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1
Content available remote Rigorous Numerics for PDEs with Indefinite Tail : Existence of a Periodic Solution of the Boussinesq Equation with Time-dependent Forcing
100%
Czechowski A. , Zgliczyński P.
Schedae Informaticae
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2015
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tom Vol. 24
143--158
EN
We consider the Boussinesq PDE perturbed by a time-dependent forcing. Even though there is no smoothing effect for arbitrary smooth initial data, we are able to apply the method of self-consistent bounds to deduce the existence of smooth classical periodic solutions in the vicinity of 0. The proof is non-perturbative and relies on construction of periodic isolating segments in the Galerkin projections.
2
Content available A new form of Boussinesq equations for long waves in water of non-uniform depth
100%
Szmidt J. , Hedzielski B.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2012
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tom Vol. 60, nr 3
631-643
EN
The paper describes the non-linear transformation of long waves in shallow water of variable depth. Governing equations of the problem are derived under the assumption that the non-viscous fluid is incompressible and the fluid flow is a rotation free. A new form of Boussinesq-type equations is derived employing a power series expansion of the fluid velocity components with respect to the water depth. These non-linear partial differential equations correspond to the conservation of mass and momentum. In order to find the dispersion characteristic of the description, a linear approximation of these equations is derived. A second order approximation of the governing equations is applied to study a time dependent transformation of waves in a rectangular basin of water of variable depth. Such a case corresponds to a spatially periodic problem of sea waves approaching a near-shore zone. In order to overcome difficulties in integrating these equations, the finite difference method is applied to transform them into a set of non-linear ordinary differential equations with respect to the time variable. This final set of these equations is integrated numerically by employing the fourth order Runge - Kutta method.
3
Content available remote Dynamics of Shallow Water Waves with Various Boussinesq Equations
100%
Kumar H. , Malik A. , Singh Gautam M. , Chand F.
Acta Physica Polonica A
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2017
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tom 131
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nr 2
275-282
EN
Attempt has been made to construct the solitary waves and shock wave solutions or domain walls (in higher dimension) for various Boussinesq equations. The method of undetermined coefficients have been used to explore the exact analytical solitary waves and shock wave solutions in terms of bell-shaped sech^p function and kink-shaped tanh^p function for the considered equations. The Boussinesq equation in the (1+1)-dimensional, the (2+1)-dimensional and the (3+1)-dimensional equations are studied and the parametric constraint conditions and uniqueness in view of both solitary waves and shock wave solutions are determined. Such solutions can be valuable and desirable for explaining some nonlinear physical phenomena in nonlinear science described by the Boussinesq equations. The effect of the varying parameters on the development of solitary waves and shock wave solutions have been demonstrated by direct numerical simulation technique.
4
Content available When the supersymmetry is not enough; the parasupersymmetric algebras of the Boussines equations
100%
Yourova A. A. , Yourov A. V. , Yourov V. A.
TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk
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2016
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tom Vol. 20, No 2
241--248
EN
In this article we look at a conundrum that the Boussinesq-type equations pose for mathematicians allowing a Miura-type transformation while at the same time exhibiting no trace of a supersymmetric structure. We demonstrate that this riddle should be unraveled by dropping the standard supersymmetric approach in favor of its generalization: the “parasupersymmetry”.
5
Content available Exact and approximate distributed controllability of processes described by KdV and Boussinesq equations: The Green’s function approach
63%
Klamka J. , Avetisyan A. S. , Khurshudyan A. Z. , Klamka J. , Avetisyan A. S. , Khurshudyan A. Z.
Archives of Control Sciences
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2020
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tom 30
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nr 1
6
Content available Exact and approximate distributed controllability of processes described by KdV and Boussinesq equations: The Green’s function approach
63%
Klamka J. , Avetisyan A. S. , Khurshudyan A. Z.
Archives of Control Sciences
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2020
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tom Vol. 30, No. 1
177--193
EN
In this paper, we study the constrained exact and approximate controllability of traveling wave solutions of Korteweg-de Vries (third order) and Boussinesq (fourth order) semi-linear equations using the Green’s function approach. Control is carried out by a moving external source. Representing the general solution of those equations in terms of the Frasca’s short time expansion, system of constraints on the distributed control is derived for both types of controllability. Due to the possibility of explicit solution provided by the heuristic method, the controllability analysis becomes straightforward. Numerical analysis confirms theoretical derivations.
7
Forecast of influence of planned sand output from under water from Obora sandpit on level and quality of underground water
63%
Chalfen M. , Szulczewski W.
Electronic Journal of Polish Agricultural Universities. Series Environmental Development
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1999
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tom 02
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nr 1
EN
The analysis of the influence of planned increase of sand output from under water in “Obora” sandpit on water relations of neighboring areas is conducted in this paper. The bases for the performed simulations were mathematical models based on Boussinesq equations and hydrodynamic dispersion equations. The model includes all most important factors occurring in a given area influencing the hydroisohipses system and lines of chemical pollutant concentrations. Apart from typical hydrogeological elements such as water-courses, wells, and supplying from the aeration zone, the terrain’s depression caused by copper ore exploitation was also considered. As results from the carried out multi-variant simulation computations, the planned sand output will not cause the formation of depression hopper and will not significantly deteriorate the quality of underground water in the area surrounding the sandpit.
8
Content available remote Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations
51%
Slimani A. , Bouzettouta L. , Guesmia A.
Demonstratio Mathematica
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2021
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tom Vol. 54, nr 1
558--575
EN
Keller-Segel chemotaxis model is described by a system of nonlinear partial differential equations: a convection diffusion equation for the cell density coupled with a reaction-diffusion equation for chemoattractant concentration. In this work, we study the phenomenon of Keller-Segel model coupled with Boussinesq equations. The main objective of this work is to study the global existence and uniqueness and boundedness of the weak solution for the problem, which is carried out by the Galerkin method.
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