×
Mohammadi, Maryam; Mokhtari, Reza

Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. (English) Zbl 1220.65143

J. Comput. Appl. Math. 235, No. 14, 4003-4014 (2011).
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
Reviewer: Wilhelm Heinrichs (Essen)

Cited in 51 Documents

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35C10 Series solutions to PDEs
35L75 Higher-order nonlinear hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)

Keywords:

generalized regularized long wave equation; reproducing kernel space; series solution; convergence; evolution equations
PDF BibTeX XML Cite
\textit{M. Mohammadi} and \textit{R. Mokhtari}, J. Comput. Appl. Math. 235, No. 14, 4003--4014 (2011; Zbl 1220.65143)
Full Text: DOI
  • logo of hbz
  • logo of WorldCat

References:

[1] Peregrine, D.H., Calculations of the development of an undular bore, J. fluid mech., 25, 321-330, (1966)
[2] Zhang, L., A finite difference scheme for generalized regularized long-wave equation, Appl. math. comput., 168, 962-972, (2005) · Zbl 1080.65079
[3] Durán, A.; López-Marcos, M.A., Numerical behaviour of stable and unstable solitary waves, Appl. numer. math., 42, 95-116, (2002) · Zbl 1002.65134
[4] Bona, J.L.; McKinney, W.R.; Restrepo, J.M., Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation, J. nonlinear sci., 10, 603-638, (2000) · Zbl 0972.35131
[5] 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. methods eng., 19, 847-863, (2003) · Zbl 1035.65110
[6] Bhardwaj, D.; Shankar, R., A computational method for regularized long wave equation, Comput. math. appl., 40, 1397-1404, (2000) · Zbl 0965.65108
[7] Eilbeck, J.C.; McGuire, G.R., Numerical study of the RLW equation I, J. comput. phys., 19, 43-57, (1975) · Zbl 0325.65054
[8] Eilbeck, J.C.; McGuire, G.R., Numerical study of the RLW equation II, J. comput. phys., 23, 63-73, (1977) · Zbl 0361.65100
[9] Jain, P.C.; Iskandar, L., Numerical solution of the RLW equation, Comput. methods appl. mech. engg., 20, 195-200, (1979)
[10] Lin, J.; Xie, Z.; Zhou, J., High-order compact difference scheme for the regularized longwave equation, Commun. numer. methods eng., 23, 135-156, (2007) · Zbl 1111.65074
[11] Ramos, J.I., Explicit finite difference methods for the EW and RLW equations, Appl. math. comput., 179, 622-638, (2007) · Zbl 1102.65092
[12] Guo, B.; Manoranjan, V.S., A spectral method for solving the RLW equation, IMA J. numer. anal., 5, 307-318, (1985) · Zbl 0577.65106
[13] Daǧ, I., Least-square quadratic \(B\)-spline finite element method for the regularized long wave equation, Comput. methods appl. mech. engrg., 182, 205-215, (2000) · Zbl 0964.76042
[14] Daǧ, I.; Özer, M.N., Approximation of the RLW equation by the least square cubic \(B\)-spline finite element method, Appl. math. model., 25, 221-231, (2001) · Zbl 0990.65110
[15] Gardner, L.R.T.; Gardner, G.A.; Dogan, A., A least-square finite element method for the RLW equation, Commun. numer. methods eng., 12, 795-804, (1996) · Zbl 0867.76040
[16] Daǧ, I.; Dogan, A.; Saka, B., \(B\)-spline collocation methods for numerical solutions of the RLW equation, Int. J. comput. math., 80, 743-757, (2003) · Zbl 1047.65088
[17] Daǧ, 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
[18] Dogan, A., Numerical solution of RLW equation using linear finite elements within galerkin’s method, Appl. math. model., 26, 771-783, (2002) · Zbl 1016.76046
[19] Esen, A.; Kutluay, S., Application of a lumped Galerkin method to the regularized long wave equation, Appl. math. comput., 174, 833-845, (2006) · Zbl 1090.65114
[20] Gardner, L.R.T.; Gardner, G.A.; Daǧ, I., A \(B\)-spline finite element method for the regularized long wave equation, Commun. numer. methods eng., 11, 59-68, (1995) · Zbl 0819.65125
[21] Raslan, K.R., A computational method for the regularized long wave equation, Appl. math. comput., 167, 1101-1118, (2005) · Zbl 1082.65582
[22] Saka, B.; Daǧ, I., A collocation method for the numerical solution of the RLW equation using cubic \(B\)-spline basis, Arab. J. sci. eng., 30, 39-50, (2005)
[23] Saka, B.; Daǧ, I.; Dogan, A., Galerkin method for the numerical solution of the RLW equation using quadratic \(B\)-spline, Int. J. comput. math., 81, 727-739, (2004) · Zbl 1060.65109
[24] Soliman, A.A.; Hussien, M.H., Collocation solution for RLW equation with septic spline, Appl. math. comput., 161, 623-636, (2005) · Zbl 1061.65102
[25] Dogan, A., Numerical solution of regularized long wave equation using petrov – galerkin method, Commun. numer. methods eng., 17, 485-494, (2001) · Zbl 0985.65121
[26] Dehghan, M.; Shokri, A., A meshless method using the radial basis functions for numerical solution of the regularized long wave equation, Numer. methods partial differential equations, 26, 4, 807-825, (2010) · Zbl 1195.65142
[27] Siraj-ul-Islam; Sirajul, H.; Arshed, A., A meshfree method for the numerical solution of the RLW equation, J. comput. appl. math., 223, 2, 997-1012, (2009) · Zbl 1156.65090
[28] Mokhtari, R.; Mohammadi, M., Numerical solution of GRLW equation using sinc-collocation method, Comput. phys. comm., 181, 1266-1274, (2010) · Zbl 1219.65113
[29] Gardner, L.R.T.; Gardner, G.A.; Ayoub, F.A.; Amein, N.K., Approximations of solitary waves of the MRLW equation by \(B\)-spline finite element, Arab. J. sci. eng., 22, 183-193, (1997) · Zbl 0893.35113
[30] Daǧ, I.; Saka, B.; Irk, D., Application of cubic \(B\)-splines for numerical solution of the RLW equation, Appl. math. comput., 159, 373-389, (2004) · Zbl 1060.65110
[31] Khalifa, A.K.; Raslan, K.R.; Alzubaidi, H.M., A collocation method with cubic \(B\)-splines for solving the MRLW equation, Comput. appl. math., 212, 406-418, (2008) · Zbl 1133.65085
[32] Aronszajn, N., Theory of reproducing kernels, Trans. amer. math. soc., 68, 337-404, (1950) · Zbl 0037.20701
[33] Cui, M.G.; Geng, F.Z., An efficient computational method for second order boundary value problems of nonlinear differential equations, Appl. math. comput., 194, 354-365, (2007) · Zbl 1193.65134
[34] Cui, M.G.; Geng, F.Z., Solving singular two-point boundary value problem in reproducing kernel space, J. comput. appl. math., 205, 6-15, (2007) · Zbl 1149.65057
[35] Cui, M.G.; Geng, F.Z., Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space, Appl. math. comput., 192, 389-398, (2007) · Zbl 1193.34017
[36] Cui, M.G.; Geng, F.Z., Solving singular nonlinear two-point boundary value problems in the reproducing kernel space, J. Korean math. soc., 45, 3, 77-87, (2008)
[37] Cui, M.G.; Geng, F.Z., Solving a nonlinear system of second order boundary value problems, J. math. anal. appl., 327, 1167-1181, (2007) · Zbl 1113.34009
[38] Geng, F.Z., Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method, Appl. math. comput., 215, 2095-2102, (2009) · Zbl 1178.65085
[39] Cui, M.G.; Geng, F.Z., A computational method for solving one-dimensional variable-coefficient Burgers equation, Appl. math. comput., 188, 1389-1401, (2007) · Zbl 1118.35348
[40] Cui, M.G.; Chen, Z., The exact solution of nonlinear age-structured population model, Nonlinear anal. RWA, 8, 1096-1112, (2007) · Zbl 1124.35030
[41] Li, C.L.; Cui, M.G., The exact solution for solving a class nonlinear operator equations in the reproducing kernel space, Appl. math. comput., 143, 2-3, 393-399, (2003) · Zbl 1034.47030
[42] Xie, S.S.; Heo, S.; Kim, S.; Woo, G.; Yi, S., Numerical solution of one-dimensional Burgers equation using reproducing kernel function, J. comput. appl. math., 214, 417-434, (2008) · Zbl 1140.65069
[43] Yao, H.; Lin, Y., New algorithm for solving a nonlinear hyperbolic telegraph equation with an integral condition, Internat. J. numer. methods engrg., (2010)
[44] Zhou, Y.; Cui, M.G.; Lin, Y., Numerical algorithm for parabolic problems with non-classical conditions, J. comput. appl. math., 230, 770-780, (2009) · Zbl 1190.65136
[45] Zhou, Y.; Cui, M.G.; Lin, Y., The solution of a parabolic differential equation with non-local boundary conditions in the reproducing kernel space, Appl. math. comput., 202, 708-714, (2008)
[46] Chen, Z.; Zhou, Y., An efficient algorithm for solving Hilbert type singular integral equations of the second kind, Comput. math. appl., 58, 632-640, (2009) · Zbl 1189.65308
[47] Du, H.; Cui, M.G., Approximate solution of the Fredholm integral equation of the first kind in a reproducing kernel Hilbert space, Appl. math. lett., 21, 617-623, (2008) · Zbl 1145.65113
[48] Du, H.; Shen, J., Reproducing kernel method of solving singular integral equation with cosecant kernel, J. math. anal. appl., 348, 308-314, (2008) · Zbl 1152.45007
[49] Yang, L.; Cui, M.G., New algorithm for a class of nonlinear integro-differential equations in the reproducing kernel space, Appl. math. comput., 174, 942-960, (2006) · Zbl 1094.65136
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.