The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas. (English) Zbl 1263.76047

Summary: This paper investigates the solitary wave solutions of the two-dimensional regularized long-wave equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas. The main idea behind the numerical solution is to use a combination of boundary knot method and the analog equation method. The boundary knot method is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution, the boundary knot method uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to method of fundamental solution, the radial basis function is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method. According to the analog equation method, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Furthermore, in order to show the efficiency and accuracy of the proposed method, the present work is compared with finite difference scheme. The new method is analyzed for the local truncation error and the conservation properties. The results of several numerical experiments are given for both the single and double-soliton waves.


76M20 Finite difference methods applied to problems in fluid mechanics
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
76B25 Solitary waves for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI


[1] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in non-linear dispersive systems, Philos. trans. R. soc. lond. ser. A, 272, 47-78, (1972) · Zbl 0229.35013
[2] Chen, W.; Tanaka, M., New insights in boundary-only and domain-type RBF methods, Int. J. nonlinear sci. numer. simul., 1, 145-152, (2000) · Zbl 0954.65084
[3] Chen, W.; Tanaka, M., A meshfree, integration-free, and boundary-only RBF technique, Comput. math. appl., 43, 379-391, (2002) · Zbl 0999.65142
[4] Chen, W.; Shen, L.J.; Shen, Z.J.; Yuan, G.W., Boundary knot method for Poisson equations, Engrg. anal. bound. elem., 29, 756-760, (2005) · Zbl 1182.74250
[5] Chen, W., Symmetric boundary knot method, Engrg. anal. bound. elem., 26, 489-494, (2002) · Zbl 1006.65500
[6] Dag, I., Least squares quadratic B-spline finite element method for the regularized long wave equation, Comput. methods mech. engng., 182, 205-215, (2000) · Zbl 0964.76042
[7] Dag, I.; Saka, B.; Irk, D., Galerkin method for the numerical solution of the RLW equation using quintic B-splines, J. comput. appl. math., 190, 532-547, (2006) · Zbl 1086.65094
[8] Dehghan, M., Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices, Math. comput. simulation, 71, 16-30, (2006) · Zbl 1089.65085
[9] Dehghan, M., On the solution of an initial-boundary value problem that combines Neumann and integral condition for the wave equation, Numer. methods partial differential equations, 21, 24-40, (2005) · Zbl 1059.65072
[10] Dehghan, M., A computational study of the one-dimensional parabolic equation subject to nonclassical boundary specifications, Numer. methods partial differential equations, 22, 220-257, (2006) · Zbl 1084.65099
[11] Dehghan, M., The one-dimensional heat equation subject to a boundary integral specification, Chaos solitons fractals, 32, 661-675, (2007) · Zbl 1139.35352
[12] Dehghan, M.; Shokri, A., A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions, Math. comput. simulation, 79, 700-715, (2008) · Zbl 1155.65379
[13] M. Dehghan, H. Hosseinzadeh, Calculation of 2D singular and near singular integrals of boundary element method based on the complex space C, Applied Mathematical Modelling (2011), in press, doi:10.1016/j.apm.2011.07.036. · Zbl 1236.65148
[14] Djidjeli, K.; Price, W.G.; Twizell, E.H.; Cao, Q., A linearized implicit pseudo-spectral method for some model equations—the regularized long wave equations, Commun. numer. meth. engng., 19, 847-863, (2003) · Zbl 1035.65110
[15] Eilbeck, J.C.; McGuire, G.R., Numerical study of RLW equation I: numerical methods, J. comput. phys., 19, 43-57, (1975) · Zbl 0325.65054
[16] Eilbeck, J.C.; McGuire, G.R., Numerical study of the regularized long-wave equation: II—interaction of solitary waves, J. comput. phys., 23, 63-73, (1977) · Zbl 0361.65100
[17] Guo, B.Y.; Cao, W.M., The Fourier pseudospectral method with a restrain operator for the RLW equation, J. comput. phys., 74, 110-126, (1988) · Zbl 0684.65097
[18] Goldstein, J.; Kajikiya, A.; Oharu, S., On some nonlinear dispersive equations in several space variables, Differential integral equations, 3, 617-632, (1990) · Zbl 0735.35103
[19] Helal, M.A., Soliton solution of some nonlinear partial differential equations and its applications in fluid mechanics, Chaos solitons fractals, 13, 1917-1929, (2002) · Zbl 0997.35063
[20] Horton, W.; Hasegawa, A., Quasi-two-dimensional dynamics of plasmas and fluids, Chaos, 4, 227-251, (1994)
[21] Hon, Y.C.; Chen, W., Boundary knot method for 2D and 3D Helmholtz and convection-diffusion problems under complicated geometry, Int. J. numer. meth. engrg., 56, 1931-1948, (2003) · Zbl 1072.76048
[22] Huang, Z., On Cauchy problems for the RLW equation in two space dimensions, Appl. math. mech., 23, 159-164, (2002)
[23] Jin, B.; Zheng, Y., Boundary knot method for the Cauchy problem associated with the inhomogeneous Helmholtz equation, Engrg. anal. bound. elem., 29, 925-935, (2005) · Zbl 1182.65179
[24] Jin, B.; Chen, W., Boundary knot method based on geodesic distance for anisotropic problems, J. comput. phys., 215, 614-629, (2006) · Zbl 1093.65116
[25] Hu, S.P.; Fan, C.M.; Young, D.L., The meshless analog equation method for solving heat transfer to molten polymer flow in tubes, Int. J. heat mass transfer, 53, 2240-2247, (2010) · Zbl 1190.80013
[26] Katsikadelis, J.T., The analog equation method—A powerful BEM-based solution technique for solving linear and nonlinear engineering problems, (), 167 · Zbl 0813.65124
[27] Katsikadelis, J.T.; Nerantzaki, M.S., The boundary element method for nonlinear problems, Engrg. anal. bound. elem., 23, 365-373, (1999) · Zbl 0945.65132
[28] Katsikadelis, J.T.; Tsiatas, G.C., Nonlinear dynamic analysis of heterogeneous orthotropic membranes by the analog equation method, Engrg. anal. bound. elem., 27, 115-124, (2003) · Zbl 1080.74559
[29] Katsikadelis, J.T., The 2D elastostatic problem in inhomogeneous anisotropic bodies by the meshless analog equation method (MAEM), Engrg. anal. bound. elem., 32, 997-1005, (2008) · Zbl 1244.74220
[30] Katsikadelis, J.T., The analog equation method, A boundary-only integral equation method for nonlinear static and dynamic problems in general bodies, Int. J. theor. appl. mech., 27, 13-38, (2002) · Zbl 1051.74052
[31] Katsikadelis, J.T., Numerical solution of multi-term fractional differential equations, Zamm, 89, 593-608, (2009) · Zbl 1175.26013
[32] Katsikadelis, J.T.; Nerantzaki, M.S., A boundary element solution to the soap bubble problem, Comput. mech., 27, 154-159, (2001) · Zbl 0980.65142
[33] Katsikadelis, J.T., The meshless analog equation method: I. solution of elliptic partial differential equations, Arch. appl. mech., 79, 557-578, (2009) · Zbl 1264.74283
[34] Katsikadelis, J.T., The fractional wave-diffusion equation in bounded inhomogeneous anisotropic media. an AEM solution, (), 255-276 · Zbl 1161.74509
[35] Kawahara, T.; Araki, K.; Toh, S., Interactions of two-dimensionally localized pulses of the regularized-long-wave equation, Phys. D, 59, 79-89, (1992) · Zbl 0810.35011
[36] Khalifa, A.K.; Raslan, K.R.; Alzubaidi, H.M., A finite difference scheme for the MRLW and solitary wave interactions, Appl. math. comput., 189, 346-354, (2007) · Zbl 1123.65085
[37] Lin, J.; Xie, Z.; Zhou, J., High-order compact difference scheme for the regularized long wave equation, Comm. numer. methods engrg., 23, 135-156, (2007) · Zbl 1111.65074
[38] Mirzaei, D.; Dehghan, M., A meshless based method for solution of integral equations, Appl. numer. math., 60, 245-262, (2010) · Zbl 1202.65174
[39] Mirzaei, D.; Dehghan, M., MLPG method for transient heat conduction problem with MLS as trial approximation in both time and space domains, computer modeling in engineering and sciences, Cmes, 72, 185-210, (2011) · Zbl 1231.80047
[40] Prahovic, M.G.; Hazeltine, R.D.; Morrison, P.J., Exact solutions for a system of nonlinear plasma fluid equations, Phys. fluids B, 4, 831-840, (1992)
[41] Peregrine, D.H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[42] Saka, B.; Dag, I., A numerical solution of the RLW equation by Galerkin method using quartic B-splines, Comm. numer. methods engrg., 24, 1339-1361, (2008) · Zbl 1156.65085
[43] Shokri, A.; Dehghan, M., A meshless method using the radial basis functions for numerical solution of the regularized long wave equation, Numer. methods partial differential equations, 26, 807-825, (2010) · Zbl 1195.65142
[44] Shang, Y.; Niu, P., Explicit exact solutions for the RLW equation and the SRLW equation in two space dimensions, Math. appl., 11, 1-5, (1988) · Zbl 0935.35161
[45] Su, X.N.; Horton, W.; Morrison, P.J., Drift wave vortices in inhomogeneous plasmas, Phys. fluids B, 3, 921-930, (1991)
[46] Tatari, M.; Dehghan, M., On the solution of the non-local parabolic partial differential equations via radial basis functions, Appl. math. model., 33, 1729-1738, (2009) · Zbl 1168.65403
[47] Tian, B.; Li, W.; Gao, Y.T., On the two-dimensional regularized long-wave equation in fluids and plasmas, Acta mech., 160, 235-239, (2003) · Zbl 1064.76018
[48] K. Wiklund, Non-linear wave interaction in a Hamiltonian family of Hasegawa-Mima related equation, in: 27th EPS Conference on Contr. Fusion and Plasma Phys., Budapest 24B, 2000, pp. 1280-1283.
[49] Wang, Q.; Huang, M., The numerical simulation of solitary wave on regular long wave equation, Acta sci. nat. univ. jilinensis, 3, 1-13, (1997)
[50] Wang, F.; Chen, W.; Jiang, Xi., Investigation of regularized techniques for boundary knot method, Int. J. numer. meth. biomed. engn., 26, 1868-1877, (2010) · Zbl 1208.65173
[51] Wazwaz, A.M., Partial differential equations and solitary waves theory, (2009), Springer New York · Zbl 1175.35001
[52] Wazwaz, A.M., Analytic study on nonlinear variants of the RLW and the phi-four equations, Commun. nonlinear sci. numer. simul., 12, 314-327, (2007) · Zbl 1109.35099
[53] Zheng-hong, H., On Cauchy problems for the RLW equation in two space dimensional, Appl. math. mech., 23, 169-177, (2002)
[54] Zhang, L., A finite difference scheme for generalized regularized long-wave equation, Appl. math. comput., 168, 962-972, (2005) · Zbl 1080.65079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.