Analytical study of the mathematical model of wave propagation in shallow water by the Galerkin method

E. V. Bychkov

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 14 (2021) no. 1, pp. 26-38

Résumé

Of concern is an initial-boundary value problem for the modified Boussinesq equation (IMBq equation) is considered. The equation is often used to describe the propagation of waves in shallow water under the condition of mass conservation in the layer and taking into account capillary effects. In addition, it is used in the study of shock waves. The modified Boussinesq equation belongs to the Sobolev type equations. Earlier, using the theory of relatively $p$-bounded operators, the theorem of existence and uniqueness of the solution to the initial-boundary value problem was proved. In this paper, we will prove that the solution constructed by the Galerkin method using the system orthornormal eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator converges $^*$-weakly to an precise solution. Based on the compactness method and Gronwall's inequality, the existence and uniqueness of solutions to the Cauchy–Dirichlet and the Showalter–Sidorov–Dirichlet problems for the modified Boussinesq equation are proved.

Détail

Keywords: modified Boussinesq equation, initial-boundary value problem, Galerkin method, $^*$-weak convergence.
Mots-clés : Sobolev type equation

Bibliographie

[1] I.L. Bogolubsky, “Some Examples of Inelastic Soliton Interaction”, Computational Physics Communications, 13:2 (1977), 49–55 | DOI

[2] P.A. Clarkson, R.J. Leveque, R. Saxton, “Solitary Wave Interactions in Elastic Rods”, Studies in Applied Mathematics, 75:1 (1986), 95–122 | DOI

[3] Shubin Wang, Guowang Chen, “Small Amplitude Solutions of the Generalized IMBq Equation”, Journal Mathematical Analysis Applied, 274 (2002), 846–866 | DOI

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

[5] Weiguo Zhang, Wenxiu Ma, “Explicit Solitary–Wave Solutions to Generalized Pochhammer–Chree Equations”, Applied Mathematics and Mechanics, 20:6 (1999), 625–632 | DOI

[6] Xu Runzhang, Liu Yacheng, “Global Existence and Blow-Up of Solutions for Generalized Pochhammer–Chree Equations”, Acta Mathematica Scientia, 30:5 (2010), 1793–1807 | DOI

[7] R.E. Showalter, “Sobolev Equations for Nonlinear Dispersive Systems”, Applicable Analysis, 7:4 (1977), 279–287 | DOI

[8] Sviridyuk G. A., Sukacheva T. G., “Phase Spaces of a Class of Operator Equations”, Differential Equations, 26:2 (1990), 250–258 (in Russian)

[9] Sviriduyk G. A., Zamyshlyaeva A. A., “The Phase Space of a Class of Linear Higher Order Sobolev Type Equations”, Differential Equations, 42:2 (2006), 269–278 | DOI

[10] Zamyshlyaeva A. A., Bychkov E. V., “The Cauchy Problem for the Sobolev Type Equation of Higher Order”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 11:1 (2018), 5–14 | DOI

[11] Sviridyuk G. A., Zamyshlyaeva A. A., “The Phase Spaces of a Class of Linear Higher-Order Sobolev Type Equations”, Differential Equations, 42:2 (2006), 269–278 | DOI

[12] Zamyshlyaeva A. A., “The Higher-Order Sobolev-Type Models”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 7:2 (2014), 5–28 (in Russian) | DOI

[13] Sviridyuk G. A., “On the Solvability of a Singular System of Ordinary Differential Equations”, Differential Equations, 23:9 (1987), 1637–1639 (in Russian)

[14] Chistyakov V. F., Chistyakova E. V., “Linear Differential-Algebraic Equations Perturbed by Volterra Integral Operators”, Differential Equations, 53:10 (2017), 1274–1287 | DOI

[15] Zamyshlyaeva A. A., Lut A. V., “Numerical Investigation of the Boussinesq–Löve Mathematical Models on Geometrical Graphs”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 10:2 (2017), 137–143 | DOI

[16] D.E. Shafranov, O.G. Kitaeva, “The Barenblatt – Zheltov – Kochina Model with the Showalter – Sidorov Condition and Additive \flqq White Noise\frqq in Spaces of Differential Forms on Riemannian Manifolds without Boundary”, Global and Stochastic Analysis, 5:2 (2018), 145–159

[17] Manakova N. A., Bogatyreva E. A., “On a Solution of the Dirichlet–Cauchy Problem for the Barenblatt–Gilman Equation”, The Bulletin of Irkutsk State University. Series: Mathematics, 7 (2014), 52–60 (in Russian)

[18] Bogatyreva E. A., Manakova N. A., “Numerical Simulation of the Process of Nonequilibrium Counterflow Capillary Imbibition”, Computational Mathematics and Mathematical Physics, 56:1 (2016), 132–139 | DOI

[19] Sveshnikov A. G., Al'shin A. B., Korpusov M. O., Pletner Yu.D., Linear and Nonlinear Sobolev Type Equation, Fizmatlit, M., 2007 (in Russian)

[20] A.V. Keller, “On the Computational Efficiency of the Algorithm of the Numerical Solution of Optimal Control Problems for Models of Leontieff Type”, Journal of Computational and Engineering Mathematics, 2:2 (2015), 39–59 | DOI

[21] Lions J. L., Sur Quelques Methodes de Resolution des Problemes aux Limites non Linears, Dunod, Gauthier Villars, Paris, 1969 (in French)

[22] Hartman P., Ordinary Differential Equations, John Wiley and Sons, New York–London–Sydney, 1964

[23] Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978 (in German)

[24] Sviridyuk G. A., Zagrebina S. A., “The Showalter–Sidorov Problem as a Phenomena of the Sobolev-Type Equations”, The Bulletin of Irkutsk State University. Series: Mathematics, 3:1 (2010), 104–125 (in Russian)