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Journal articles on the topic 'Boussinesque Equation'

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Published: 11 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Li, Biao, and Yong Chen. "Nonlinear Partial Differential Equations Solved by Projective Riccati Equations Ansatz." Zeitschrift für Naturforschung A 58, no. 9-10 (October 1, 2003): 511–19. http://dx.doi.org/10.1515/zna-2003-9-1007.

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Based on the general projective Riccati equations method and symbolic computation, some new exact travelling wave solutions are obtained for a nonlinear reaction-diffusion equation, the highorder modified Boussinesq equation and the variant Boussinesq equation. The obtained solutions contain solitary waves, singular solitary waves, periodic and rational solutions. From our results, we can not only recover the known solitary wave solutions of these equations found by existing various tanh methods and other sophisticated methods, but also obtain some new and more general travelling wave solutions.

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12

Fan, Fei, Bing Chen Liang, and Xiu Li Lv. "Study of Wave Models of Parabolic Mild Slope Equation and Boussinesq Equation." Applied Mechanics and Materials 204-208 (October 2012): 2334–40. http://dx.doi.org/10.4028/www.scientific.net/amm.204-208.2334.

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The Parabolic mild slope equation and Boussinesq equation wave models are studied in this paper. First, the wave models Funwave and REF/DIF, which based on Boussinesq equations and the parabolic mild slope equation, respectively, are introduced. And then, two experiments are used to study these two wave models, one is the non-breaking shoal experiment of University of Delaware and the other is the breaking undertow test experiment, which was finished in Ocean University of China by author. Last, the simulation data of two wave models are compared with the measured data. The results show that both Boussinesq equation and the parabolic mild slope equation wave models can simulated nearshore wave condition precisely, but Boussinesq equation wave models has a disadvantage in catching the variation of wave height caused by wave breaking.

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13

Wayne, C. Eugene, and J. Douglas Wright. "Higher Order Modulation Equations for a Boussinesq Equation." SIAM Journal on Applied Dynamical Systems 1, no. 2 (January 2002): 271–302. http://dx.doi.org/10.1137/s1111111102411298.

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14

Jang, T. S. "A Regular Integral Equation Formalism for Solving the Standard Boussinesq’s Equations for Variable Water Depth." Journal of Scientific Computing 75, no. 3 (November 22, 2017): 1721–56. http://dx.doi.org/10.1007/s10915-017-0605-6.

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Abstract This paper begins with a question of existence of a regular integral equation formalism, but different from the existing usual ones, for solving the standard Boussinesq’s equations for variable water depth (or Peregrine’s model). For the question, a pseudo-water depth parameter, suggested by Jang (Commun Nonlinear Sci Numer Simul 43:118–138, 2017), is introduced to alter the standard Boussinesq’s equations into an integral formalism. This enables us to construct a regular (nonlinear) integral equations of second kind (as required), being equivalent to the standard Boussinesq’s equations (of Peregrine’s model). The (constructed) integral equations are, of course, inherently different from the usual integral equation formalisms. For solving them, the successive approximation (or the fixed point iteration) is applied (Jang 2017), whereby a new iterative formula is immediately derived, in this paper, for numerical solutions of the standard Boussinesq’s equations for variable water depth. The formula, semi-analytic and derivative-free, is shown to be useful to observe especially the nonlinear wave phenomena of shallow water waves on a beach. In fact, a numerical experiment is performed on a solitary wave approaching a sloping beach. It shows clearly the main feature of nonlinear wave characteristics, which has reached good agreement with the known (numerical) solutions. Hence, while being theoretical but fundamental in nonlinear computational partial differential equations, the question raised in the study may be solved.

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15

Yang, Zonghang, and Benny Y. C. Hon. "An Improved Modified Extended tanh-Function Method." Zeitschrift für Naturforschung A 61, no. 3-4 (April 1, 2006): 103–15. http://dx.doi.org/10.1515/zna-2006-3-401.

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In this paper we further improve the modified extended tanh-function method to obtain new exact solutions for nonlinear partial differential equations. Numerical applications of the proposed method are verified by solving the improved Boussinesq equation and the system of variant Boussinesq equations. The new exact solutions for these equations include Jacobi elliptic doubly periodic type,Weierstrass elliptic doubly periodic type, triangular type and solitary wave solutions

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16

Tao, Zhao-Ling. "Variational Principles for Some Nonlinear Wave Equations." Zeitschrift für Naturforschung A 63, no. 5-6 (June 1, 2008): 237–40. http://dx.doi.org/10.1515/zna-2008-5-601.

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Using the semi-inverse method proposed by Ji-Huan He, variational principles are established for some nonlinear wave equations arising in physics, including the Pochhammer-Chree equation, Zakharov-Kuznetsov equation, Korteweg-de Vries equation, Zhiber-Shabat equation, Kawahara equation, and Boussinesq equation.

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17

IVANOV, E., S. KRIVONOS, and R. P. MALIK. "N = 2 SUPER W3 ALGEBRA AND N = 2 SUPER BOUSSINESQ EQUATIONS." International Journal of Modern Physics A 10, no. 02 (January 20, 1995): 253–88. http://dx.doi.org/10.1142/s0217751x95000127.

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We study classical N=2 super W3 algebra and its interplay with N=2 supersymmetric extensions of the Boussinesq equation in the framework of the nonlinear realization method and the inverse Higgs-covariant reduction approach. These techniques have been previously used by us in the bosonic W3 case to give a new geometric interpretation of the Boussinesq hierarchy. Here we deduce the most general N=2 super Boussinesq equation and two kinds of the modified N=2 super Boussinesq equations, as well as the super Miura maps relating these systems to each other, by applying the covariant reduction to certain coset manifolds of linear [Formula: see text] symmetry associated with N=2 super W3. We discuss the integrability properties of the equations obtained and their correspondence with the formulation based on the notion of the second Hamiltonian structure.

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18

YOU, FUCAI, TIECHENG XIA, and JIAO ZHANG. "FROBENIUS INTEGRABLE DECOMPOSITIONS FOR TWO CLASSES OF NONLINEAR EVOLUTION EQUATIONS WITH VARIABLE COEFFICIENTS." Modern Physics Letters B 23, no. 12 (May 20, 2009): 1519–24. http://dx.doi.org/10.1142/s0217984909019764.

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Frobenius integrable decompositions are introduced for partial differential equations with variable coefficients. Two classes of partial differential equations with variable coefficients are transformed into Frobenius integrable ordinary differential equations. The resulting solutions are illustrated to describe the solution phenomena shared with the KdV and potential KdV equations, the Boussinesq equation and the Camassa–Holm equation with variable coefficients.

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19

Wan, Renhui. "Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space Hs." Communications in Contemporary Mathematics 22, no. 03 (September 10, 2018): 1850063. http://dx.doi.org/10.1142/s0219199718500633.

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Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

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20

Wang, Deng-Shan, and Xiaodong Zhu. "Long-time asymptotics of the good Boussinesq equation with q_xx-term and its modified version." Journal of Mathematical Physics 63, no. 12 (December 1, 2022): 123501. http://dx.doi.org/10.1063/5.0118374.

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Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with q xx-term and its modified version.

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21

Shakhmurov, Veli, and Rishad Shahmurov. "The regularity properties and blow-up of the solutions for improved Boussinesq equations." Electronic Journal of Qualitative Theory of Differential Equations, no. 89 (2021): 1–21. http://dx.doi.org/10.14232/ejqtde.2021.1.89.

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In this paper, we study the Cauchy problem for linear and nonlinear Boussinesq type equations that include the general differential operators. First, by virtue of the Fourier multipliers, embedding theorems in Sobolev and Besov spaces, the existence, uniqueness, and regularity properties of the solution of the Cauchy problem for the corresponding linear equation are established. Here, L p -estimates for a~solution with respect to space variables are obtained uniformly in time depending on the given data functions. Then, the estimates for the solution of linearized equation and perturbation of operators can be used to obtain the existence, uniqueness, regularity properties, and blow-up of solution at the finite time of the Cauchy for nonlinear for same classes of Boussinesq equations. Here, the existence, uniqueness, L p -regularity, and blow-up properties of the solution of the Cauchy problem for Boussinesq equations with differential operators coefficients are handled associated with the growth nature of symbols of these differential operators and their interrelationships. We can obtain the existence, uniqueness, and qualitative properties of different classes of improved Boussinesq equations by choosing the given differential operators, which occur in a wide variety of physical systems.

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22

Gandarias, M. L., and M. S. Bruzón. "Conservation laws for a Boussinesq equation." Applied Mathematics and Nonlinear Sciences 2, no. 2 (November 12, 2017): 465–72. http://dx.doi.org/10.21042/amns.2017.2.00037.

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AbstractIn this work, we study a generalized Boussinesq equation from the point of view of the Lie theory. We determine all the low-order conservation laws by using the multiplier method. Taking into account the relationship between symmetries and conservation laws and applying the multiplier method to a reduced ordinary differential equation, we obtain directly a second order ordinary differential equation and two third order ordinary differential equations.

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23

Mohamad- Jawad, Anwar. "The Sine-Cosine Function Method for Exact Solutions of Nonlinear Partial Differential Equations." Journal of Al-Rafidain University College For Sciences ( Print ISSN: 1681-6870 ,Online ISSN: 2790-2293 ), no. 2 (October 17, 2021): 120–39. http://dx.doi.org/10.55562/jrucs.v32i2.327.

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The Sine-Cosine function algorithm is applied for solving nonlinear partial differential equations. The method is used to obtain the exact solutions for different types of nonlinear partial differential equations such as, The K(n + 1, n + 1) equation, Schrödinger-Hirota equation, Gardner equation, the modified KdV equation, perturbed Burgers equation, general Burger’s-Fisher equation, and Cubic modified Boussinesq equation which are the important Soliton equations.Keywords: Nonlinear PDEs, Exact Solutions, Nonlinear Waves, Gardner equation, Sine-Cosine function method, The Schrödinger-Hirota equation.

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24

Demiray, Hilmi. "An Application of the Modified Reductive Perturbation Method to a Generalized Boussinesq Equation." International Journal of Nonlinear Sciences and Numerical Simulation 14, no. 1 (February 21, 2013): 27–31. http://dx.doi.org/10.1515/ijnsns-2011-0088.

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Abstract In this work, we apply “the modified reductive perturbation method” to the generalized Boussinesq equation and obtain various form of generalized KdV equations as the evolution equations. Seeking a localized travelling wave solutions for these evolution equations we determine the scale parameters g 1 and g 2, which corresponds to the correction terms in the wave speed, so as to remove the possible secularities that might occur. Depending on the sign and the values of certain parameters the resulting solutions are shown to be a solitary wave or a periodic solution. The suitability of the method is also shown by comparing the results with the exact travelling wave solution for the generalized Boussinesq equation.

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25

Choi, Jin Hyuk, and Hyunsoo Kim. "Coupled Fractional Traveling Wave Solutions of the Extended Boussinesq–Whitham–Broer–Kaup-Type Equations with Variable Coefficients and Fractional Order." Symmetry 13, no. 8 (August 1, 2021): 1396. http://dx.doi.org/10.3390/sym13081396.

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In this paper, we propose the extended Boussinesq–Whitham–Broer–Kaup (BWBK)-type equations with variable coefficients and fractional order. We consider the fractional BWBK equations, the fractional Whitham–Broer–Kaup (WBK) equations and the fractional Boussinesq equations with variable coefficients by setting proper smooth functions that are derived from the proposed equation. We obtain uniformly coupled fractional traveling wave solutions of the considered equations by employing the improved system method, and subsequently their asymmetric behaviors are visualized graphically. The result shows that the improved system method is effective and powerful to find explicit traveling wave solutions of the fractional nonlinear evolution equations.

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26

Young, William R. "Dynamic Enthalpy, Conservative Temperature, and the Seawater Boussinesq Approximation." Journal of Physical Oceanography 40, no. 2 (February 1, 2010): 394–400. http://dx.doi.org/10.1175/2009jpo4294.1.

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Abstract A new seawater Boussinesq system is introduced, and it is shown that this approximation to the equations of motion of a compressible binary solution has an energy conservation law that is a consistent approximation to the Bernoulli equation of the full system. The seawater Boussinesq approximation simplifies the mass conservation equation to ∇ · u = 0, employs the nonlinear equation of state of seawater to obtain the buoyancy force, and uses the conservative temperature introduced by McDougall as a thermal variable. The conserved energy consists of the kinetic energy plus the Boussinesq dynamic enthalpy h‡, which is the integral of the buoyancy with respect to geopotential height Z at a fixed conservative temperature and salinity. In the Boussinesq approximation, the full specific enthalpy h is the sum of four terms: McDougall’s potential enthalpy, minus the geopotential g0Z, plus the Boussinesq dynamic enthalpy h‡, and plus the dynamic pressure. The seawater Boussinesq approximation removes the large and dynamically inert contributions to h, and it reveals the important conversions between kinetic energy and h‡.

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27

Hwang, Sooncheol, Sangyoung Son, and Patrick J. Lynett. "A GPU-ACCELERATED MODELING OF SCALAR TRANSPORT BASED ON BOUSSINESQ-TYPE EQUATIONS." Coastal Engineering Proceedings, no. 36v (December 28, 2020): 11. http://dx.doi.org/10.9753/icce.v36v.waves.11.

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This paper describes a two-dimensional scalar transport model solving advection-diffusion equation based on GPU-accelerated Boussinesq model called Celeris. Celeris is the firstly-developed Boussinesq-type model that is equipped with an interactive system between user and computing unit. Celeris provides greatly advantageous user-interface that one can change not only water level, topography but also model parameters while the simulation is running. In this study, an advection-diffusion equation for scalar transport was coupled with extended Boussinesq equations to simulate scalar transport in the nearshore.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/aHvMmdz3wps

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28

Gözükızıl, Ömer Faruk, and Şamil Akçağıl. "Travelling Wave Solutions to the Benney-Luke and the Higher-Order Improved Boussinesq Equations of Sobolev Type." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/890574.

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By using the tanh-coth method, we obtained some travelling wave solutions of two well-known nonlinear Sobolev type partial differential equations, namely, the Benney-Luke equation and the higher-order improved Boussinesq equation. We show that the tanh-coth method is a useful, reliable, and concise method to solve these types of equations.

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29

SINGH, S. V., and N. N. RAO. "Adiabatic dust-acoustic waves with dust-charge fluctuations." Journal of Plasma Physics 60, no. 3 (October 1998): 541–50. http://dx.doi.org/10.1017/s0022377898006916.

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We study the effect of charge fluctuations on the propagation of adiabatic linear and nonlinear dust-acoustic waves by considering the electrons and ions to be in Boltzmann equilibria, and the dust grains to satisfy the fluid equations with full adiabatic equation of state. Linear dust-acoustic waves are damped owing to the dust-charge fluctuations, and the damping rate decreases with increasing adiabatic dust pressure. Nonlinear dust-acoustic waves are governed by the set of coupled Boussinesq-like and dust-charge perturbation equations. It is shown that for unidirectional propagation, the Boussinesq-like equation reduces to usual Korteweg–de Vries (KdV) equation. At early times, the localized solutions of the KdV equation are damped owing to the dust-charge perturbations. The soliton amplitude decreases with increasing adiabatic dust plasma pressure and increases with Mach number. Soliton solutions are found only in the supersonic regime.

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30

Gandarias, María Luz, and María Rosa. "Symmetries and conservation laws of a damped Boussinesq equation." International Journal of Modern Physics B 30, no. 28n29 (November 10, 2016): 1640012. http://dx.doi.org/10.1142/s0217979216400129.

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In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

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31

ROY, P. K. "SUPERSYMMETRIC GENERALIZATION OF BOUSSINESQ–BURGER SYSTEM." International Journal of Modern Physics A 13, no. 10 (April 20, 1998): 1623–27. http://dx.doi.org/10.1142/s0217751x98000706.

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We generalize the Boussinesq–Burger coupled equation by constructing supersymmetric two boson system that is integrable. We obtain two super Hamiltonians and consequently two different set of supersymmetric coupled equations corresponding to the two Hamiltonians.

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32

Yang, Wei, and Chunguang Li. "General Propagation Lattice Boltzmann Model for the Boussinesq Equation." Entropy 24, no. 4 (March 30, 2022): 486. http://dx.doi.org/10.3390/e24040486.

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A general propagation lattice Boltzmann model is used to solve Boussinesq equations. Different local equilibrium distribution functions are selected, and the macroscopic equation is recovered with second order accuracy by means of the Chapman–Enskog multi-scale analysis and the Taylor expansion technique. To verify the effectiveness of the present model, some Boussinesq equations with initial boundary value problems are simulated. It is shown that our model can remain stable and accurate, which is an effective algorithm worthy of promotion and application.

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Barannik, L. F., and H. O. Lahno. "Symmetry reduction of the Boussinesq equation to ordinary differential equations." Reports on Mathematical Physics 38, no. 1 (August 1996): 1–9. http://dx.doi.org/10.1016/0034-4877(96)87674-9.

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Zeng, Libing, Keding Qin, and Shengqiang Tang. "Exact Explicit Traveling Wave Solution for the Generalized (2+1)-Dimensional Boussinesq Equation." ISRN Applied Mathematics 2011 (June 21, 2011): 1–10. http://dx.doi.org/10.5402/2011/419678.

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The sine-cosine method and the extended tanh method are used to construct exact solitary patterns solution and compactons solutions of the generalized (2+1)-dimensional Boussinesq equation. The compactons solutions and solitary patterns solutions of the generalized (2+1)-dimensional Boussinesq equation are successfully obtained. These solutions may be important and of significance for the explanation of some practical physical problems. It is shown that the sine-cosine and the extended tanh methods provide a powerful mathematical tool for solving great many nonlinear partial differential equations in mathematical physics.

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Mukhtar, Safyan. "Numerical Analysis of the Time-Fractional Boussinesq Equation in Gradient Unconfined Aquifers with the Mittag-Leffler Derivative." Symmetry 15, no. 3 (February 27, 2023): 608. http://dx.doi.org/10.3390/sym15030608.

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In this study, two numerical methods—the variational iteration transform method (VITM) and the Adomian decomposition (ADM) method—were used to solve the second- and fourth-order fractional Boussinesq equations. Both methods are helpful in approximating non-linear problems effectively, easily, and accurately. The fractional Atangana–Baleanu operator and ZZ transform were utilized to derive solutions for the equation. Two examples are discussed to validate the methods and solutions. The results demonstrate that both the VITM and ADM methods are effective in obtaining accurate and reliable solutions for the time-fractional Boussinesq equation.

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Park, H. M., and W. J. Lee. "Recursive Identification of Thermal Convection." Journal of Dynamic Systems, Measurement, and Control 125, no. 1 (March 1, 2003): 1–10. http://dx.doi.org/10.1115/1.1540116.

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A method is developed for the recursive identification of thermal convection system governed by the Boussinesq equation using an extended Kalman filter. A computationally feasible Kalman filter is constructed by reducing the Boussinesq equation to a small number of ordinary differential equations by means of the Karhunen-Loe`ve Galerkin procedure which is a type of Galerkin method employing the empirical eigenfunctions of the Karhunen-Loe`ve decomposition. Employing the Kalman filter constructed by using the reduced order model, the thermal convection induced by a spatially varying heat flux at the bottom is identified recursively by using either the Boussinesq equation or the reduced order model itself. The recursive identification technique developed in the present work is found to yield accurate results for thermal convection even with approximate covariance equation and noisy measurements. It is also shown that a reasonably accurate and computationally feasible method of recursive identification can be constructed even with a relatively inaccurate reduced order model.

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Michael Mueller, Thomas. "The Boussinesq Debate: Reversibility, Instability, and Free Will." Science in Context 28, no. 4 (November 11, 2015): 613–35. http://dx.doi.org/10.1017/s0269889715000290.

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ArgumentIn 1877, a young mathematician named Joseph Boussinesq presented amémoireto theAcadémiedes sciences which demonstrated that some differential equations may have more than one solution. Boussinesq linked this fact to indeterminism and to a possible solution to the free will versus determinism debate. Boussinesq's main interest was to reconcile his philosophical and religious views with science by showing that matter and motion do not suffice to explain all there is in the world. His argument received mixed criticism that addressed both his philosophical views and the scientific content of his work, pointing to the physical “realisticness” of multiple solutions. While Boussinesq proved to be able to face the philosophical criticism, the scientific objections became a serious problem, thus slowly moving the focus of the debate from the philosophical plane to the scientific one. This change of perspective implied a wide discussion on topics such as instability, the sensitivity to initial conditions, and the conservation of energy. The Boussinesq debate is an example of a philosophically motivated debate that transforms into a scientific one, an example of the influence of philosophy on the development of science.

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GHADIMI, PARVIZ, MOHAMMAD HADI JABBARI, and ARSHAM REISINEZHAD. "FINITE ELEMENT MODELING OF ONE-DIMENSIONAL BOUSSINESQ EQUATIONS." International Journal of Modeling, Simulation, and Scientific Computing 02, no. 02 (June 2011): 207–35. http://dx.doi.org/10.1142/s1793962311000414.

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Finite element modeling of one-dimensional Beji and Nadaoka Boussinesq equation is presented. The continuous equations are spatially discretized using standard Galerkin method. Since the extended Boussinesq equations contain high-order derivatives, two different numerical techniques are proposed in this paper in order to simplify the discretization task of the third-order terms. In the first technique, an auxiliary equation is introduced to eliminate the third-order derivatives of the momentum equation while non-overlapping elements with linear interpolating functions are employed to account for the dependent variables. However, in the second method, overlapping elements with quadratic interpolating functions are applied for discretizing the governing equations. Time integration is performed using the Adams–Bashforth–Moulton predictor–corrector method. By considering the truncation error and theoretical analysis for both of the numerical techniques, accuracy and stability of the adopted finite element schemes have been studied. Finally, a computer code is developed based on the proposed schemes. To show the validity as well as the practicality of the developed code, five different test cases are presented, and the results are compared with some analytical solutions and experimental data. Favorable agreements have been achieved in all cases.

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ONORATO, M., A. R. OSBORNE, P. A. E. M. JANSSEN, and D. RESIO. "Four-wave resonant interactions in the classical quadratic Boussinesq equations." Journal of Fluid Mechanics 618 (January 10, 2009): 263–77. http://dx.doi.org/10.1017/s0022112008004229.

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We investigate theoretically the irreversibile energy transfer in flat bottom shallow water waves. Starting from the oldest weakly nonlinear dispersive wave model in shallow water, i.e. the original quadratic Boussinesq equations, and by developing a statistical theory (kinetic equation) of the aforementioned equations, we show that the four-wave resonant interactions are naturally part of the shallow water wave dynamics. These interactions are responsible for a constant flux of energy in the wave spectrum, i.e. an energy cascade towards high wavenumbers, leading to a power law in the wave spectrum of the form of k−3/4. The nonlinear time scale of the interaction is found to be of the order of (h/a)4 wave periods, with a the wave amplitude and h the water depth. We also compare the kinetic equation arising from the Boussinesq equations with the arbitrary-depth Hasselmann equation and show that, in the limit of shallow water, the two equations coincide. It is found that in the narrow band case, both in one-dimensional propagation and in the weakly two-dimensional case, there is no irreversible energy transfer because the coupling coefficient in the kinetic equation turns out to be identically zero on the resonant manifold.

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Aminikhah, Hossein, Amir Hosein Refahi Sheikhani, and Hadi Rezazadeh. "Travelling wave solutions of nonlinear systems of PDEs by using the functional variable method." Boletim da Sociedade Paranaense de Matemática 34, no. 2 (September 3, 2015): 213–29. http://dx.doi.org/10.5269/bspm.v34i2.25501.

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In this paper, we will use the functional variable method to construct exact solutions of some nonlinear systems of partial differential equations, including, the (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation, the WhithamBroer-Kaup-Like systems and the Kaup-Boussinesq system. This approach can also be applied to other nonlinear systems of partial differential equations which can be converted to a second-order ordinary differential equation through the travelling wave transformation.

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Nakata, Kenta. "Integrable delay-difference and delay-differential analogs of the KdV, Boussinesq, and KP equations." Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 113505. http://dx.doi.org/10.1063/5.0125308.

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Delay-difference and delay-differential analogs of the KdV and Boussinesq (BSQ) equations are presented. Each of them has the N-soliton solution and reduces to an already known soliton equation as the delay parameter approaches 0. In addition, a delay-differential analog of the KP equation is proposed. We discuss its N-soliton solution and the limit as the delay parameter approaches 0. Finally, the relationship between the delay-differential analogs of the KdV, BSQ, and KP equations is clarified. Namely, reductions of the delay KP equation yield the delay KdV and delay BSQ equations.

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Chen, Guiying, Xiangpeng Xin, and Hanze Liu. "The Improvedexp⁡-Φξ-Expansion Method and New Exact Solutions of Nonlinear Evolution Equations in Mathematical Physics." Advances in Mathematical Physics 2019 (April 4, 2019): 1–8. http://dx.doi.org/10.1155/2019/4354310.

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Theexp(-Φ(ξ))-expansion method is improved by presenting a new auxiliary ordinary differential equation forΦ(ξ). By using this method, new exact traveling wave solutions of two important nonlinear evolution equations, i.e., the ill-posed Boussinesq equation and the unstable nonlinear Schrödinger equation, are constructed. The obtained solutions contain Jacobi elliptic function solutions which can be degenerated to the hyperbolic function solutions and the trigonometric function solutions. The present method is very concise and effective and can be applied to other types of nonlinear evolution equations.

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Son, Sangyoung, and Patrick Lynett. "INTER-COUPLED TSUNAMI MODELLING THROUGH AN ABSORBING-GENERATING BOUNDARY." Coastal Engineering Proceedings, no. 36v (December 28, 2020): 38. http://dx.doi.org/10.9753/icce.v36v.currents.38.

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For many practical and theoretical purposes, various types of tsunami wave models have been developed and utilized so far. Some distinction among them can be drawn based on governing equations used by the model. Shallow water equations and Boussinesq equations are probably most typical ones among others since those are computationally efficient and relatively accurate compared to 3D Navier-Stokes models. From this idea, some coupling effort between Boussinesq model and shallow water equation model have been made (e.g., Son et al. (2011)). In the present study, we couple two different types of tsunami models, i.e., nondispersive shallow water model of characteristic form(MOST ver.4) and dispersive Boussinesq model of non-characteristic form(Son and Lynett (2014)) in an attempt to improve modelling accuracy and efficiency.Recorded Presentation from the vICCE (YouTube Link): https://youtu.be/cTXybDEnfsQ

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Bekir, Ahmet, Adem C. Cevikel, Ozkan Guner, and Sait San. "BRIGHT AND DARK SOLITON SOLUTIONS OF THE (2 + 1)-DIMENSIONAL EVOLUTION EQUATIONS." Mathematical Modelling and Analysis 19, no. 1 (February 20, 2014): 118–26. http://dx.doi.org/10.3846/13926292.2014.893456.

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In this paper, we obtained the 1-soliton solutions of the (2+1)-dimensional Boussinesq equation and the Camassa–Holm–KP equation. By using a solitary wave ansatz in the form of sechp function, we obtain exact bright soliton solutions and another wave ansatz in the form of tanhp function we obtain exact dark soliton solutions for these equations. The physical parameters in the soliton solutions are obtained nonlinear equations with constant coefficients.

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Miles, John, and Rick Salmon. "Weakly dispersive nonlinear gravity waves." Journal of Fluid Mechanics 157 (August 1985): 519–31. http://dx.doi.org/10.1017/s0022112085002488.

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The equations for gravity waves on the free surface of a laterally unbounded inviscid fluid of uniform density and variable depth under the action of an external pressure are derived through Hamilton's principle on the assumption that the fluid moves in vertical columns. The resulting equations are equivalent to those of Green & Naghdi (1976). The conservation laws for energy, momentum and potential vorticity are inferred directly from symmetries of the Lagrangian. The potential vorticity vanishes in any flow that originates from rest; this leads to a canonical formulation in which the evolution equations are equivalent, for uniform depth, to Whitham's (1967) generalization of the Boussinesq equations, in which dispersion, but not nonlinearity, is assumed to be weak. The further approximation that nonlinearity and dispersion are comparably weak leads to a canonical form of Boussinesq's equations that conserves consistent approximations to energy, momentum (for a level bottom) and potential vorticity.

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Hu, Hengchun, and Yihui Lu. "Lie group analysis and invariant solutions of (3+1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation." Modern Physics Letters B 34, no. 11 (February 3, 2020): 2050106. http://dx.doi.org/10.1142/s0217984920501067.

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We study the (3[Formula: see text]+[Formula: see text]1)-dimensional B-type Kadomtsev–Petviashvili–Boussinesq equation by using the Lie group analysis method. The reduction equations are given out by selecting different constants and new explicit solutions are also presented by the Riccati equation expansion and the power series expansion.

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Abdel-Salam, Emad A. B., and Eltayeb A. Yousif. "Solution of Nonlinear Space-Time Fractional Differential Equations Using the Fractional Riccati Expansion Method." Mathematical Problems in Engineering 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/846283.

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The fractional Riccati expansion method is proposed to solve fractional differential equations. To illustrate the effectiveness of the method, space-time fractional Korteweg-de Vries equation, regularized long-wave equation, Boussinesq equation, and Klein-Gordon equation are considered. As a result, abundant types of exact analytical solutions are obtained. These solutions include generalized trigonometric and hyperbolic functions solutions which may be useful for further understanding of the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The periodic and kink solutions are founded as special case.

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Shi, Jing, Xin Li Yan, and Qing Tian Deng. "Solutions for the Finite Deformation Elastic Rod Nonlinear Wave Equation." Applied Mechanics and Materials 457-458 (October 2013): 358–61. http://dx.doi.org/10.4028/www.scientific.net/amm.457-458.358.

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The solving process of the hyperbola function method is introduced in this paper. By using hyperbola function method, the analytical solutions of nonlinear wave equation of a finite deformation elastic circular rod and the variant Boussinesq equations are studied. The more physical specifications of these nonlinear equations have been identified.

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Wagner, G. L., G. Ferrando, and W. R. Young. "An asymptotic model for the propagation of oceanic internal tides through quasi-geostrophic flow." Journal of Fluid Mechanics 828 (September 12, 2017): 779–811. http://dx.doi.org/10.1017/jfm.2017.509.

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We derive a time-averaged ‘hydrostatic wave equation’ from the hydrostatic Boussinesq equations that describes the propagation of inertia–gravity internal waves through quasi-geostrophic flow. The derivation uses a multiple-scale asymptotic method to isolate wave field evolution over intervals much longer than a wave period, assumes the wave field has a well-defined non-inertial frequency such as that of the mid-latitude semi-diurnal lunar tide, assumes that the wave field and quasi-geostrophic flow have comparable spatial scales and neglects nonlinear wave–wave dynamics. As a result the hydrostatic wave equation is a reduced model applicable to the propagation of large-scale internal tides through the inhomogeneous and moving ocean. A numerical comparison with the linearized and hydrostatic Boussinesq equations demonstrates the validity of the hydrostatic wave equation model and illustrates how the model fails when the quasi-geostrophic flow is too strong and the wave frequency is too close to inertial. The hydrostatic wave equation provides a first step toward a coupled model for energy transfer between oceanic internal tides and quasi-geostrophic eddies and currents.

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Slimani, Ali, Lamine Bouzettouta, and Amar Guesmia. "Existence and uniqueness of the weak solution for Keller-Segel model coupled with Boussinesq equations." Demonstratio Mathematica 54, no. 1 (January 1, 2021): 558–75. http://dx.doi.org/10.1515/dema-2021-0027.

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Abstract 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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