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Добірка наукової літератури з теми "Boussinesque Equation"

Автор: Grafiati

Опубліковано: 11 березня 2023

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Статті в журналах з теми "Boussinesque Equation"

Ike, CC, HN Onah, and CU Nwoji. "BESSEL FUNCTIONS FOR AXISYMMETRIC ELASTICITY PROBLEMS OF THE ELASTIC HALF SPACE SOIL: A POTENTIAL FUNCTION METHOD." Nigerian Journal of Technology 36, no. 3 (June 30, 2017): 773–81. http://dx.doi.org/10.4314/njt.v36i3.16.

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Elasticity problems are formulated using displacement methods or stress methods. In this paper a displacement formulation of axisymmetric elasticity problem is presented. The formulation uses the Boussinesqâ€“ Papkovich â€“ Neuber potential function. The problem is then solved by assuming Boussinesq â€“ Papkovich - Neuber potential functions in the form of Bessel functions of order zero and of the first kind. The potential functions are then made to satisfy the governing field equations and the associated boundary conditions for the particular problem of a point load at the origin of the semi-infinite linear elastic isotropic soil mass. The unknown parameters of the function are thus determined and used to find the stresses, strains and displacement fields in the loaded soil. The results obtained were identical with the results obtained by Boussinesq.Â http://dx.doi.org/10.4314/njt.v36i3.16

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CLARKSON, PETER A. "RATIONAL SOLUTIONS OF THE BOUSSINESQ EQUATION." Analysis and Applications 06, no. 04 (October 2008): 349–69. http://dx.doi.org/10.1142/s0219530508001250.

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Rational solutions of the Boussinesq equation are expressed in terms of special polynomials associated with rational solutions of the second and fourth Painlevé equations, which arise as symmetry reductions of the Boussinesq equation. Further generalized rational solutions of the Boussinesq equation, which involve an infinite number of arbitrary constants, are derived. The generalized rational solutions are analogs of such solutions for the Korteweg–de Vries and nonlinear Schrödinger equations.

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Clarkson, Peter A. "New exact solutions of the Boussinesq equation." European Journal of Applied Mathematics 1, no. 3 (September 1990): 279–300. http://dx.doi.org/10.1017/s095679250000022x.

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In this paper new exact solutions are derived for the physically and mathematically significant Boussinesq equation. These are obtained in two different ways: first, by generating exact solutions to the ordinary differential equations which arise from (classical and nonclassical) similarity reductions of the Boussinesq equation (these ordinary differential equations are solvable in terms of the first, second and fourth Painlevé equations); and second, by deriving new space-independent similarity reductions of the Boussinesq equation. Extensive sets of exact solutions for both the second and fourth Painlevé equations are also generated. The symbolic manipulation language MACSYMA is employed to facilitate the calculations involved.

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Melinand, Benjamin. "Long wave approximation for water waves under a Coriolis forcing and the Ostrovsky equation." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 6 (July 19, 2018): 1201–37. http://dx.doi.org/10.1017/s0308210518000136.

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This paper is devoted to the study of the long wave approximation for water waves under the influence of the gravity and a Coriolis forcing. We start by deriving a generalization of the Boussinesq equations in one (spatial) dimension and we rigorously justify them as an asymptotic model of water wave equations. These new Boussinesq equations are not the classical Boussinesq equations: a new term due to the vorticity and the Coriolis forcing appears that cannot be neglected. We study the Boussinesq regime and derive and fully justify different asymptotic models when the bottom is flat: a linear equation linked to the Klein–Gordon equation admitting the so-called Poincaré waves; the Ostrovsky equation, which is a generalization of the Korteweg–de Vries (KdV) equation in the presence of a Coriolis forcing, when the rotation is weak; and the KdV equation when the rotation is very weak. Therefore, this work provides the first mathematical justification of the Ostrovsky equation. Finally, we derive a generalization of the Green–Naghdi equations in one spatial dimension for small topography variations and we show that this model is consistent with the water wave equations.

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Jafari, Hossein, Nematollah Kadkhoda, and Chaudry Massod Khalique. "Application of Lie Symmetry Analysis and Simplest Equation Method for Finding Exact Solutions of Boussinesq Equations." Mathematical Problems in Engineering 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/452576.

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The Lie symmetry approach with simplest equation method is used to construct exact solutions of the bad Boussinesq and good Boussinesq equations. As the simplest equation, we have used the equation of Riccati.

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Xu, Fei, Yixian Gao, and Weipeng Zhang. "Construction of Analytic Solution for Time-Fractional Boussinesq Equation Using Iterative Method." Advances in Mathematical Physics 2015 (2015): 1–7. http://dx.doi.org/10.1155/2015/506140.

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This paper is aimed at constructing analytical solution for both linear and nonlinear time-fractional Boussinesq equations by an iterative method. By the iterative process, we can obtain the analytic solution of the fourth-order time-fractional Boussinesq equation inR,R2, andRn, the sixth-order time-fractional Boussinesq equation, and the2nth-order time-fractional Boussinesq equation inR. Through these examples, it shows that the method is simple and effective.

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Rashidi, Saeede, and S. Reza Hejazi. "Symmetry properties, similarity reduction and exact solutions of fractional Boussinesq equation." International Journal of Geometric Methods in Modern Physics 14, no. 06 (May 4, 2017): 1750083. http://dx.doi.org/10.1142/s0219887817500839.

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In this paper, some properties of the time fractional Boussinesq equation are presented. Group analysis of the time fractional Boussinesq equation with Riemann–Liouville derivative is performed and the corresponding optimal system of subgroups are determined. Next, we apply the obtained optimal systems for constructing reduced fractional ordinary differential equations (FODEs). Finally, we show how to derive exact solutions to time fractional Boussinesq equation via invariant subspace method.

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Abazari, Reza, and Adem Kılıçman. "Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form." Mathematical Problems in Engineering 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/468206.

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This paper presents the general case study of previous works on generalized Boussinesq equations, (Abazari, 2011) and (Kılıcman and Abazari, 2012), that focuses on the application ofG′/G-expansion method with the aid of Maple to construct more general exact solutions for the coupled Boussinesq equations. In this work, the mentioned method is applied to construct more general exact solutions of Boussinesq equation and improved Boussinesq equation, which the French scientistJoseph Valentin Boussinesq(1842–1929) described in the 1870s model equations for the propagation of long waves on the surface of water with small amplitude. Our work is motivated by the fact that theG′/G-expansion method provides not only more general forms of solutions but also periodic, solitary waves and rational solutions. The method appears to be easier and faster by means of a symbolic computation.

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Johnson, R. S. "A Two-dimensional Boussinesq equation for water waves and some of its solutions." Journal of Fluid Mechanics 323 (September 25, 1996): 65–78. http://dx.doi.org/10.1017/s0022112096000845.

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A two-dimensional Boussinesq equation, \[u_{tt} - u_{xx} + 3(u^2)_{xx} - u_{xxxx} - u_{yy} = 0,\] is introduced to describe the propagation of gravity waves on the surface of water, in particular the head-on collision of oblique waves. This equation combines the two-way propagation of the classical Boussinesq equation with the (weak) dependence on a second spatial variable, as occurs in the two-dimensional Korteweg-de Vries (2D KdV) (or KPII) equation. Exact and general solitary-wave, two-soliton and resonant solutions are obtained from the Hirota bilinear form of the equation. The existence of a distributed-soliton solution is investigated, but it is shown that this is not a possibility. However the connection with the classical 2D KdV equation (which does possess such a solution) is explored via a suitable parametric representation of the dispersion relation.A three-soliton solution is also constructed, but this exists only if an auxiliary constraint among the six parameters is satisfied; thus the two-dimensional Boussinesq equation is not one of the class of completely integrable equations, confirming the analysis of Hietarinta (1987). This constraint is automatically satisfied for the classical Boussinesq equation (which is completely integrable). Graphical reproductions of some of the solutions of the two-dimensional Boussinesq equations are also presented.

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Bulut, Hasan, Münevver Tuz, and Tolga Akturk. "New Multiple Solution to the Boussinesq Equation and the Burgers-Like Equation." Journal of Applied Mathematics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/952614.

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By considering an improved tanh function method, we found some exact solutions of Boussinesq and Burgers-like equations. The main idea of this method is to take full advantage of the Riccati equation which has more new solutions. We found some exact solutions of the Boussinesq equation and the Burgers-like equation.

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Дисертації з теми "Boussinesque Equation"

SANTELLI, LUCA. "Thermally driven flows in spherical geometries." Doctoral thesis, Gran Sasso Science Institute, 2021. http://hdl.handle.net/20.500.12571/23841.

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In this manuscript we describe an efficient numerical scheme for simu- lations of three-dimensional Navier-Stokes equations for incompressible viscous flows in spherical coordinates. The code is second order accurate in space and time and relies on a finite–difference discretization in space. The nonphysical singularities induced by the change of coordinates are addressed by exploiting a change of variables and special treatments of few discrete terms. Thanks to these precautions the time–step restrictions caused by the region around the po- lar axis are alleviated and the sphere center is source of limitations only in very unfavorable flow configurations. We test the code and compare results with literature, always obtaining an excellent agreement. The flexibility due to the structure of the code allows it to perform efficiently in several applications without requiring changes in the structure: the mesh can be stretched (in two of the three directions), complex boundary conditions can be implemented, and in addition to full spheres, also spherical shells and sectors can be easily simulated. Characterization of the behaviour of fluids between spherical shells is the focus of the second part of the manuscript. We firstly explored the low-Rayleigh number regime for non rotating Rayleigh-B ́enard convection. Various radial gravity profiles are analysed for both air and water. We observe how the effect of the different gravity can be reabsorbed by the introduction of an effective Rayleigh number, yielding a critical Rac ≈ 1750 for the onset of convection regardless of the specific gravity profile. The exploration of higher values of Ra shows that the system is subjected to hysteresis, i.e. the dynamic has a very strong dependence on initial conditions and flow parameters. We then explore the effect of an offset between the sphere center and the gravity center, which might be used to simulate the effect of a dishomogeneity in the Earth core. Even a small displacement between the two points gives rise to a distorted temperature profile, with a hot jet emerging from the inner sphere in the direction opposite to the shift. Nevertheless, while the local heat flux and temperature profile are greatly modified, the global heat flux seems to be mostly unaffected by these changes. Lastly, we analysed the diffusion–free scaling regime for slowly rotating Rayleigh- B ́enard convection between spherical shells. This regime is characterized by a bulk–dominated flow and its emergence, for the parameters used, is due to the peculiar properties of the spherical geometry.

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Sitanggang, Khairil Irfan. "Boussinesq-equation and rans hybrid wave model." [College Station, Tex. : Texas A&M University, 2008. http://hdl.handle.net/1969.1/ETD-TAMU-2795.

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Sjölander, Filip. "Numerical solutions to the Boussinesq equation and the Korteweg-de Vries equation." Thesis, KTH, Skolan för teknikvetenskap (SCI), 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297544.

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The aim of the report is to numerically construct solutions to two analytically solvable non-linear differential equations: the Korteweg–De Vries equation and the Boussinesq equation. To accomplish this, a range of numerical methods where implemented, including Galerkin methods. To asses the accuracy of the solutions, analytic solutions were derived for reference. Characteristic of both equations is that they support a certain type of wave-solutions called "soliton solutions", which admit an intuitive physical interpretation as solitary traveling waves. Theses solutions are the ones simulated. The solitons are also qualitatively studied in the report.

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Liu, Fang-Lan. "Some asymptotic stability results for the Boussinesq equation." Diss., Virginia Tech, 1993. http://hdl.handle.net/10919/40052.

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Sun, Weizhou. "LOCAL DISCONTINUOUS GALERKIN METHOD FOR KHOKHLOV-ZABOLOTSKAYA-KUZNETZOV EQUATION AND IMPROVED BOUSSINESQ EQUATION." The Ohio State University, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=osu1480327264817905.

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Moore, Kieron R. "Coupled Boussinesq equations and nonlinear waves in layered waveguides." Thesis, Loughborough University, 2013. https://dspace.lboro.ac.uk/2134/13636.

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There exists substantial applications motivating the study of nonlinear longitudinal wave propagation in layered (or laminated) elastic waveguides, in particular within areas related to non-destructive testing, where there is a demand to understand, reinforce, and improve deformation properties of such structures. It has been shown [76] that long longitudinal waves in such structures can be accurately modelled by coupled regularised Boussinesq (cRB) equations, provided the bonding between layers is sufficiently soft. The work in this thesis firstly examines the initial-value problem (IVP) for the system of cRB equations in [76] on the infinite line, for localised or sufficiently rapidly decaying initial conditions. Using asymptotic multiple-scales expansions, a nonsecular weakly nonlinear solution of the IVP is constructed, up to the accuracy of the problem formulation. The asymptotic theory is supported with numerical simulations of the cRB equations. The weakly nonlinear solution for the equivalent IVP for a single regularised Boussinesq equation is then constructed; constituting an extension of the classical d'Alembert's formula for the leading order wave equation. The initial conditions are also extended to allow one to separately specify an O(1) and O(ε) part. Large classes of solutions are derived and several particular examples are explicitly analysed with numerical simulations. The weakly nonlinear solution is then improved by considering the IVP for a single regularised Boussinesq-type equation, in order to further develop the higher order terms in the solution. More specifically, it enables one to now correctly specify the higher order term's time dependence. Numerical simulations of the IVP are compared with several examples to justify the improvement of the solution. Finally an asymptotic procedure is developed to describe the class of radiating solitary wave solutions which exist as solutions to cRB equations under particular regimes of the parameters. The validity of the analytical solution is examined with numerical simulations of the cRB equations. Numerical simulations throughout this work are derived and implemented via developments of several finite difference schemes and pseudo-spectral methods, explained in detail in the appendices.

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Dickson, Ronald. "Algebro-geometric solutions of the Boussinesq hierarchy /." free to MU campus, to others for purchase, 1998. http://wwwlib.umi.com/cr/mo/fullcit?p9904841.

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Hu, Weiwei. "Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems." Diss., Virginia Tech, 2012. http://hdl.handle.net/10919/38664.

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In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory.
Ph. D.

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Li, Shenghao. "Non-homogeneous Boundary Value Problems for Boussinesq-type Equations." University of Cincinnati / OhioLINK, 2016. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1468512590.

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Lin, Qun. "The well-posedness and solutions of Boussinesq-type equations." Thesis, Curtin University, 2009. http://hdl.handle.net/20.500.11937/2247.

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We develop well-posedness theory and analytical and numerical solution techniques for Boussinesq-type equations. Firstly, we consider the Cauchy problem for a generalized Boussinesq equation. We show that under suitable conditions, a global solution for this problem exists. In addition, we derive sufficient conditions for solution blow-up in finite time.Secondly, a generalized Jacobi/exponential expansion method for finding exact solutions of non-linear partial differential equations is discussed. We use the proposed expansion method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified Korteweg-de Vries equations. We also apply it to the shallow water long wave approximate equations. New solutions are deduced for this system of partial differential equations.Finally, we develop and validate a numerical procedure for solving a class of initial boundary value problems for the improved Boussinesq equation. The finite element method with linear B-spline basis functions is used to discretize the equation in space and derive a second order system involving only ordinary derivatives. It is shown that the coefficient matrix for the second order term in this system is invertible. Consequently, for the first time, the initial boundary value problem can be reduced to an explicit initial value problem, which can be solved using many accurate numerical methods. Various examples are presented to validate this technique and demonstrate its capacity to simulate wave splitting, wave interaction and blow-up behavior.

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Книги з теми "Boussinesque Equation"

A, Babin, and Institute for Computer Applications in Science and Engineering., eds. On the asymptotic regimes and the strongly stratified limit of rotating Boussinesq equations. Hampton, Va: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, 1997.

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National Aeronautics and Space Administration (NASA) Staff. On the Asymptotic Regimes and the Strongly Stratified Limit of Rotating Boussinesq Equations. Independently Published, 2018.

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Частини книг з теми "Boussinesque Equation"

Pedersen, Geir K. "Boussinesq Equations." In Encyclopedia of Applied and Computational Mathematics, 155–56. Berlin, Heidelberg: Springer Berlin Heidelberg, 2015. http://dx.doi.org/10.1007/978-3-540-70529-1_405.

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Xu, Xiaoping. "Boussinesq Equations in Geophysics." In Algebraic Approaches to Partial Differential Equations, 231–67. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-36874-5_8.

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Hietarinta, Jarmo, and Da-jun Zhang. "Discrete Boussinesq-type equations." In Nonlinear Systems and Their Remarkable Mathematical Structures, 54–101. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003087670-3.

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Deville, Michel O. "Incompressible Newtonian Fluid Mechanics." In An Introduction to the Mechanics of Incompressible Fluids, 1–32. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-04683-4_1.

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AbstractThis chapter, presents the basic concepts of fluid mechanics such as velocity, acceleration, material derivative and the governing equations obtained from the conservation laws of mass, momentum, angular momentum and energy. The introduction of the constitutive relation for viscous incompressible Newtonian fluid leads to the Navier–Stokes equations. Boundary and initial conditions are discussed. Special attention is devoted to the meaning and differences between incompressible and compressible fluids. The Boussinesq equations are described. The chapter ends with considerations on the control volume method, a very efficient tool to solve fluid problems.

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Berger, Marsha. "Asteroid-Generated Tsunamis: A Review." In SEMA SIMAI Springer Series, 3–17. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-030-86236-7_1.

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AbstractWe study ocean waves caused by an asteroid airburst located over the ocean. The concern is that the waves would damage distant coastal cities. Simple qualitative analysis suggests that the wave energy is proportional to the ocean depth and the strength and speed of the blast. Computational simulations using GeoClaw and the shallow water equations show that explosions from realistic asteroids do not endanger distant cities. We explore the validity of the shallow water, Boussinesq, and linearized Euler equations to model these water waves.

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Kim, Tujin, and Daomin Cao. "The Steady Boussinesq System." In Equations of Motion for Incompressible Viscous Fluids, 227–50. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-78659-5_7.

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Zeytounian, Radyadour Kh. "A Typical RAM Approach: Boussinesq Model Equations." In Navier-Stokes-Fourier Equations, 81–94. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-20746-4_4.

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Kim, Tujin, and Daomin Cao. "The Non-steady Boussinesq System." In Equations of Motion for Incompressible Viscous Fluids, 251–84. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-78659-5_8.

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Wazwaz, Abdul-Majid. "Boussinesq, Klein-Gordon and Liouville Equations." In Nonlinear Physical Science, 639–63. Berlin, Heidelberg: Springer Berlin Heidelberg, 2009. http://dx.doi.org/10.1007/978-3-642-00251-9_16.

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Zhang, Bing-Yu. "Exact Controllability of the Generalized Boussinesq Equation." In Control and Estimation of Distributed Parameter Systems, 297–310. Basel: Birkhäuser Basel, 1998. http://dx.doi.org/10.1007/978-3-0348-8849-3_23.

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Тези доповідей конференцій з теми "Boussinesque Equation"

Petrova, Z., Michail D. Todorov, and Christo I. Christov. "Oscillation Properties of the Equation of Boussinesq and Comparison with Other Fourth Order Equations." In 1ST INTERNATIONAL CONFERENCE ON APPLICATIONS OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES. AIP, 2009. http://dx.doi.org/10.1063/1.3265341.

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Choudhury, Jayanta. "2D Solitary Waves of Boussinesq Equation." In ISIS INTERNATIONAL SYMPOSIUM ON INTERDISCIPLINARY SCIENCE. AIP, 2005. http://dx.doi.org/10.1063/1.1900395.

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Christou, M. A., Michail D. Todorov, and Christo I. Christov. "Some Boussinesq Equations with Saturation." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: Proceedings of the 2nd International Conference. AIP, 2010. http://dx.doi.org/10.1063/1.3526657.

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Vucheva, V., and N. Kolkovska. "A symplectic numerical method for Boussinesq equation." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 10th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’18. Author(s), 2018. http://dx.doi.org/10.1063/1.5064941.

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Kolkovska, N., and V. Vucheva. "Numerical investigation of sixth order Boussinesq equation." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 9th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’17. Author(s), 2017. http://dx.doi.org/10.1063/1.5007409.

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Veeramony, Jayaram, and James M. Kaihatu. "Spectral Models Based on Boussinesq Equations." In Fifth International Conference on Coastal Dynamics. Reston, VA: American Society of Civil Engineers, 2006. http://dx.doi.org/10.1061/40855(214)115.

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Kennedy, Andrew B., James T. Kirby, and Mauricio F. Gobbi. "Improved Performance in Boussinesq-Type Equations." In 27th International Conference on Coastal Engineering (ICCE). Reston, VA: American Society of Civil Engineers, 2001. http://dx.doi.org/10.1061/40549(276)53.

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salmei, H., and F. salimi. "Modified Homotopy Pertutbation Method for solving Boussinesq Equation." In ICMS INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCE. American Institute of Physics, 2010. http://dx.doi.org/10.1063/1.3525215.

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Kudryashov, Nikolay A., and Alexandr K. Volkov. "On analytical solutions of the generalized Boussinesq equation." In INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2015 (ICNAAM 2015). Author(s), 2016. http://dx.doi.org/10.1063/1.4952014.

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Servi, Sema, Yildiray Keskin, and Galip Oturanç. "Reduced differential transform method for improved Boussinesq equation." In PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON NUMERICAL ANALYSIS AND APPLIED MATHEMATICS 2014 (ICNAAM-2014). AIP Publishing LLC, 2015. http://dx.doi.org/10.1063/1.4912601.

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Звіти організацій з теми "Boussinesque Equation"

M. A. Jafarizadeh and A. R. Esfandyari. Exact Solutions of Boussinesq Equation. GIQ, 2012. http://dx.doi.org/10.7546/giq-3-2002-304-314.

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Walker, David T. Variational Data Assimilation for Near-Shore Waves Using the Extended Boussinesq Equations. Fort Belvoir, VA: Defense Technical Information Center, October 2005. http://dx.doi.org/10.21236/ada441232.

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Dimova, Milena, Natalia Kolkovska, and Nikolay Kutev. Orbital Stability or Instability of Solitary Waves to Generalized Boussinesq Equation with Quadratic-cubic Nonlinearity. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, August 2018. http://dx.doi.org/10.7546/crabs.2018.08.01.

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