions, Appl. Math. Modelling. 32 (2008), 535-546.
M. Dehghan, A. Shokri, A numerical method for KdV equation using collocation and radial basis functions, Nonlinear Dyn. 50 (2007), 111-120.
M. Dehghan, A. Shokri, A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Mathematics and Computers in Simulation. 79 (2008), 700-715.
K. Djidjeli, W.G. Price, E.H. Twizell, Q. Cao, A linearized implicit pseudo-spectral method for some model equations: the regularized long wave equations, Commun. Numer. Meth. Engng. 19 (2003), 847-863.
R.K. Dodd, J.C. Eilbeck, J.D. Gibbon, H.C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York, 1982.
D.J. Evans, K.R. Raslan, Solitary waves for the generalized equal width (GEW) equation, Int. J. Comput. Math. 82 (4) (2005), 445-455.
B. Fornberg, J. Zuev, The Runge phenomenon and spatially variable shape parameters in RBF interpolation, Comput. Math. Appl. 54 (2007), 379-398.
L.R.T. Gardner, G.A. Gardner, Solitary waves of the equal width wave equation, Comput. Phys. 101 (1992), 218-223.
L.R.T. Gardner, G.A. Gardner, F.A. Ayoub, N.K. Amein, Simulations of the EWE undular bore, Commun. Num. Methods Eng. 13 (1997), 583-592.
L.R.T. Gardner, G. Gardner, T. Geyikli, The boundary forced MKdV equation, J. Comput. Phys. 11 (1994), 5-12.
Y.C. Hon, R. Schaback, On unsymmetric collocation by radial basis functions, Appl. Math. Comput. 119 (2001), 177-186.
E.J. Kansa, Multiquadrics, A scattered data approximation scheme with applications to computational fluid-dynamics, I- Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (1990), 127-145.
E.J. Kansa, Multiquadrics, A scattered data approximation scheme with applications to computational fluid-dynamics, II-Solutions to parabolic, hyperbolic and elliptic partial differential equations, Comput. Math. Appl. 19 (1990), 147-161.
A. Karageorghis, C.S. Chen and Y.S. Smyrlis, A matrix decomposition RBF algorithm: Approximation of functions and their derivatives, Appl. Numer. Math. 57 (2007), 304-319.
D. Kaya, A numerical simulation of solitary-wave solutions of the generalized long-wave equation, Appl. Math. Comput. 88 (1997), 153-175.
D. Kaya, S.M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos Solitons Fractals. 17 (2003), 869-877.
J.C. Lewis, J.A. Tjon, Resonant production of solitons in the RLW equation, Phys. Lett. A. 73 (1979), 275-279.
C.A. Micchelli, Interpolation of scattered data: distance matrix and conditionally positive functions, Construct. Approx. 2 (1986), 11-22.
D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966), 321-330.
D.H. Peregrine, Long waves on a beach, J. Fluid Mech. 27 (1967), 815-827.
J.I. Ramos, Explicit finite difference methods for the EW and RLW equations, Appl. Math. Comput. 179 (2006), 622-638.
K.R. Raslan, A computational method for the equal width equation, Int. J. Comput. Math. 81(1) (2004), 63-72.
K.R. Raslan, Collocation method using quartic B-spline for the equal width (EW) equation, Appl. Math. Comput. 168 (2005), 795-805.
K.R. Raslan, Numerical methods for partial differential equations, Dissertation, Univ. of Al-Azhar, Cairo, Egypt, (1999).
B. Saka, A finite element method for equal width equation, Appl. Math. Comput. 175 (2006), 730-747.
B. Saka, Algorithms for numerical solution of the modified equal width wave equation using collocation method, Math. Comput. Modeling. 45 (2007), 1096-1117.
Siraj-ul-Islam, Sirajul Haq, Arshed Ali, A meshfree method for the numerical solution of the RLW equation, J. Comput. Appl. Math. 223(2) (2009), 997-1012.
S.I. Zaki, A least-squares finite element scheme for the EW equation, Comput. Methods Appl. Mech. Engrg. 189 (2000), 587-594.
S.I. Zaki, Solitary waves induced by the boundary forced EW equation, Comput. Methods Appl. Mech. Engrg. 190 (2001), 4881-4887.
S.I. Zaki, Solitary wave interactions for the modified equal width equation, Comput. Phys. 126 (2000), 219-231.