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Keywords:
nonlinear Love equation; Faedo-Galerkin method; convergence of order $N$
Summary:
In this paper, a high-order iterative scheme is established for a nonlinear Love equation associated with homogeneous Dirichlet boundary conditions. This is a development based on recent results (L. T. P. Ngoc, N. T. Long (2011); L. X. Truong, L. T. P. Ngoc, N. T. Long (2009)) to get a convergent sequence at a rate of order $N \geq 2$ to a local unique weak solution of the above mentioned equation.
References:
[1] Albert, J.: On the decay of solutions of the generalized Benjamin-Bona-Mahony equation. J. Math. Anal. Appl. 141 (1989), 527-537. DOI 10.1016/0022-247X(89)90195-9 | MR 1009061 | Zbl 0697.35116
[2] Amick, C. J., Bona, J. L., Schonbek, M. E.: Decay of solutions of some nonlinear wave equations. J. Differ. Equations 81 (1989), 1-49. DOI 10.1016/0022-0396(89)90176-9 | MR 1012198 | Zbl 0689.35081
[3] Chattopadhyay, A., Gupta, S., Singh, A. K., Sahu, S. A.: Propagation of shear waves in an irregular magnetoelastic monoclinic layer sandwiched between two isotropic half-spaces. Internat. J. Engrg., Sci. Technol. 1 (2009), 228-244.
[4] Clarkson, P. A.: New similarity reductions and Painlevé analysis for the symmetric regularised long wave and modified Benjamin-Bona-Mahoney equations. J. Phys. A, Math. Gen. 22 (1989), 3821-3848. DOI 10.1088/0305-4470/22/18/020 | MR 1015235 | Zbl 0711.35113
[5] Deimling, K.: Nonlinear Functional Analysis. Springer Berlin (1985). MR 0787404 | Zbl 0559.47040
[6] Dutta, S.: On the propagation of Love type waves in an infinite cylinder with rigidity and density varying linearly with the radial distance. Pure Applied Geophys. 98 (1972), 35-39. DOI 10.1007/BF00875578
[7] Lakshmikantham, V., Leela, S.: Differential and Integral Inequalities. Theory and Applications. Vol. I: Ordinary Differential Equations. Mathematics in Science and Engineering. Vol. 55 Academic Press, New York (1969). MR 0379933 | Zbl 0177.12403
[8] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites nonlinéaires. French Etudes mathematiques Dunod; Gauthier-Villars, Paris (1969). MR 0259693
[9] Makhankov, V. G.: Dynamics of classical solitons (in nonintegrable systems). Phys. Rep. 35 (1978), 1-128. DOI 10.1016/0370-1573(78)90074-1 | MR 0481361
[10] Ngoc, L. T. P., Duy, N. T., Long, N. T.: A linear recursive scheme associated with the Love equation. Acta Math. Vietnam. 38 (2013), 551-562. DOI 10.1007/s40306-013-0034-z | MR 3129917 | Zbl 1310.35174
[11] Ngoc, L. T. P., Duy, N. T., Long, N. T.: Existence and properties of solutions of a boundary problem for a Love's equation. Bull. Malays. Math. Sci. Soc. (2) 37 (2014), 997-1016. MR 3295564 | Zbl 1304.35231
[12] Ngoc, L. T. P., Long, N. T.: A high order iterative scheme for a nonlinear Kirchhoff wave equation in the unit membrane. Int. J. Differ. Equ. 2011 (2011), Article ID 679528, 31 pages. MR 2854955 | Zbl 1242.35014
[13] Ngoc, L. T. P., Truong, L. X., Long, N. T.: An $N$-order iterative scheme for a nonlinear Kirchhoff-Carrier wave equation associated with mixed homogeneous conditions. Acta Math. Vietnam. 35 (2010), 207-227. MR 2731324 | Zbl 1233.35134
[14] Ogino, T., Takeda, S.: Computer simulation and analysis for the spherical and cylindrical ion-acoustic solitons. J. Phys. Soc. Jpn. 41 (1976), 257-264. DOI 10.1143/JPSJ.41.257
[15] Parida, P. K., Gupta, D. K.: Recurrence relations for a Newton-like method in Banach spaces. J. Comput. Appl. Math. 206 (2007), 873-887. DOI 10.1016/j.cam.2006.08.027 | MR 2333719 | Zbl 1119.47063
[16] Paul, M. K.: On propagation of Love-type waves on a spherical model with rigidity and density both varying exponentially with the radial distance. Pure Applied Geophys. 59 (1964), 33-37. DOI 10.1007/BF00880505 | Zbl 0135.23902
[17] Radochová, V.: Remark to the comparison of solution properties of Love's equation with those of wave equation. Apl. Mat. 23 (1978), 199-207. MR 0492985
[18] Seyler, C. E., Fenstermacher, D. L.: A symmetric regularized-long-wave equation. Phys. Fluids 27 (1984), 4-7. DOI 10.1063/1.864487 | Zbl 0544.76170
[19] Truong, L. X., Ngoc, L. T. P., Long, N. T.: High-order iterative schemes for a nonlinear Kirchhoff-Carrier wave equation associated with the mixed homogeneous conditions. Nonlinear Anal., Theory Mathods Appl., Ser. A, Theory Methods 71 (2009), 467-484. DOI 10.1016/j.na.2008.10.086 | MR 2518053 | Zbl 1173.35603
[20] Truong, L. X., Ngoc, L. T. P., Long, N. T.: The $N$-order iterative schemes for a nonlinear Kirchhoff-Carrier wave equation associated with the mixed inhomogeneous conditions. Appl. Math. Comput. 215 (2009), 1908-1925. DOI 10.1016/j.amc.2009.07.056 | MR 2557432 | Zbl 1191.65122
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