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

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

### 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)
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### 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
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