A numerical method for KdV equation using collocation and radial basis functions

Nonlinear Dynamics

Volumn 50, Issue 1-2, 2007, Pages 111-120

  Dehghan, Mehdi ^a   Shokri, Ali ^a  

^a AMIRKABIR UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Collocation; KdV equation; Multiquardric (MQ); Radial basis function (RBF); Soliton

Indexed keywords

FINITE DIFFERENCE METHOD; NONLINEAR EQUATIONS; NUMERICAL METHODS; RADIAL BASIS FUNCTION NETWORKS;

COLLOCATION POINTS; KDV EQUATION;

BOUNDARY VALUE PROBLEMS;

EID: 34548359104     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-006-9146-5     Document Type: Article

References (48)

1. Liu, IL, Yan, J.: A local discontinuous Galerkin method for the Kortewegde Vries equation with boundary effect. J. Comput. Phys. 215(1), 197-218 (2006)
2. Kansa, EJ.: Multiquadrics - a scattered data approximation scheme with applications to computational fluid dynamics - I. Comput. Math. Appl. 19, 127-145 (1990)
3. Zerroukat, M., Power, H., Chen, C.S.: A numerical, method for heat transfer problem using collocation and radial, basis functions. Int. J. Numer. Methods Eng. 42, 1263-1278 (1992)
4. Wazwaz, A.M. : Two reliable methods for solving variants of the KdV equation with compact and noncompact structures. Chaos Solitons Fractals 28, 457-462 (2006)
5. Das, G., Sarma, J.: A new mathematical approach, for finding the solitary waves in dusty plasma. Phys. Plasmas 6, 4394-4397 (1999)
6. Osborne, A.: The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves. Chaos Solitons Fractals 5, 2623-2637 (1995)
7. Ostrovsky, L., Stepanyants, Y.A.: Do internal solitons exist in the ocean? Rev. Geophys. 27(3), 293-310 (1989)
8. Ludu, A., Draayer, J.P.: Nonlinear modes of liquid drops as solitary waves. Phys. Rev. Lett. 80, 2125-2128 (1998)
9. Reatto, L., Galli, D.: What is a ROTON? Int. J. Mod. Phys. B 13, 607-616 (1999)
10. Turitsyn, S., Aceves, A., Jones, C., Zharnitsky, V.: Average dynamics of the optical soliton in communication lines with dispersion management: Analytical results. Phys. Rev. E 58, R48-R51 (1998)
11. Coffey, M.W.: Nonlinear dynamics of votices in ultraclean type-II superconductors: Integrable wave equations in cylindrical, geometry. Phys. Rev. B 54, 1279-1285 (1996)
12. Singh, K., Gupta, R.K.: Lie symmetries and exact solutions of a new generalized Hirota-Satsuma coupled KdV system with variable coefficients. Int. J. Eng. Sci. 44(3-4), 241-255 (2006)
13. Hereman, W., Banerjee, P.P., Korpel, A., Assnto, J., Van Immerzeele, A., Meerpoel, A.: Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method. J. Phys. A 607-628 (1986)
14. Lei, Y., Fajianj, Z., Yinghai, W.: The homogenous balance method, Lax pair, Hirota transformation and a general, fifth-order KdV equation. Chaos Solitons Fractals 13, 337-340 (2002)
15. Coley, A., Levi, D., Milson, R., Rogers, C.: Backlund and Darboux transformations, the geometry of solitons. AARMS-CRM workshop, Halifax, Canada, June 4-9, 1999. CRM Proceedings and Lecture Notes. 29. Providence, RI. American Mathematical Society (2001)
16. Hirota, R.: Exact solution of the Korteweg-de Vries equation for multiple collision of solitons. Phys. Rev. Lett. 27, 1192-1194 (1971)
17. Malfliet, W.: Solitary wave solutions of nonlinear wave equations. Am. J. Phys. 60, 650-654 (1992)
18. Yan, C.T.: New explicit solitary wave solutions and periodic wave solutions for Whitham-Broer-Kaup equation in. shallow water. Phys. Lett. A 224, 77-84 (1996)
19. Yan, Z.Y., Zhang, H.Q.: Symbolic computation and new families of exact soliton like solutions to the integrable Broer-Kaup (BK) equations in (2 + 1)-dimenslonal spaces. J. Phys. A 34, 1785-1792 (2001)
20. Wang, M.L.: Exact solutions for a compound KdV-Burgers equation. Phys. Lett. A 213(5-6), 279-287 (1996)
21. Yan, Z.Y., Zhang, H.Q.: A simple transformation for nonlinear waves. Phys. Lett. A 285, 355-362 (2001)
22. Gardner, CS., Greene, J.M., Kruskal, M.D., Miura, R.M.: Method for solving Korteweg-de Vries equation. Phys. Rev. Lett. 19, 1095-1097 (1967)
23. Guo, B.Y., Shen, J.: On spectral, approximations using modified Legendre rational, functions: Application to the Korteweg-de Vries equation on the half line (Special Issue). Indiana Univ. Math. J. 50, 181-204 (2001)
24. Huang, W.Z., Sloan, D.M.: The pseudospectral, method for third-order differential equations. SIAM J. Numer. Anal. 29(6), 1626-1647 (1992)
25. Ma, H.P., Sun, W.W: A Legendre-Petrov-Galerkin and Chebyshev collocation method for third-order differential, equations. SIAM J. Numer. Anal. 38(5), 1425-1438 (2000)
26. Ma, H.P., Sun, W.W.: Optimal error estimates of the Legendre-Petrov- Galerkin method for the Korteweg-de Vries equation. SIAM J. Numer. Anal. 39(4), 1380-1394 (2001)
27. Refik Bahadir, A.: Exponential finite-difference method applied to Kortewegñ-de Vries for small times. Appl. Math. Comput. 160(3), 675-682 (2005)
28. Kansa, E.J.: Muitiquadrics - A scattered data approximation scheme with applications to computational fluid dynamics - II. Comput. Math. Appl. 19, 147-161 (1990)
29. Hardy, R.L.: Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res. 176, 1905-1915 (1971)
30. Franke, R.: Scattered data interpolation: Test of some methods. Math. Comput. 38, 181-200 (1982)
31. Hon, Y.C., Mao, X.Z.: An efficient numerical scheme for Burgersí equation. Appl. Math. Comput. 95(1), 37-50 (1998)
32. Hon, Y.C., Cheung, K.F., Mao, X.Z., Kansa, E.J.: Multi-quadric solution for shallow water equations. ASCE J. Hydraul. Eng. 125(5), 524-533 (1999)
33. Hon, Y.C., Mao, X.Z.: A radial, basis function method for solving options pricing model. Financ. Eng. 8(1), 31-49 (1999)
34. Marcozzi, M., Choi, S., Chen, C.S.: On the use of boundary conditions for variational, formulations arising in financial, mathematics. Appl. Math. Comput. 124, 197-214 (2001)
35. Fasshauer, G.E.: Solving partial differential equations by collocation with radial basis functions. In: Surface Fitting and Muitiresoiution Methods, Le Méhauté, A., Rabut, C., Schumaker, L.L. (eds.), pp. 131-138. Vanderbilt University Press, Nashville, TN (1997)
36. Wendland, H.: Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv. Comput. Math. 4, 389-396 (1995)
37. Madych, W.R., Nelson, S.A.: Multivariate interpolation and conditionally positive definite functions II. Math. Comput. 54(189), 211-230 (1990)
38. Madych, W.R., Nelson, S.A.: Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation. J. Approx. Theory 70, 94-114 (1992)
39. Franke, C., Schaback, R.: Convergence orders of meshiess collocation methods using radial, basis functions. Adv. Comput. Math. 8(4), 381-399 (1997)
40. Franke, C., Schaback, R.: Solving partial differential equations by collocation using radial basis functions. Appl. Math. Comput. 93(1), 73-82 (1998)
41. Hon, Y.C., Wu, Z.: Additive Schwarz domain decomposition with radial basis approximation. Int. J. Appl. Math. 4(1), 599-606 (2000)
42. 2 in multiquadric interpolation. Comput. Math. Appl. 21, 29-42 (1991)
43. Tarwater, A.E.: A parameter study of Hardy's multiquadric method for scattered, data interpolation, Lawrence Livermore National Laboratory, Technical Report UCRL-54670 (1985)
44. Irk, D., Daǧ, İ., Saka, B.: A small time solution for the Kortewegde Vries equation, using spline approximation. Appl. Math. Comput. 173(2), 834-846 (2006)
45. Dehghan, M.: Finite difference procedueres for solving a problem arising in modeling and design of certain optoecletronic devices. Math. Comput. Simul. 71, 16-30 (2006)
46. Chantrasirivan, S.: Cartesian, grid method using radial basis functions for solving Poisson, Helmholtz, and diffusion-convection equations. Eng. Anal. Boundary Elements 28, 1417-1425 (2004)
47. Liu, G.R., Gu, Y.T.: Boundary meshfree methods based on the boundary point methods. Eng. Anal. Boundary Elements 28, 475-487 (2004)
48. Dehghan, M., Tatari, M.: Determination of a control parameter in a one-dimensional parabolic equation using the method, of radial basis functions, Mathematical, and Computer Modelling 44 (2006) 1160-1168