Geodesic
Finite difference method for modified Boussinesq equation
Journal of computational and engineering mathematics, Tome 5 (2018) no. 4, pp. 58-63.

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In this paper a numerical of the solution of the Cauchy problem for the nonlinear modified Boussinesq equation (or IMBq equation) is studied. This equation with the boundary conditions models the propagation of waves in shallow water, taking into account capillary effects and the preservation of mass in the layer, filtration of water in the soil, as well as shock waves. In the case when the equation is nondegenerate, a global solution and a solution in the form of solitons are obtained. In the degenerate case, the existence of a unique local solution was proved by the methods of phase space and the theory of relatively limited ones developed by G. Sviridyuk and his students, as well as the theory of differentiable Banach manifolds. A numerical study of this problem by the modified Galerkin method has already been carried out earlier. However, the operation time of algorithms based on modified Galerkin method it rapidly increases with an increase amount of Galerkin sum. In this article, a numerical study is carried out by the finite difference method. The Cauchy – Dirichlet problem for the IMBq equation is reduced to an implicit difference problem. A comparison is made of the speed of the modified Galerkin method and the finite difference method.
Keywords: Sobolev type equations, finite difference method, Galerkins method. 
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E. V. Bychkov. Finite difference method for modified Boussinesq equation. Journal of computational and engineering mathematics, Tome 5 (2018) no. 4, pp. 58-63. http://geodesic.mathdoc.fr/item/JCEM_2018_5_4_a4/

[1] D. G. Arkhipov, G. A. Khabakhpashev, “New Equation for the Description of Inelastic Interaction of Nonlinear Localized Waves in Dispersive Media”, JETP Letters, 93:8 (2011), 423–426 | DOI

[2] A. A. Zamyshlyaeva, “Matematicheskie modeli sobolevskogo tipa vysokogo poryadka”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 7:2 (2014), 5–28 | DOI | Zbl

[3] S. Wang, G. Chen, “Small Amplitude Solutions of the Generalized IMBq Equation”, Journal of Mathematical Analysis and Applications, 274:2 (2002), 846–866 | DOI | MR | Zbl

[4] A. A. Zamyshlyaeva, E. V. Bychkov, “Fazovoe prostranstvo modifitsirovannogo uravneniya Bussineska”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 2012, no. 12, 13–19 | MR | Zbl

[5] A. A. Zamyshlyaeva, S. V. Surovtsev, “Numerical Investigation of One Sobolev Type Mathematical Model”, J. Comp. Eng. Math., 2:3 (2015), 72–80 | DOI | Zbl

[6] A. A. Zamyshlyaeva, E. V. Bychkov, “The Cauchy Problem for the Sobolev Type Equation of Higher Order”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 11:1 (2018), 5–14 | DOI | MR | Zbl

[7] G. A. Sviridyuk, M. M. Yakupov, “The Phase Space of the Initial-Boundary Value Problem for the Oskolkov System”, Differ. Equ., 32:11 (1996), 1535–1540 | MR | Zbl