Boussinesq evolution equations
DEFF Research Database (Denmark)
Bredmose, Henrik; Schaffer, H.; Madsen, Per A.
2004-01-01
This paper deals with the possibility of using methods and ideas from time domain Boussinesq formulations in the corresponding frequency domain formulations. We term such frequency domain models "evolution equations". First, we demonstrate that the numerical efficiency of the deterministic...... Boussinesq evolution equations of Madsen and Sorensen [Madsen, P.A., Sorensen, O.R., 1993. Bound waves and triad interactions in shallow water. Ocean Eng. 20 359-388] can be improved by using Fast Fourier Transforms to evaluate the nonlinear terms. For a practical example of irregular waves propagating over...... a submerged bar, it is demonstrated that evolution equations utilising FFT can be solved around 100 times faster than the corresponding time domain model. Use of FFT provides an efficient bridge between the frequency domain and the time domain. We utilise this by adapting the surface roller model for wave...
Exponential Attractor for a Nonlinear Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
Ahmed Y. Abdallah
2006-01-01
This paper is devoted to prove the existence of an exponential attractor for the semiflow generated by a nonlinear Boussinesq equation. We formulate the Boussinesq equation as an abstract equation in the Hilbert space H20(0, 1) × L2(0, 1). The main step in this research is to show that there exists an absorbing set for the solution semiflow in the Hilbert space H03(0, 1) × H10(0, 1).
Analytical solutions of the extended Boussinesq equation
International Nuclear Information System (INIS)
The extended Boussinesq equation for the description of the Fermi-Pasta-Ulam problem has been studied and analyzed with the Painleve test. It has been shown that the equation does not pass the Painleve test, but the necessary condition for the existence of meromorphic solutions is satisfied
Numerical Solutions of Fractional Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
WANG Qi
2007-01-01
Based upon the Adomian decomposition method,a scheme is developed to obtain numerical solutions of a fractional Boussinesq equation with initial condition,which is introduced by replacing some order time and space derivatives by fractional derivatives.The fractional derivatives are described in the Caputo sense.So the traditional Adomian decomposition method for differential equations of integer order is directly extended to derive explicit and numerical solutions of the fractional differential equations.The solutions of our model equation are calculated in the form of convergent series with easily computable components.
PERTURBED PERIODIC SOLUTION FOR BOUSSINESQ EQUATION
Institute of Scientific and Technical Information of China (English)
Jiang Xinhua; Wang Zhen
2009-01-01
We consider the solution of the good Boussinesq equation Utt- Uxx + Uxxxx> = (U2)xx - ∞ 0, the difference between the true solution u(x, t; ε) and the N-th partial sum of the asymptotic series is bounded by εN+1 multiplied by a constant depending on T and N, for all -∞ < x < ∞ 0 ≤｜ε｜t ≤ T and 0 ≤ ｜ε｜≤ε0.
Comparison between characteristics of mild slope equations and Boussinesq equations
Institute of Scientific and Technical Information of China (English)
无
2005-01-01
Boussinesq-type equations and mild-slope equations are compared in terms of their basic forms and characteristics. It is concluded that linear mild-slope equations on dispersion relation are better than non-linear Boussinesq equations. In addition, Berkhoff experiments are computed and compared by the two models, and agreement between model results and available experimental data is found to be quite reasonable, which demonstrates the two models' capacity to simulate wave transformation. However they can deal with different physical processes respectively, and they have their own characteristics.
problem for the damped Boussinesq equation
Directory of Open Access Journals (Sweden)
Vladimir V. Varlamov
1997-01-01
Full Text Available For the damped Boussinesq equation utt−2butxx=−αuxxxx+uxx+β(u2xx,x∈(0,π,t>0;α,b=const>0,β=const∈R1, the second initial-boundary value problem is considered with small initial data. Its classical solution is constructed in the form of a series in small parameter present in the initial conditions and the uniqueness of solutions is proved. The long-time asymptotics is obtained in the explicit form and the question of the blow up of the solution in a certain case is examined. The possibility of passing to the limit b→+0 in the constructed solution is investigated.
COMPARISON BETWEEN BOUSSINESQ EQUATIONS AND MILD-SLOPE EQUATIONS MODEL
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
In this paper, the Boussinesq equations and mild-slope equation of wave transformation in near-shore shallow water were introduced and the characteristics of the two forms of equations were compared and analyzed. Meanwhile, a Boussinesq wave model which includes effects of bottom friction, wave breaking and subgrid turbulent mixing is established, slot technique dealing with moving boundary and damping layer dealing with absorbing boundary were established. By adopting empirical nonlinear dispersion relation and including nonlinear term, the mild-slope equation model was modified to take nonlinear effects into account. The two types of models were validated with the experiment results given by Berkhoff and their accuracy was analysed and compared with that of correlated methods.
Incompressible Boussinesq equations and spaces of borderline Besov type
Glenn-Levin, Jacob
2011-01-01
We prove local-in-time existence and uniqueness of an inviscid Boussinesq-type system. We assume the density equation contains nonzero diffusion and that our initial vorticity and density belong to a space of borderline Besov type.
Whitham modulation equations, coalescing characteristics, and dispersive Boussinesq dynamics
Ratliff, Daniel J.; Bridges, Thomas J.
2016-10-01
Whitham modulation theory with degeneracy in wave action is considered. The case where all components of the wave action conservation law, when evaluated on a family of periodic travelling waves, have vanishing derivative with respect to wavenumber is considered. It is shown that Whitham modulation equations morph, on a slower time scale, into the two way Boussinesq equation. Both the 1 + 1 and 2 + 1 cases are considered. The resulting Boussinesq equation arises in a universal form, in that the coefficients are determined from the abstract properties of the Lagrangian and do not depend on particular equations. One curious by-product of the analysis is that the theory can be used to confirm that the two-way Boussinesq equation is not a valid model in shallow water hydrodynamics. Modulation of nonlinear travelling waves of the complex Klein-Gordon equation is used to illustrate the theory.
Solitons induced by boundary conditions from the Boussinesq equation
Chou, Ru Ling; Chu, C. K.
1990-01-01
The behavior of solitons induced by boundary excitation is investigated at various time-dependent conditions and different unperturbed water depths, using the Korteweg-de Vries (KdV) equation. Then, solitons induced from Boussinesq equations under similar conditions were studied, making it possible to remove the restriction in the KdV equation and to treat soliton head-on collisions (as well as overtaking collisions) and reflections. It is found that the results obtained from the KdV and the Boussinesq equations are in good agreement.
Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation
Institute of Scientific and Technical Information of China (English)
XU Rui
2008-01-01
Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq-Burgers equation are presented in this letter.They are used to generate new solutions of the classical Boussinesq-Burgers equation.
Blow-up of solution for a generalized Boussinesq equation
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method. Moreover, it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.
Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form
Reza Abazari; Adem Kılıçman
2013-01-01
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 of G′/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 scientist Joseph Valentin Bo...
Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation
Yang, Duo; Lou, Sen-Yue; Yu, Wei-Feng
2013-10-01
The Boussinesq equation is one of important prototypic models in nonlinear physics. Various nonlinear excitations of the Boussinesq equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this peper, two equivalent very simple methods, the truncated Painlevé analysis and the generalized tanh function expansion approaches, are developed to find interaction solutions between solitons and any other types of Boussinesq waves.
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq Equations
Brenier, Yann
2009-10-01
We establish a connection between optimal transport theory (see Villani in Topics in optimal transportation. Graduate studies in mathematics, vol. 58, AMS, Providence, 2003, for instance) and classical convection theory for geophysical flows (Pedlosky, in Geophysical fluid dynamics, Springer, New York, 1979). Our starting point is the model designed few years ago by Angenent, Haker, and Tannenbaum (SIAM J. Math. Anal. 35:61-97, 2003) to solve some optimal transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized hydrostatic-Boussinesq equations) to various models involving optimal transport (and the related Monge-Ampère equation, Brenier in Commun. Pure Appl. Math. 64:375-417, 1991; Caffarelli in Commun. Pure Appl. Math. 45:1141-1151, 1992). This includes the 2D semi-geostrophic equations (Hoskins in Annual review of fluid mechanics, vol. 14, pp. 131-151, Palo Alto, 1982; Cullen et al. in SIAM J. Appl. Math. 51:20-31, 1991, Arch. Ration. Mech. Anal. 185:341-363, 2007; Benamou and Brenier in SIAM J. Appl. Math. 58:1450-1461, 1998; Loeper in SIAM J. Math. Anal. 38:795-823, 2006) and some fully nonlinear versions of the so-called high-field limit of the Vlasov-Poisson system (Nieto et al. in Arch. Ration. Mech. Anal. 158:29-59, 2001) and of the Keller-Segel for Chemotaxis (Keller and Segel in J. Theor. Biol. 30:225-234, 1971; Jäger and Luckhaus in Trans. Am. Math. Soc. 329:819-824, 1992; Chalub et al. in Mon. Math. 142:123-141, 2004). Mathematically speaking, we establish some existence theorems for local smooth, global smooth or global weak solutions of the different models. We also justify that the inertia terms can be rigorously neglected under appropriate scaling assumptions in the generalized Navier-Stokes-Boussinesq equations
A Variational Iteration Solving Method for a Class of Generalized Boussinesq Equations
Institute of Scientific and Technical Information of China (English)
MO Jia-Qi
2009-01-01
We study a generalized nonlinear Boussinesq equation by introducing a proper functional and constructing the variational iteration sequence with suitable initial approximation.The approximate solution is obtained for the solitary wave of the Boussinesq equation with the variational iteration method.
Single-peak solitary wave solutions for the variant Boussinesq equations
Indian Academy of Sciences (India)
Hong Li; Lilin Ma; Dahe Feng
2013-06-01
This paper presents all possible smooth, cusped solitary wave solutions for the variant Boussinesq equations under the inhomogeneous boundary condition. The parametric conditions for the existence of smooth, cusped solitary wave solutions are given using the phase portrait analytical technique. Asymptotic analysis and numerical simulations are provided for smooth, cusped solitary wave solutions of the variant Boussinesq equations.
Solitary Wave Solutions of the Boussinesq Equation and Its Improved Form
Directory of Open Access Journals (Sweden)
Reza Abazari
2013-01-01
Full Text Available 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 of G′/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 scientist Joseph 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 the G′/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.
A Study of Enhanced, Higher Order Boussinesq-Type Equations and Their Numerical Modelling
DEFF Research Database (Denmark)
Banijamali, Babak
This project has encompassed efforts in two separate veins: on the one hand, the acquiring of highly accurate model equations of the Boussinesq-type, and on the other hand, the theoretical and practical work in implementing such equations in the form of conventional numerical models, with obvious...... potential for applications to the realm of numerical modelling in coastal engineering. The derivation and analysis of several forms of higher-order in dispersion and non-linearity Boussinesq-type equations have been undertaken, obtaining and investigating the properties of a new and generalised class...... of Boussinesq-type equations. The equations emerge from a study of arbitrarily higher-order Boussinesq-type equations in several customary choices of velocity variables. In doing so, a generalised horizontal velocity variable is defined corresponding to optimal properties, in the sense of the Padé...
Multi-soliton solution, rational solution of the Boussinesq-Burgers equations
Abdel Rady, A. S.; Osman, E. S.; Khalfallah, Mohammed
2010-05-01
In this paper we consider the Boussinesq-Burgers equations and establish the transformation which turns the Boussinesq-Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.
Unstructured nodal DG-FEM solution of high-order Boussinesq-type equations
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter
2007-01-01
of today, the high-order Boussinesq equations represent the most advanced set of Boussinesq-type equations capable of modelling nonlinear and dispersive waves from shallow to deep water without the practical limitations of classical Boussinesq-type equations. The high-order Boussinesq equations constitute...... time-dependent linear system; and the numerical treatment of high-order and cross-differential derivatives. The suggested solution strategy of the current work is based on a collocation approach where the DG-FEM is used to approximate spatial derivatives and the boundary conditions are imposed weakly...... using a symmetry technique. Since collocation methods are prone to aliasing errors, various anti-aliasing strategies are applied for the stabilization of the models. A practical and relatively straightforward discretization is applied, which is based on a simple treatment of slip boundary conditions...
The classification of the single travelling wave solutions to the variant Boussinesq equations
Indian Academy of Sciences (India)
YUE KAI
2016-10-01
The discrimination system for the polynomial method is applied to variant Boussinesq equations to classify single travelling wave solutions. In particular, we construct corresponding solutions to the concrete parameters to show that each solution in the classification can be realized.
EXACT EXPLICIT SOLUTIONS OF THE NONLINEAR SCHR(O)DINGER EQUATION COUPLED TO THE BOUSSINESQ EQUATION
Institute of Scientific and Technical Information of China (English)
姚若侠; 李忠斌
2003-01-01
A system comprised of the nonlinear Schrodinger equation coupled to theBoussinesq equation (S-B equations) which dealing with the stationary propagation of cou-pled non-linear upper-hybrid and magnetosonic waves in magnetized plasma is proposed.To examine its solitary wave solutions, a reduced set of ordinary differential equations areconsidered by a simple traveling wave transformation. It is then shown that several newsolutions (either functional or parametrical) can be obtained systematically, in addition torederiving all known ones by means of our simple and direct algebra method with the helpof the computer algebra system Maple.
Fully Nonlinear Boussinesq-Type Equations with Optimized Parameters for Water Wave Propagation
Institute of Scientific and Technical Information of China (English)
荆海晓; 刘长根; 龙文; 陶建华
2015-01-01
For simulating water wave propagation in coastal areas, various Boussinesq-type equations with improved properties in intermediate or deep water have been presented in the past several decades. How to choose proper Boussinesq-type equations has been a practical problem for engineers. In this paper, approaches of improving the characteristics of the equations, i.e. linear dispersion, shoaling gradient and nonlinearity, are reviewed and the advantages and disadvantages of several different Boussinesq-type equations are compared for the applications of these Boussinesq-type equations in coastal engineering with relatively large sea areas. Then for improving the properties of Boussinesq-type equations, a new set of fully nonlinear Boussinseq-type equations with modified representative velocity are derived, which can be used for better linear dispersion and nonlinearity. Based on the method of minimizing the overall error in different ranges of applications, sets of parameters are determined with optimized linear dispersion, linear shoaling and nonlinearity, respectively. Finally, a test example is given for validating the results of this study. Both results show that the equations with optimized parameters display better characteristics than the ones obtained by matching with padé approximation.
A generalized simplest equation method and its application to the Boussinesq-Burgers equation.
Directory of Open Access Journals (Sweden)
Bilige Sudao
Full Text Available In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs. In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.
A generalized simplest equation method and its application to the Boussinesq-Burgers equation.
Sudao, Bilige; Wang, Xiaomin
2015-01-01
In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.
New application of Exp-function method for improved Boussinesq equation
International Nuclear Information System (INIS)
The Exp-function method is used to obtain generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computation method, namely, the improved Boussinesq equation. The method is straightforward and concise, and its applications is promising for other nonlinear evolution equations in mathematical physics
New application of Exp-function method for improved Boussinesq equation
Energy Technology Data Exchange (ETDEWEB)
Abdou, M.A. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt); Department of Physics, Faculty of Education for Girls, Science Departments, King Khalid University, Bisha (Saudi Arabia)], E-mail: m_abdou_eg@yahoo.com; Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish) Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College (Bisha), King Khalid University, Bisha, PO Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com; El-Basyony, S.T. [Theoretical Research Group, Physics Department, Faculty of Science, Mansoura University, 35516 Mansoura (Egypt)
2007-10-01
The Exp-function method is used to obtain generalized solitary solutions and periodic solutions for nonlinear evolution equations arising in mathematical physics with the aid of symbolic computation method, namely, the improved Boussinesq equation. The method is straightforward and concise, and its applications is promising for other nonlinear evolution equations in mathematical physics.
DEFF Research Database (Denmark)
Ganji, S. S.; Barari, Amin; Sfahani, M. G.;
2011-01-01
The phenomenon of stream–aquifer interaction was investigated via mathematical modeling using the Boussinesq equation. A new approximate solution of the one-dimensional Boussinesq equation is presented for a semi-infinite aquifer when the hydraulic head at the source is an arbitrary function...... of time. The differential equations were solved using the method of Homotopy Perturbation. The simplicity and accuracy of the approximation are compared with “exact” solution and illustrated numerically and graphically. The results reveal that the HPM is very effective and simple and provides highly...
Unstructured nodal DG-FEM solution of high-order Boussinesq-type equations
Engsig-Karup, Allan Peter; Madsen, Per A.; Bingham, Harry B.; Thomsen, Per Grove
2007-01-01
The main objective of the present study has been to develop a numerical model and investigate solution techniques for solving the recently derived high-order Boussinesq equations of \\cite{MBL02} in irregular domains in one and two horizontal dimensions. The Boussinesq-type methods are the simplest alternative to solving full three-dimensional wave problems by e.g. Navier-Stokes equations, which can capture all the important wave phenomena such as diffraction, refraction, nonlinear wave-wave i...
NUMERICAL SIMULATON OF IMPROVED BOUSSINESQ EQUATIONS BY A FINITE ELEMENT METHOD
Institute of Scientific and Technical Information of China (English)
Zhao Ming; Teng Bin; Liu Shu-xue
2003-01-01
The improved Boussinesq equations for varying depth derived by Beji and Nadaoka[1]significantly improved the linear dispersive properties of wave models in intermediate water depths. In this study, a finite element method was developed to solve the improved Boussinesq equations. A spongy layer was applied at the open boundary of the computational domain to absorb the wave energy. The fourth-order predictor-corrector method was employed in the time integration. Several test cases were illustrated. The numerical results of this model were compared with laboratory data and those from other numerical models. It turns out that the present numerical model is capable of giving satisactory prediction for wave propagation.
DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Hesthaven, Jan; Bingham, Harry B.;
2008-01-01
waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry......We present a high-order nodal Discontinuous Galerkin Finite Element Method (DG-FEM) solution based on a set of highly accurate Boussinesq-type equations for solving general water-wave problems in complex geometries. A nodal DG-FEM is used for the spatial discretization to solve the Boussinesq...... equations in complex and curvilinear geometries which amends the application range of previous numerical models that have been based on structured Cartesian grids. The Boussinesq method provides the basis for the accurate description of fully nonlinear and dispersive water waves in both shallow and deep...
Chae, Dongho; Constantin, Peter; Wu, Jiahong
2014-09-01
We give an example of a well posed, finite energy, 2D incompressible active scalar equation with the same scaling as the surface quasi-geostrophic equation and prove that it can produce finite time singularities. In spite of its simplicity, this seems to be the first such example. Further, we construct explicit solutions of the 2D Boussinesq equations whose gradients grow exponentially in time for all time. In addition, we introduce a variant of the 2D Boussinesq equations which is perhaps a more faithful companion of the 3D axisymmetric Euler equations than the usual 2D Boussinesq equations.
Hydraulic Modeling of A Curtain-Walled Dissipater by the Coupling of RANS and Boussinesq Equations
Institute of Scientific and Technical Information of China (English)
齐鹏; 王永学
2002-01-01
A hybrid numerical method for the hydraulic modeling of a curtain-walled dissipater of reflected waves from breakwa-ters is presented. In this method, a zonal approach that combines a nonlinear weakly dispersive wave (Boussinesq-typeequation) method and a Reynolds-Averaged Navier-Stokes (RANS) method is used. The Boussinesq-type equation issolved in the far field to describe wave transformation in shallow water. The RANS method is used in the near field to re-solve the turbulent boundary layer and vortex flows around the structure. Suitable matching conditions are enforced at theinterface between the viscous and the Boussinesq region. The Coupled RANS and Boussinesq method successfully resolvesthe vortex characteristics of flow in the vicinity of the structure, while unexpected phenomena like wave re-reflection areeffectively controlled by lengthening the Boussinesq region. Extensive results on hydraulic performance of a curtain-walleddissipater and the mechanism of dissipation of reflected waves are presented, providing a reference for minimization of thebreadth of the water chamber and for determination of the submerged depth of the curtain wall.
Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation
Institute of Scientific and Technical Information of China (English)
MA Zhi-Min; SUN Yu-Huai; LIU Fu-Sheng
2013-01-01
In this paper,the generalized Boussinesq wave equation utt-uxx + a(um)xx + bu =0 is investigated by using the bifurcation theory and the method of phase portraits analysis.Under the different parameter conditions,the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.
Periodic Wave Solution to the (3+1)-Dimensional Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
WU Yong-Qi
2008-01-01
@@ One- and two-periodic wave solutions for (3+1)-dimensional Boussinesq equation are presented by means of Hirota's bilinear method and the Riemann theta function. The soliton solution can be obtained from the periodic wave solution in an appropriate limiting procedure.
Dissipative Boussinesq equations on non-cylindrical domains in R^n
Directory of Open Access Journals (Sweden)
Haroldo R. Clark
2010-01-01
Full Text Available This article concerns the initial-boundary value problem for the nonlinear Boussinesq equations on time dependent domains in $mathbb{R}^n$ with $1leq n leq 4$. Global solvability, uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense.
On quasi-periodic solutions for generalized Boussinesq equation with quadratic nonlinearity
Shi, Yanling; Xu, Junxiang; Xu, Xindong
2015-02-01
In this paper, one-dimensional generalized Boussinesq equation: utt - uxx + (u2 + uxx)xx = 0 with boundary conditions ux(0, t) = ux(π, t) = uxxx(0, t) = uxxx(π, t) = 0 is considered. It is proved that the equation admits a Whitney smooth family of small-amplitude quasi-periodic solutions with 2-dimensional Diophantine frequencies. The proof is based on an infinite dimensional Kolmogorov-Arnold-Moser theorem and Birkhoff normal form.
Painlevé properties and exact solutions for the high-dimensional Schwartz Boussinesq equation
Institute of Scientific and Technical Information of China (English)
Ren Bo; Lin Ji
2009-01-01
The usual(l+l)-dimensional Schwartz Boussinesq equation is extended to the (l+l)-dimensional space-time sym-metric form and the general (n+1)-dimensional space-time symmetric form. These extensions are Painlevé integrable in the sense that they possess the Painleve property. The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space-time symmetric form are obtained by the Painleve-Backlund transformation.
Application of homotopy perturbation method to the RLW and generalized modified Boussinesq equations
Energy Technology Data Exchange (ETDEWEB)
Rafei, M. [Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol (Iran, Islamic Republic of)]. E-mail: salammorteza@yahoo.com; Ganji, D.D. [Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol (Iran, Islamic Republic of); Mohammadi Daniali, H.R. [Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol (Iran, Islamic Republic of); Pashaei, H. [Department of Mechanical Engineering, Mazandaran University, P.O. Box 484, Babol (Iran, Islamic Republic of)
2007-04-16
In this Letter, He's homotopy perturbation method (HPM) is implemented for finding the solitary-wave solutions of the regularized long-wave (RLW) and generalized modified Boussinesq (GMB) equations. We obtain numerical solutions of these equations for the initial conditions. We will show that the convergence of the HPM is faster than those obtained by the Adomian decomposition method (ADM). The obtained solutions, in comparison with the exact solutions admit a remarkable accuracy. A clear conclusion can be drawn from the numerical results that the HPM provides highly accurate numerical solutions for nonlinear differential equations.
Nodal DG-FEM solution of high-order Boussinesq-type equations
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.;
2006-01-01
We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis...... and convergence of the model with both h (grid size) and p (order) refinement are verified for the linearized equations, and calculations are provided for two nonlinear test cases in one horizontal dimension: harmonic generation over a submerged bar; and reflection of a steep solitary wave from a vertical wall...
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
An approximate homotopy symmetry method for nonlinear problems is proposed and applied to the sixth-order Boussinesq equation,which arises from fluid dynamics.We summarize the general formulas for similarity reduction solutions and similarity reduction equations of different orders,educing the related homotopy series solutions.Zero-order similarity reduction equations are equivalent to the Painlevé IV type equation or Weierstrass elliptic equation.Higher order similarity solutions can be obtained by solving linear variable coefficients ordinary differential equations.The auxiliary parameter has an effect on the convergence of homotopy series solutions.Series solutions and similarity reduction equations from the approximate symmetry method can be retrieved from the approximate homotopy symmetry method.
Inclined periodic homoclinic breather and rogue waves for the (1+1)-dimensional Boussinesq equation
Indian Academy of Sciences (India)
Zhengde Dai; Chuanjian Wang; Jun Liu
2014-10-01
A new method, homoclinic (heteroclinic) breather limit method (HBLM), for seeking rogue wave solution to nonlinear evolution equation (NEE) is proposed. (1+1)-dimensional Boussinesq equation is used as an example to illustrate the effectiveness of the suggested method. Rational homoclinic wave solution, a new family of two-wave solution, is obtained by inclined periodic homoclinic breather wave solution and is just a rogue wave solution. This result shows that rogue wave originates by the extreme behaviour of homoclinic breather wave in (1+1)-dimensional nonlinear wave fields.
Institute of Scientific and Technical Information of China (English)
ZHANG Huan; TIAN Bo; ZHANG Hai-Qiang; GENG Tao; MENG Xiang-Hua; LIU Wen-Jun; CAI Ke-Jie
2008-01-01
For describing various complex nonlinear phenomena in the realistic world, the higher-dimensional nonlinear evolution equations appear more attractive in many fields of physical and engineering sciences. In this paper, by virtue of the Hirota bilinear method and Riemann theta functions, the periodic wave solutions for the (2+1)-dimensional Boussinesq equation and (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation are obtained. Furthermore, it is shown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.
An Improved Nearshore Wave Breaking Model Based on the Fully Nonlinear Boussinesq Equations
Institute of Scientific and Technical Information of China (English)
LI Shao-wu; LI Chun-ying; SHI Zhong; GU Han-bin
2005-01-01
This paper aims to propose an improved numerical model for wave breaking in the nearshore region based on the fully nonlinear form of Boussinesq equations. The model uses the κ equation turbulence scheme to determine the eddy viscosity in the Boussinesq equations. To calculate the turbulence production term in the equation, a new formula is derived based on the concept of surface roller. By use of this formula, the turbulence production in the one-equation turbulence scheme is directly related to the difference between the water particle velocity and the wave celerity. The model is verified by Hansen and Svendsen's experimental data (1979) in terms of wave height and setup and setdown. The comparison between the model and experimental results of wave height and setup and setdown shows satisfactory agreement. The modeled turbulence energy decreases as waves attenuate in the surf zone. The modeled production term peaks at the breaking point and decreases as waves propagate shoreward. It is also suggested that both convection and diffusion play their important roles in the transport of turbulence energy immediately after wave breaking. When waves approach to the shoreline, the production and dissipation of turbulence energy are almost balanced. By use of the slot technique for the simulation of the movable shoreline boundary, wave runup in the swash zone is well simulated by the present model.
Boundary Conditions for 2D Boussinesq-type Wave-Current Interaction Equations
Directory of Open Access Journals (Sweden)
Mera M.
2011-01-01
Full Text Available This research focuses on the development of a set of two-dimensional boundary conditions for specific governing equations. The governing equations are existing Boussinesqtype equations which is capable of simulating wave-current interaction. The present boundary conditions consist of for waves only case and for currents only case. To simulate wave-current interaction, the two kinds of the present boundary conditions are then combined. A numerical model based on both the existing governing equations and the present boundary conditions is applied to simulation of currents only and of wave-current interaction propagating over a basin with a submerged shoal. The results of the numerical model show that the present boundary conditions go well with the existing Boussinesq-type wave-current interaction equations.
On Triply Periodic Wave Solutions for （2d-1）-Dimensional Boussinesq Equation
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王军民
2012-01-01
By employing Hirota bilinear method and Riemann theta functions of genus one,explicit triply periodic wave solutions for the（2＋1）-dimensional Boussinesq equation are constructed under the Backlund transformation u =（1 /6）（u0 1） ＋ 2[ln f（x,y,t）] xx,four kinds of triply periodic wave solutions are derived,and their long wave limit are discussed.The properties of one of the solutions are shown in Fig.1.
Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
YANG Jian-Rong; ZHANG Yi; MAO Jie-Jian; YE Ling-Ya
2008-01-01
Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.
Mechdene, Mohamed; Gala, Sadek; Guo, Zhengguang; Ragusa, Alessandra Maria
2016-10-01
This work establishes a sufficient condition for the regularity criterion of the Boussinesq equation in terms of the derivative of the pressure in one direction. It is shown that if the partial derivative of the pressure {partial 3π } satisfies the logarithmical Serrin-type condition int0TVert partial 3π (s)Vert_{L^{λ }}q/1+ln (1+Vert θ Vert_{L4)} {d}s < ∞ quad {with}quad2/q+3/λ =7/4quad {and}quad12/7 < λ ≤ ∞, then the solution {(u,θ )} remains smooth on {[0,T]}. Compared to the Navier-Stokes result, there is a logarithmic correction involving {θ} in the denominator.
On Chorin's Method for Stationary Solutions of the Oberbeck-Boussinesq Equation
Kagei, Yoshiyuki; Nishida, Takaaki
2016-08-01
Stability of stationary solutions of the Oberbeck-Boussinesq system (OB) and the corresponding artificial compressible system is considered. The latter system is obtained by adding the time derivative of the pressure with small parameter {ɛ > 0} to the continuity equation of (OB), which was proposed by A. Chorin to find stationary solutions of (OB) numerically. Both systems have the same sets of stationary solutions and the system (OB) is obtained from the artificial compressible one as the limit {ɛ to 0} which is a singular limit. It is proved that if a stationary solution of the artificial compressible system is stable for sufficiently small {ɛ > 0} , then it is also stable as a solution of (OB). The converse is proved provided that the velocity field of the stationary solution satisfies some smallness condition.
Yang, Xiao-Feng; Deng, Zi-Chen; Li, Qing-Jun; Wei, Yi
2016-07-01
The homogeneous balance of undetermined coefficients method (HBUCM) is firstly proposed to construct not only the exact traveling wave solutions, three-wave solutions, homoclinic solutions, N-soliton solutions, but also multi-symplectic structures of some nonlinear partial differential equations (NLPDEs). By applying the proposed method to the variant Boussinesq equations (VBEs), the exact combined traveling wave solutions and a multi-symplectic structure of the VBEs are obtained directly. Then, the definition and a multi-symplectic structure of the variant Boussinesq-Whitham-Broer-Kaup type equations (VBWBKTEs) which can degenerate to the VBEs, the Whitham-Broer-Kaup equations (WBKEs) and the Broer-Kaup equations (BKEs) are given in the multi-symplectic sense. The HBUCM is also a standard and computable method, which can be generalized to obtain the exact solutions and multi-symplectic structures for some types of NLPDEs.
Institute of Scientific and Technical Information of China (English)
张卫国; 刘强; 李正明; 李想
2014-01-01
This article studies bounded traveling wave solutions of variant Boussinesq equa-tion with a dissipation term and dissipation effect on them. Firstly, we make qualitative analysis to the bounded traveling wave solutions for the above equation by the theory and method of planar dynamical systems, and obtain their existent conditions, number, and gen-eral shape. Secondly, we investigate the dissipation effect on the shape evolution of bounded traveling wave solutions. We find out a critical value r* which can characterize the scale of dissipation effect, and prove that the bounded traveling wave solutions appear as kink profile waves if |r|≥r*; while they appear as damped oscillatory waves if |r|
Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary
Tang, Yuehao; Jiang, Qinghui; Zhou, Chuangbing
2016-04-01
An approximate solution is presented to the 1-D Boussinesq equation (BEQ) characterizing transient groundwater flow in an unconfined aquifer subject to a constant water variation at the sloping water-land boundary. The flow equation is decomposed to a linearized BEQ and a head correction equation. The linearized BEQ is solved using a Laplace transform. By means of the frozen-coefficient technique and Gauss function method, the approximate solution for the head correction equation can be obtained, which is further simplified to a closed-form expression under the condition of local energy equilibrium. The solutions of the linearized and head correction equations are discussed from physical concepts. Especially for the head correction equation, the well posedness of the approximate solution obtained by the frozen-coefficient method is verified to demonstrate its boundedness, which can be further embodied as the upper and lower error bounds to the exact solution of the head correction by statistical analysis. The advantage of this approximate solution is in its simplicity while preserving the inherent nonlinearity of the physical phenomenon. Comparisons between the analytical and numerical solutions of the BEQ validate that the approximation method can achieve desirable precisions, even in the cases with strong nonlinearity. The proposed approximate solution is applied to various hydrological problems, in which the algebraic expressions that quantify the water flow processes are derived from its basic solutions. The results are useful for the quantification of stream-aquifer exchange flow rates, aquifer response due to the sudden reservoir release, bank storage and depletion, and front position and propagation speed.
Directory of Open Access Journals (Sweden)
Harun-Or- Roshid
2014-01-01
Full Text Available Periodic and soliton solutions are presented for the (1+1-dimensional classical Boussinesq equation which governs the evolution of nonlinear dispersive long gravity wave traveling in two horizontal directions on shallow water of uniform depth. The equation is handled via the exp(−Φ(η-expansion method. It is worth declaring that the method is more effective and useful for solving the nonlinear evolution equations. In particular, mathematical analysis and numerical graph are provided for those solitons, periodic, singular kink and bell type solitary wave solutions to visualize the dynamics of the equation.
Institute of Scientific and Technical Information of China (English)
YAN ZhenYa; XIE FuDing; ZHANG HongQing
2001-01-01
Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of AblowRz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation.``
On 2-D Boussinesq equations for MHD convection with stratification effects
Bian, Dongfen; Gui, Guilong
2016-08-01
This paper is concerned with the two-dimensional magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. The first progress on this topic was made independently by Chae and Hou-Li [8,26] where the Boussinesq system with partial constant viscosity is obtained. Recently, Wang-Zhang [45] considered the temperature-dependent viscosity and thermal diffusivity, and Li-Xu [16] generalized the Wang-Zhang's result to the inviscid case with temperature-dependent thermal diffusivity. In this paper, we include the stratification and magnetic effects and consider the full system, in the framework of low regularity. We prove that, without any smallness assumption on the initial data, the full system is globally well-posed. Moreover, by applying the uniformly bounded generalized Oseen operator, time decay estimate of the solution is obtained.
Pauwels, Valentijn R. N.; Verhoest, Niko E. C.; de Troch, FrançOis P.
2002-12-01
In hydrology the slow, subsurface component of the discharge is usually referred to as base flow. One method to model base flow is the conceptual approach, in which the complex physical reality is simplified using hypotheses and assumptions, and the various physical processes are described mathematically. The purpose of this paper is to develop and validate a conceptual method, based on hydraulic theory, to calculate the base flow of a catchment, under observed precipitation rates. The governing groundwater equation, the Boussinesq equation, valid for a unit width sloping aquifer, is linearized and solved for a temporally variable recharge rate. The solution allows the calculation of the transient water table profile in and the outflow from an aquifer under temporally variable recharge rates. When a catchment is considered a metahillslope, the solution can be used, when coupled to a routing model, to calculate the catchment base flow. The model is applied to the Zwalm catchment and four subcatchments in Belgium. The results suggest that it is possible to model base flow at the catchment scale, using a Boussinesq-based metahillslope model. The results further indicate that it is sufficient to use a relatively simple formulation of the infiltration, overland flow, and base flow processes to obtain reasonable estimates of the total catchment discharge.
Paniconi, C.; Troch, P.A.A.; Loon, van E.E.; Hilberts, A.G.J.
2003-01-01
The Boussinesq equation for subsurface flow in an idealized sloping aquifer of unit width has recently been extended to hillslopes of arbitrary geometry by incorporating the hillslope width function w(x) into the governing equation, where x is the flow distance along the length of the hillslope [ Tr
DEFF Research Database (Denmark)
Fuhrmann, David R.; Bingham, Harry B.; Madsen, Per A.;
2004-01-01
This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly nonlinear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...... of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water nonlinearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only...... moderately non-normal, suggesting that the eigenvalues are likely suitable for analysis purposes. Numerical experiments demonstrate excellent agreement with the linear analysis, and good qualitative agreement with the local nonlinear analysis. The various methods of analysis combine to provide significant...
Institute of Scientific and Technical Information of China (English)
Zuo Jin-Ming; Zhang Yao-Ming
2011-01-01
This paper studies the coupled Burgers equation and the high-order Bonssinesq-Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.
Solitary-wave propagation and interactions for a sixth-order generalized Boussinesq equation
Directory of Open Access Journals (Sweden)
Bao-Feng Feng
2005-01-01
based on the phase plane analysis around the equilibrium point, is used to construct the solitary-wave solutions for this nonintegrable equation. A symmetric three-level implicit finite difference scheme with a free parameter θ is proposed to study the propagation and interactions of solitary waves. Numerical simulations show the propagation of a single solitary wave of SGBE, and two solitary waves pass by each other without changing their shapes in the head-on collisions.
Institute of Scientific and Technical Information of China (English)
TIAN; Xiangjun; XIE; Zhenghui; ZHANG; Shenglei
2006-01-01
Subsurface runoff in a land surface model is usually parameterized as a single-valued function of total storage in a basin aquifer reservoir. This kind of parameterization is often single-valued function of storage-discharge under a steady or "quasi-steady" state, which cannot represent the influence of aquifer recharge on subsurface runoff generation. In this paper, a new subsurface runoff parameterization with water storage and recharge based on the Boussinesq-storage equation is developed. This model is validated by a subsurface flow separation algorithm for an example river basin, which shows that the new model can simulate the subsurface flow reasonably.
Institute of Scientific and Technical Information of China (English)
沈春; 孙梅娜
2005-01-01
Based on the theory of stratification, the well-posedness of the initial and boundary value problems for the system of two-dimensional non-hydrostatic Boussinesq equations was discussed. The sufficient and necessary conditions of the existence and uniqueness for the solution of the equations were given for some representative initial and boundary value problems. Several special cases were discussed.
Institute of Scientific and Technical Information of China (English)
王倩
2013-01-01
T he method of constructing approximate conserved vectors and conserved law s for perturbed (2+1)-dimensional Boussinesq equation are concretely described .In terms of the partial Lagrangian ap-proach ,the conserved law s are constructed by using approximate Noether method ,then the approximate Noether-type symmetry operators and approximate conserved law s are obtained .%利用近似Noether-type对称算子构造了具有扰动项的（2＋1）维Boussinesq方程的近似守恒向量和近似守恒律，在（2＋1）维Boussinesq方程允许的拉格朗日函数的情况下，利用近似Noether法研究了该方程的守恒律，给出了（2＋1）维扰动Boussinesq方程的近似Noether对称算子、近似守恒向量以及近似守恒律。
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry B.; Madsen, Per A.;
2004-01-01
of rotational and irrotational formulations in two horizontal dimensions provides evidence that the irrotational formulation has significantly better stability properties when the deep-water non-linearity is high, particularly on refined grids. Computation of matrix pseudospectra shows that the system is only...... insight into the numerical behaviour of this rather complicated system of non-linear PDEs.......This paper considers a method of lines stability analysis for finite difference discretizations of a recently published Boussinesq method for the study of highly non-linear and extremely dispersive water waves. The analysis demonstrates the near-equivalence of classical linear Fourier (von Neumann...
一类Boussinesq方程的同宿轨和周期孤立子%Homoclinic Orbits and Periodic Solitons for a Class of Boussinesq Equations
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李正彪; 戴正德
2005-01-01
This paper considers homoclinic orbits and periodic solitons for a class of Boussinesq equations with periodic boundary condition and even constraint condition. At first,by the linearized stability analysis,the authors get the existence of homoclinic orbits for "bad" Bq equation and periodic solitons for "good" Bq equation. Then, by the Hirota' s bilinear method, the exact expressions of homoclinic orbits and periodic solitons are obtained respectively,and the authors find there is blow-up phenomenon for the soliton solutions.%研究了一类具有周期边界条件和偶约束的Boussinesq方程.首先,通过线性稳定性分析,证明了"坏"Boussinesq方程存在同宿轨解,而"好"Boussinesq方程存在孤立子解.然后,利用Hirota双线性方法,分别获得了同宿轨和孤立子的显式表达式,而且发现孤立子解存在爆破现象.
Study on Solitary Waves of a General Boussinesq Model
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, we employ the bifurcation method of dynamical systems to study the solitary waves and periodic waves of a generalized Boussinesq equations. All possible phase portraits in the parameter plane for the travelling wave systems are obtained. The possible solitary wave solutions, periodic wave solutions and cusp waves for the general Boussinesq type fluid model are also investigated.
Boussinesq modeling of surface waves due to underwater landslides
Directory of Open Access Journals (Sweden)
D. Dutykh
2013-05-01
Full Text Available Consideration is given to the influence of an underwater landslide on waves at the surface of a shallow body of fluid. The equations of motion that govern the evolution of the barycenter of the landslide mass include various dissipative effects due to bottom friction, internal energy dissipation, and viscous drag. The surface waves are studied in the Boussinesq scaling, with time-dependent bathymetry. A numerical model for the Boussinesq equations is introduced that is able to handle time-dependent bottom topography, and the equations of motion for the landslide and surface waves are solved simultaneously. The numerical solver for the Boussinesq equations can also be restricted to implement a shallow-water solver, and the shallow-water and Boussinesq configurations are compared. A particular bathymetry is chosen to illustrate the general method, and it is found that the Boussinesq system predicts larger wave run-up than the shallow-water theory in the example treated in this paper. It is also found that the finite fluid domain has a significant impact on the behavior of the wave run-up.
BOUSSINESQ MODELLING OF NEARSHORE WAVES UNDER BODY FITTED COORDINATE
Institute of Scientific and Technical Information of China (English)
FANG Ke-zhao; ZOU Zhi-li; LIU Zhong-bo; YIN Ji-wei
2012-01-01
A set of nonlinear Boussinesq equations with fully nonlinearity property is solved numerically in generalized coordinates,to develop a Boussinesq-type wave model in dealing with irregular computation boundaries in complex nearshore regions and to facilitate the grid refinements in simulations.The governing equations expressed in contravariant components of velocity vectors under curv ilinear coordinates are derived and a high order finite difference scheme on a staggered grid is employed for the numerical implementation.The developed model is used to simulate nearshore wave propagations under curvilinear coordinates,the numerical results are compared against analytical or experimental data with a good agreement.
The Boussinesq Debate: Reversibility, Instability, and Free Will.
Michael Mueller, Thomas
2015-12-01
In 1877, a young mathematician named Joseph Boussinesq presented a mémoire to the Académie des 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.
The Boussinesq Debate: Reversibility, Instability, and Free Will.
Michael Mueller, Thomas
2015-12-01
In 1877, a young mathematician named Joseph Boussinesq presented a mémoire to the Académie des 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. PMID:26554644
Random attractor of non-autonomous stochastic Boussinesq lattice system
International Nuclear Information System (INIS)
In this paper, we first consider the existence of tempered random attractor for second-order non-autonomous stochastic lattice dynamical system of nonlinear Boussinesq equations effected by time-dependent coupled coefficients and deterministic forces and multiplicative white noise. Then, we establish the upper semicontinuity of random attractors as the intensity of noise approaches zero
Spectral element modelling of floating bodies in a Boussinesq framework
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter; Eskilsson, Claes; Ricchiuto, Mario
a possible middle way between the highly simplified and fast linear hydrodynamics and the very complete but slow VOF-RANS simulations is to use nonlinear, dispersive wave equations of Boussinesq-type. Jiang (2001) presented a unified approach for including bodies into the Boussinesq framework and solved...... the system with finite differences. In the unified approach the pressure working on the body are solved for using the instantaneous draft. In this study we will outline how to implement the approach of Jiang in a spectral/hp element setting, and simulate the heave motion of a body using different asymptotic...... wave equations. We will especially focus on the stabilization of the coupled system....
STUDY OF NON-BOUSSINESQ EFFCET ON SEA SURFACE HEIGHT
Institute of Scientific and Technical Information of China (English)
CHEN Xian-yao; WANG Xuan; WANG Xiu-hong; QIAO Fang-li
2004-01-01
A set of equations was derived for a non-Boussinesq ocean model in this paper.A new time-splitting scheme was introduced which incorporates the 4th-order Runge-Kutta explicit scheme of low-frequency mode and an implicit scheme of high-frequency mode.With this model,potential temperature,salinity fields and sea surface height were calculated simultaneously such that the numerical error of extrapolation of density field from the current time level to the next one could be reduced while using the equation of mass conservation to determine sea surface height.The non-Boussinesq effect on the density field and sea surface height was estimated by numerical experiments in the final part of this paper.
On devising Boussinesq-type models with bounded eigenspectra: One horizontal dimension
DEFF Research Database (Denmark)
Eskilsson, Claes; Engsig-Karup, Allan Peter
2014-01-01
The propagation of water waves in the nearshore region can be described by depth-integrated Boussinesq-type equations. The dispersive and nonlinear characteristics of the equations are governed by tuneable parameters. We examine the associated linear eigenproblem both analytically and numerically...... requires Δt∝p−2. We derive and present conditions on the parameters under which implicitly-implicit Boussinesq-type equations will exhibit bounded eigenspectra. Two new bounded versions having comparable nonlinear and dispersive properties as the equations of Nwogu (1993) and Schäffer and Madsen (1995) are...
The consistent Riccati expansion and new interaction solution for a Boussinesq-type coupled system
Ruan, Shao-Qing; Yu, Wei-Feng; Yu, Jun; Yu, Guo-Xiang
2015-06-01
Starting from the Davey-Stewartson equation, a Boussinesq-type coupled equation system is obtained by using a variable separation approach. For the Boussinesq-type coupled equation system, its consistent Riccati expansion (CRE) solvability is studied with the help of a Riccati equation. It is significant that the soliton-cnoidal wave interaction solution, expressed explicitly by Jacobi elliptic functions and the third type of incomplete elliptic integral, of the system is also given. Project supported by the National Natural Science Foundation of China (Grant No. 11275129).
Coastal zone simulations with variational Boussinesq modelling
Adytia, Didit
2012-01-01
The main challenge in deriving a Boussinesq model for water wave is to model accurately the dispersion and nonlinearity of waves. The dispersion is a depth-dependent relation between the wave speed and the wavelength. A Boussinesq-type model can be derived from the so-called variational principle
Mitsotakis, Dimitrios
2009-01-01
Considered here are Boussinesq systems of equations of surface water wave theory over a variable bottom. A simplified such Boussinesq system is derived and solved numerically by the standard Galerkin-finite element method. We study by numerical means the generation of tsunami waves due to bottom deformation and we compare the results with analytical solutions of the linearized Euler equations. Moreover, we study tsunami wave propagation in the case of the Java 2006 event, comparing the result...
The Asymptotic Limit for the 3D Boussinesq System
Institute of Scientific and Technical Information of China (English)
LI Lin-rui; WANG Ke; HONG Ming-li
2016-01-01
In this paper, we show the asymptotic limit for the 3D Boussinesq system with zero viscosity limit or zero diffusivity limit. By the classical energy method, we prove that as viscosity(or diffusivity) coeﬃcient goes to zero the solutions of the fully viscous equations converges to those of zero viscosity(or zero diffusivity) equations, which extend the previous results on the asymptotic limit under the conditions of the zero parameter(zero viscosityν=0 or zero diffusivityη=0) in 2D case separately.
CTE Solvability, Nonlocal Symmetry and Explicit Solutions of Modified Boussinesq System
Ren, Bo; Cheng, Xue-Ping
2016-07-01
A consistent tanh expansion (CTE) method is used to study the modified Boussinesq equation. It is proved that the modified Boussinesq equation is CTE solvable. The soliton-cnoidal periodic wave is explicitly given by a nonanto-BT theorem. Furthermore, the nonlocal symmetry for the modified Boussinesq equation is obtained by the Painlevé analysis. The nonlocal symmetry is localized to the Lie point symmetry by introducing one auxiliary dependent variable. The finite symmetry transformation related with the nonlocal symemtry is obtained by solving the initial value problem of the prolonged systems. Thanks to the localization process, many interaction solutions among solitons and other complicated waves are computed through similarity reductions. Some special concrete soliton-cnoidal wave interaction behaviors are studied both in analytical and graphical ways. Supported by the National Natural Science Foundation of China under Grant Nos. 11305106 and 11505154
Modified Boussinesq System with Variable Coefficients: Classical Lie Approach and Exact Solutions
Institute of Scientific and Technical Information of China (English)
GUPTA R.K.; SINGH K.
2009-01-01
The Lie-group formalism is applied to investigate the symmetries of the modified Boussinesq system with variable coefficients. We derived the infinitesimals and the admissible forms of the coefficients that admit the classical symmetry group. The reduced systems of ordinary differential equations deduced from the optimal system of subalgebras are further studied and some exact solutions are obtained.
The Boussinesq approximation in rapidly rotating flows
Lopez, Jose M; Avila, Marc
2013-01-01
In the classical formulation of the Boussinesq approximation centrifugal buoyancy effects related to differential rotation, as well as strong vortices in the flow, are neglected. However, these may play an important role in rapidly rotating flows, such as in astrophysical and geophysical applications, and also in turbulent convection. We here provide a straightforward approach resulting in a Boussinesq-type approximation that consistently accounts for centrifugal effects. We further compare our new approach to the classical one in fluid flows confined between two differentially heated and rotating cylinders. The results justify the need of using the proposed approximation in rapidly rotating flows.
Panaches horizontaux non-Boussinesq en milieu homog\\`ene
Daddi-Moussa-Ider, Abdallah; Mehaddi, Rabah; Vauquelin, Olivier; Candelier, Fabien
2014-01-01
The environmental impact of pollutants and effluents discharged into the atmosphere or the oceans has led researchers to conduct studies related to this issue. Several works have been carried out in this context in order to reduce the effect on the local environment. These types of ejections in nature are modeled as jets in the presence of a density gradient. In this study we treated the problem of inclined round turbulent buoyant jets and plumes ejected in a homogeneous or stratified fluid, at rest or in motion. The prediction of the flow behavior, i.e. the evolution of its variables, is first treated theoretically from a model whose formalism is valid in both the Boussinesq case as well as in the non-Boussinesq general case. Solving the equations governing the plumes is performed numerically using a Runge-Kutta 4th order. To validate the model, laboratory experiments are performed with round jets of air and helium for a wide range of densities. The confrontation theory-experience aims here to fix the limits...
Hilberts, A.G.J.
2006-01-01
Key words: hillslope hydrology, low-dimensional modeling, Boussinesq equation, Richards equation, water table dynamics.In this thesis the focus is on investigating the hillslope hydrological behavior, as a crucial part in understanding the catchment hydrological response. To overcome difficulties as
Nonhydrostatic granular flow over 3-D terrain: New Boussinesq-type gravity waves?
Castro-Orgaz, Oscar; Hutter, Kolumban; Giraldez, Juan V.; Hager, Willi H.
2015-01-01
granular mass flow is a basic step in the prediction and control of natural or man-made disasters related to avalanches on the Earth. Savage and Hutter (1989) pioneered the mathematical modeling of these geophysical flows introducing Saint-Venant-type mass and momentum depth-averaged hydrostatic equations using the continuum mechanics approach. However, Denlinger and Iverson (2004) found that vertical accelerations in granular mass flows are of the same order as the gravity acceleration, requiring the consideration of nonhydrostatic modeling of granular mass flows. Although free surface water flow simulations based on nonhydrostatic depth-averaged models are commonly used since the works of Boussinesq (1872, 1877), they have not yet been applied to the modeling of debris flow. Can granular mass flow be described by Boussinesq-type gravity waves? This is a fundamental question to which an answer is required, given the potential to expand the successful Boussinesq-type water theory to granular flow over 3-D terrain. This issue is explored in this work by generalizing the basic Boussinesq-type theory used in civil and coastal engineering for more than a century to an arbitrary granular mass flow using the continuum mechanics approach. Using simple test cases, it is demonstrated that the above question can be answered in the affirmative way, thereby opening a new framework for the physical and mathematical modeling of granular mass flow in geophysics, whereby the effect of vertical motion is mathematically included without the need of ad hoc assumptions.
(1+1)-维Boussinesq-Burgers方程的Wronskian解
Institute of Scientific and Technical Information of China (English)
牛艳霞; 王曦峰; 张俊良
2013-01-01
本文利用Hirota双线性化方法,从(1+1)-维Boussinesq-Burgers保谱问题的lax对中,找到适当的函数φ、ψ,进而构造出了(1+1)-维Boussinesq-Burgers方程的Wronskian形式的精确孤子解。
Solitary wave shoaling and breaking in a regularized Boussinesq system
Senthilkumar, Amutha
2016-01-01
A coupled BBM system of equations is studied in the situation of water waves propagating over decreasing fluid depth. A conservation equation for mass and a wave breaking criterion valid in the Boussinesq approximation is found. A Fourier collocation method coupled with a 4-stage Runge-Kutta time integration scheme is employed to approximate solutions of the BBM system. The mass conservation equation is used to quantify the role of reflection in the shoaling of solitary waves on a sloping bottom. Shoaling results based on an adiabatic approximation are analyzed. Wave shoaling and the criterion of breaking solitary waves on a sloping bottom is studied. To validate the numerical model the simulation results are compared with those obtained by Grilli et al.[16] and a good agreement between them is observed. Shoaling of solitary waves of two different types of mild slope model systems in [8] and [13] are compared, and it is found that each of these models works well in their respective regimes of applicability.
Two-layer interfacial flows beyond the Boussinesq approximation: a Hamiltonian approach
Camassa, R; Ortenzi, G
2015-01-01
The theory of integrable systems of Hamiltonian PDEs and their near-integrable deformations is used to study evolution equations resulting from vertical-averages of the Euler system for two-layer stratified flows in an infinite 2D channel. The Hamiltonian structure of the averaged equations is obtained directly from that of the Euler equations through the process of Hamiltonian reduction. Long-wave asymptotics together with the Boussinesq approximation of neglecting the fluids' inertia is then applied to reduce the leading order vertically averaged equations to the shallow-water Airy system, and thence, in a non-trivial way, to the dispersionless non-linear Schr\\"odinger equation. The full non-Boussinesq system for the dispersionless limit can then be viewed as a deformation of this well known equation. In a perturbative study of this deformation, it is shown that at first order the deformed system possesses an infinite sequence of constants of the motion, thus casting this system within the framework of comp...
Inertial particle dynamics: Coherent structures in the presence of the Basset-Boussinesq memory term
Farazmand, Mohammad; Haller, George
2013-11-01
We present an equivalent formulation of the Maxey-Riley equation in the presence of the Basset-Boussinesq memory term. A physical advantage of this formulation is that it reveals drag- and pressure-type forces within the memory term. The computational advantage of the new form is that it turns the Maxey-Riley equation from an implicit differential equation into an explicit one, enabling the use of classic numerical schemes in its solution. We further simplify the Maxey-Riley equation for small particles by deriving its reduction to its attractor. The reduced equation obtained in this fashion reveals that the memory term is asymptotically of the order of St 3 / 2, with St being the Stokes number. This explains recent numerical findings on the relative importance of the Basset-Boussinesq term. Finally, we compute inertial Lagrangian coherent structures (ILCS) for vortex shedding behind a cylinder. The reduced ILCS closely capture the full inertial dynamics while providing significant savings in computational cost and complexity.
The Oberbeck-Boussinesq approximation as a constitutive limit
Kagei, Yoshiyuki; Růžička, Michael
2016-09-01
We derive the usual Oberbeck-Boussinesq approximation as a constitutive limit of the full system describing the motion of an compressible linearly viscous fluid. To this end, the starting system is written, using the Gibbs free energy, in the variables v, θ and p. The Oberbeck-Boussinesq system is then obtained as the thermal expansion coefficient α and the isothermal compressibility coefficient β tend to zero.
Non-Boussinesq Rolls in 2d Thermal Convection
Málaga, C; Peralta-Fabi, R; Arzate, C
2013-01-01
A study of convection in a circular two dimensional cell is presented. The system is heated and cooled at two diametrically opposed points on the edge of the circle, which are parallel or anti-parallel to gravity. The latter's role in the plane of the cell can be changed by tilting the cell. When the system is in a horizontal position, a non-trivial analytic solution for the temperature distribution of the quiescent fluid can be found. For a slight inclination, the projection of gravity in the plane of the cell is used as a perturbation parameter in the full hydrodynamic description, as the Boussinesq approximation is inadequate. To first order, the equations are solved for the stationary case and four symmetrical rolls become apparent, showing that a purely conductive state is impossible if gravity -however small- is present; an approximate closed analytical expression is obtained, which describes the four convection rolls. Further analysis is done by a direct numerical integration. Comparison with prelimina...
A Non-Incompressible Non-Boussinesq (NINB) framework for studying atmospheric turbulence
Yan, C.; Archer, C. L.; Xie, S.; Ghaisas, N.
2015-12-01
The incompressible assumption is widely used for studying the turbulent atmospheric boundary layer (ABL) and is generally accepted when the Mach number wind turbine blades can reach and exceed this threshold, neglecting air compressibility will introduce errors. In addition, if air incompressibility does not hold, then the Boussinesq approximation, by which air density is treated as a constant except in the gravity term of the Navier-Stokes equation, is also invalidated. Here, we propose a new theoretical framework, called NINB for Non-Incompressible Non-Boussinesq, in which air is not considered incompressible and air density is treated as a non-turbulent 4D variable. First, the NINB mass, momentum, and energy conservation equations are developed using Reynolds averaging. Second, numerical simulations of the NINB equations, coupled with a k-epsilon turbulence model, are performed with the finite-volume method. Wind turbines are modeled with the actuator-line model using SOWFA (Software for Offshore/onshore Wind Farm Applications). Third, NINB results are compared with the traditional incompressible buoyant simulations performed by SOWFA with the same set up. The results show differences between NINB and traditional simulations in the neutral atmosphere with a wind turbine. The largest differences in wind speed (up to 1 m/s), turbulent kinetic energy (~10%), dissipation rate (~5%), and shear stress (~10%) occur near the turbine tip region. The power generation differences are 5-15% (depending on setup). These preliminary results suggest that compressibility effects are non-negligible around wind turbines and should be taken into account when forecasting wind power. Since only a few extra terms are introduced, the NINB framework may be an alternative to the traditional incompressible Boussinesq framework for studying the turbulent ABL in general (i.e., without turbines) in the absence of shock waves.
Un esquema semidiscreto de elementos finitos para el sistema "bueno" de Boussinesq
Díez Fernández, Honorato
2009-01-01
El sistema "bueno" de Boussiiiesq es un sistema de ecuaciones en derivadas parciales coi1 estructura hamiltoniana. Al discrctizarlo es de interés no perder tal estructura y en este articulo proponemos un método iluinérico de elementos finitos Petrov-Galerkiii que de origen a un sistema hamiltoniano discreto. Analizamos el error y presentamos resultados numéricos. The "good" Boussinesq systein is a system of partial differential equations with a hamiltonian structure. Wlieii carryiilg ou...
Brocchini, Maurizio
2013-01-01
This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather th...
Energetics of a fluid under the Boussinesq approximation
Maruyama, Kiyoshi
2014-01-01
This paper presents a theory describing the energy budget of a fluid under the Boussinesq approximation: the theory is developed in a manner consistent with the conservation law of mass. It shows that no potential energy is available under the Boussinesq approximation, and also reveals that the work done by the buoyancy force due to changes in temperature corresponds to the conversion between kinetic and internal energy. This energy conversion, however, makes only an ignorable contribution to the distribution of temperature under the approximation. The Boussinesq approximation is, in physical oceanography, extended so that the motion of seawater can be studied. This paper considers this extended approximation as well. Under the extended approximation, the work done by the buoyancy force due to changes in salinity corresponds to the conversion between kinetic and potential energy. It also turns out that the conservation law of mass does not allow the condition $\
Larios, Adam; Titi, Edriss S
2010-01-01
We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in the horizontal direction, which arises in Ocean dynamics. This work improves the global well-posedness results established recently by R. Danchin and M. Paicu for the Boussinesq system with anisotropic viscosity and zero diffusion. Although we follow some of their ideas, in proving the uniqueness result, we have used an alternative approach by writing the transported temperature (density) as $\\theta = \\Delta\\xi$ and adapting the techniques of V. Yudovich for the 2D incompressible Euler equations. This new idea allows us to establish uniqueness results with fewer assumptions on the initial data for the transported quantity $\\theta$. Furthermore, this new technique allows us to establish uniqueness results without having to resort to the paraproduct calculus of J. Bony. We also propose an inviscid $\\alpha$-regularization for the two-dimensional inviscid, non-diffusive Boussinesq s...
Developing hillslope-based catchment models: coupling Boussinesq and regional scale flow models
Broda, S.; Paniconi, C.; Larocque, M.
2009-04-01
The gaining recognition of hillslopes as fundamental building blocks in watershed hydrology makes them appealing for incorporation into larger scale river basin models. Hillslope processes are commonly simulated by means of the Boussinesq equation and are therefore applicable to single layer flow systems only. Two coupled models are presented to simulate both local hillslope scale and regional scale groundwater flow: 1) the hillslope-storage Boussinesq (hsB) model representing unconfined flow and a steady, analytic element model representing transient regional deep groundwater flow through a succession of steady state stress periods over many hydrological years, and 2) the hsB model and a newly developed analytical solution for 1D transient confined groundwater flow. Recharge zones are defined by means of irregular geometric domains, capturing the plan form geometry of the hillslopes. Lateral flows are calculated in inclined aquifers of homogeneous thickness. Tests are conducted on i) single hillslopes of varying inclination and plan form geometry and ii) a laboratory watershed, and heads and baseflows are compared to the results from a fully coupled 3D Richards equation model. Both approaches presented provide reasonable heads and fluxes for a range of hillslope properties in comparison to the benchmark model, and are promising approaches, applicable to a range of land surface models that lack a detailed description of subsurface flow. However the coupled hsB/1D-analytical model is numerically more stable and computationally more efficient than the coupled hsB/analytic element model.
Systematic investigation of non-Boussinesq effects in variable-density groundwater flow simulations
Guevara Morel, Carlos R.; van Reeuwijk, Maarten; Graf, Thomas
2015-12-01
The validity of three mathematical models describing variable-density groundwater flow is systematically evaluated: (i) a model which invokes the Oberbeck-Boussinesq approximation (OB approximation), (ii) a model of intermediate complexity (NOB1) and (iii) a model which solves the full set of equations (NOB2). The NOB1 and NOB2 descriptions have been added to the HydroGeoSphere (HGS) model, which originally contained an implementation of the OB description. We define the Boussinesq parameter ερ = βω Δω where βω is the solutal expansivity and Δω is the characteristic difference in solute mass fraction. The Boussinesq parameter ερ is used to systematically investigate three flow scenarios covering a range of free and mixed convection problems: 1) the low Rayleigh number Elder problem (Van Reeuwijk et al., 2009), 2) a convective fingering problem (Xie et al., 2011) and 3) a mixed convective problem (Schincariol et al., 1994). Results indicate that small density differences (ερ ≤ 0.05) produce no apparent changes in the total solute mass in the system, plume penetration depth, center of mass and mass flux independent of the mathematical model used. Deviations between OB, NOB1 and NOB2 occur for large density differences (ερ > 0.12), where lower description levels will underestimate the vertical plume position and overestimate mass flux. Based on the cases considered here, we suggest the following guidelines for saline convection: the OB approximation is valid for cases with ερ 0.10. Whether NOB effects are important in the intermediate region differ from case to case.
Nonlinear Super Integrable Couplings of Super Classical-Boussinesq Hierarchy
Directory of Open Access Journals (Sweden)
Xiuzhi Xing
2014-01-01
Full Text Available Nonlinear integrable couplings of super classical-Boussinesq hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then, its super Hamiltonian structures were established by using super trace identity. As its reduction, nonlinear integrable couplings of the classical integrable hierarchy were obtained.
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
A trial equation method to nonlinear evolution equation with rank inhomogeneous is given. As applications, the exact traveling wave solutions to some higher-order nonlinear equations such as generalized Boussinesq equation,generalized Pochhammer-Chree equation, KdV-Burgers equation, and KS equation and so on, are obtained. Among these, some results are new. The proposed method is based on the idea of reduction of the order of ODE. Some mathematical details of the proposed method are discussed.
Effects of compressibility in the mantle convection Equations
Trubitsyn, V. P.; Trubitsyna, A. P.
2015-11-01
The Boussinesq approximation of thermal convection equations results from neglecting the number of the terms which are actually not small in the conditions of the Earth's mantle. However, the error of calculating the structure of the convective flows is lower than the discarded terms. In this work we analyze the causes of this fact by successively passing from the general equations for a heated viscous compressible fluid to the simpler thermal convection equations by rejecting small quantities with the parameters of the presentday Earth. We consider the anelastic liquid approximation (ALA), truncated anelastic liquid approximation (TALA), extended Boussinesq approximation (EBA), and the simplest classical Boussinesq approximation (BA) which fully disregards the compressibility of a fluid. With the parameters of the mantle, BA is only accurate when describing the flow velocities, while the temperature is predicted with an error of up to a few dozen percent. Therefore, it appears reasonable to consider an intermediate approximation between EBA and BA, in which the effects of compressibility are only taken into account for temperature. This approximation can be referred to as the superadiabatic Boussinesq approximation (SBA) for temperature T sa. The corresponding equations are structurally similar to the standard Boussinesq approximation but with a superadiabatic temperature T sa instead of total temperature T. In this simple approximation, the calculated structure of the convective flows and the distribution of total temperature (obtained by adding the known adiabatic T a to the calculated T sa) are more accurate than in the classical Boussinesq approximation.
One-Dimensional Horizontal Boussinesq Model Enhanced for Non-Breaking and Breaking Waves
Institute of Scientific and Technical Information of China (English)
DONG Guo-hai; MA Xiao-zhou; TENG Bin
2008-01-01
Based on a set of fully nonlinear Boussinesq equations up to the order of O(μ2, ε3μ2) (where ε is the ratio of wave amplitude to water depth and μ is the ratio of water depth to wave length) a numerical wave model is formulated. The model's linear dispersion is acceptably accurate to μ≌1.0, which is confirmed by comparisons between the simulated and measured time series of the regular waves propagating on a submerged bar. The moving shoreline is treated numerically by replacing the solid beach with a permeable beach. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is fulfilled by introducing an eddy term in the momentum equation to serve as the breaking wave force term to dissipate wave energy in the surf zone. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches. Comparisons of the model test results comprising spatial distribution of wave height and mean water level with experimental data are presented.
Shi, Fengyan; Kirby, James T.; Harris, Jeffrey C.; Geiman, Joseph D.; Grilli, Stephan T.
We present a high-order adaptive time-stepping TVD solver for the fully nonlinear Boussinesq model of Chen (2006), extended to include moving reference level as in Kennedy et al. (2001). The equations are reorganized in order to facilitate high-order Runge-Kutta time-stepping and a TVD type scheme with a Riemann solver. Wave breaking is modeled by locally switching to the nonlinear shallow water equations when the Froude number exceeds a certain threshold. The moving shoreline boundary condition is implemented using the wetting-drying algorithm with the adjusted wave speed of the Riemann solver. The code is parallelized using the Message Passing Interface (MPI) with non-blocking communication. Model validations show good performance in modeling wave shoaling, breaking, wave runup and wave-averaged nearshore circulation.
Velocity potential formulations of highly accurate Boussinesq-type models
DEFF Research Database (Denmark)
Bingham, Harry B.; Madsen, Per A.; Fuhrman, David R.
2009-01-01
, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487-504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective...... is of interest because it reduces the computational effort by approximately a factor of two and facilitates a coupling to other potential flow solvers. A new shoaling enhancement operator is introduced to derive new models (in both formulations) with a velocity profile which is always consistent...... processes on the weather side of reflective structures. Coast. Eng. 53, 929-945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z=(z) over cap. For practical implementation however, the solution is expanded based on a slow...
Directory of Open Access Journals (Sweden)
R. C. Cabrales
2009-01-01
Full Text Available Obtenemos cotas para el error de las soluciones fuertes de las ecuaciones de Boussinesq que modelan los fluidos incompresibles y conductores de calor, suponiendo que dichas soluciones son condicionalmente asintóticamente estables.
Stability of 3D Gaussian vortices in rotating stratified Boussinesq flows: Linear analysis
Mahdinia, Mani; Jiang, Chung-Hsiang
2016-01-01
The linear stability of three-dimensional (3D) vortices in rotating, stratified flows has been studied by analyzing the non-hydrostatic inviscid Boussinesq equations. We have focused on a widely-used model of geophysical and astrophysical vortices, which assumes an axisymmetric Gaussian structure for pressure anomalies in the horizontal and vertical directions. For a range of Rossby number ($-0.5 < Ro < 0.5$) and Burger number ($0.02 < Bu < 2.3$) relevant to observed long-lived vortices, the growth rate and spatial structure of the most unstable eigenmodes have been numerically calculated and presented as a function of $Ro-Bu$. We have found neutrally-stable vortices only over a small region of the $Ro-Bu$ parameter space: cyclones with $Ro \\sim 0.02-0.05$ and $Bu \\sim 0.85-0.95$. However, we have also found that anticyclones in general have slower growth rates compared to cyclones. In particular, growth rate of the most unstable eigenmode for anticyclones in a large region of the parameter space ...
Brocchini, Maurizio
2013-12-01
This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather than of an applied mathematician. The chronological progress of the currently available BTMs from the pioneering models of the late 1960s is given. The main applications of BTMs are illustrated, with reference to specific models and methods. The evolution in time of the numerical methods used to solve BTMs (e.g. finite differences, finite elements, finite volumes) is described, with specific focus on finite volumes. Finally, an overview of the most important BTMs currently available is presented, as well as some indications on improvements required and fields of applications that call for attention. PMID:24353475
Brocchini, Maurizio
2013-12-01
This paper, which is largely the fruit of an invited talk on the topic at the latest International Conference on Coastal Engineering, describes the state of the art of modelling by means of Boussinesq-type models (BTMs). Motivations for using BTMs as well as their fundamentals are illustrated, with special attention to the interplay between the physics to be described, the chosen model equations and the numerics in use. The perspective of the analysis is that of a physicist/engineer rather than of an applied mathematician. The chronological progress of the currently available BTMs from the pioneering models of the late 1960s is given. The main applications of BTMs are illustrated, with reference to specific models and methods. The evolution in time of the numerical methods used to solve BTMs (e.g. finite differences, finite elements, finite volumes) is described, with specific focus on finite volumes. Finally, an overview of the most important BTMs currently available is presented, as well as some indications on improvements required and fields of applications that call for attention.
On a new non-Boussinesq instability in stratified lakes and oceans
Shete, Mihir H
2016-01-01
Lakes and many other geophysical flows are shallow, density stratified, and contain a free-surface. Conventional studies on stratified shear instabilities make Boussinesq approximation. Free-surface arising due to large density variations between air and water cannot be taken into consideration under this approximation. Hence the free-surface is usually replaced by a rigid-lid, and therefore has little effect on the stability of the fluid below it. In this paper we have performed non-Boussinesq linear stability analyses of a double circulation velocity profile prevalent in two-layered density stratified lakes. One of our analyses is performed by considering the presence of wind, while the other one considers quiescent air. Both analyses have shown similar growth rates and stability boundaries. We have compared our non-Boussinesq study with a corresponding Boussinesq one. The maximum non-Boussinesq growth rate is found to be an order of magnitude greater than the maximum Boussinesq growth rate. Furthermore, th...
Directory of Open Access Journals (Sweden)
Elsayed Mohamed Elsayed ZAYED
2014-07-01
Full Text Available In this article, many new exact solutions of the (2+1-dimensional nonlinear Boussinesq-Kadomtsev-Petviashvili equation and the (1+1-dimensional nonlinear heat conduction equation are constructed using the Riccati equation mapping method. By means of this method, many new exact solutions are successfully obtained. This method can be applied to many other nonlinear evolution equations in mathematical physics.doi:10.14456/WJST.2014.14
Algebraic Approaches to Partial Differential Equations
Xu, Xiaoping
2012-01-01
Partial differential equations are fundamental tools in mathematics,sciences and engineering. This book is mainly an exposition of the various algebraic techniques of solving partial differential equations for exact solutions developed by the author in recent years, with emphasis on physical equations such as: the Calogero-Sutherland model of quantum many-body system in one-dimension, the Maxwell equations, the free Dirac equations, the generalized acoustic system, the Kortweg and de Vries (KdV) equation, the Kadomtsev and Petviashvili (KP) equation, the equation of transonic gas flows, the short-wave equation, the Khokhlov and Zabolotskaya equation in nonlinear acoustics, the equation of geopotential forecast, the nonlinear Schrodinger equation and coupled nonlinear Schrodinger equations in optics, the Davey and Stewartson equations of three-dimensional packets of surface waves, the equation of the dynamic convection in a sea, the Boussinesq equations in geophysics, the incompressible Navier-Stokes equations...
Travelling wave solutions for ( + 1)-dimensional nonlinear evolution equations
Indian Academy of Sciences (India)
Jonu Lee; Rathinasamy Sakthivel
2010-10-01
In this paper, we implement the exp-function method to obtain the exact travelling wave solutions of ( + 1)-dimensional nonlinear evolution equations. Four models, the ( + 1)-dimensional generalized Boussinesq equation, ( + 1)-dimensional sine-cosine-Gordon equation, ( + 1)-double sinh-Gordon equation and ( + 1)-sinh-cosinh-Gordon equation, are used as vehicles to conduct the analysis. New travelling wave solutions are derived.
The Boussinesq-Mindlin problem for a non-homogeneous elastic halfspace
Selvadurai, A. P. S.; Katebi, A.
2016-06-01
Boussinesq's problem for the indentation of an isotropic, homogeneous elastic halfspace by a rigid circular punch constitutes a seminal problem in the theory of contact mechanics as does Mindlin's problem for the action of a concentrated force at the interior of an isotropic, homogeneous elastic halfspace. The combined action of the surface indentation in the presence of the interior loading is referred to as the Boussinesq-Mindlin problem, which has important applications in the area of geomechanics. The Boussinesq-Mindlin problem, which represents a self-stressing loading configuration, serves as a useful model for interpreting the mechanics of indentation of geologic media for purposes of estimating their bulk elasticity properties. In this paper, the analysis of the problem is extended to include an exponential variation in the linear elastic shear modulus of the halfspace region.
High-order Boussinesq-type modelling of nonlinear wave phenomena in deep and shallow water
DEFF Research Database (Denmark)
Madsen, Per A.; Fuhrman, David R.
2010-01-01
In this work, we start with a review of the development of Boussinesq theory for water waves covering the period from 1872 to date. Previous reviews have been given by Dingemans,1 Kirby,2,3 and Madsen & Schäffer.4 Next, we present our most recent high-order Boussinesq-type formulation valid for f...... from an undular sea bed; (8) Run-up of non-breaking solitary waves on a beach; and (9) Tsunami generation from submerged landslides....
Optimized Variational 1D Boussinesq Modelling for broad-band waves over flat bottom
Lakhturov, I.; Adytia, D.; Groesen, van E.
2012-01-01
The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate disper
Optimized variational Boussinesq modelling; part 1: Broad-band waves over flat bottom
Lakhturov, I.; Groesen, van E.
2010-01-01
The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate disper
Haney, Sean
The ocean mixed layer serves as buffer through which the deep ocean and atmosphere communicate. Fluxes of heat, momentum, fresh water, and gases must pass through the mixed layer, and phytoplankton flourish most in the mixed layer where light is abundant. The dynamics of the mixed layer influence these fluxes and productivity of phytoplankton by altering the stratification and mean flow. Restratifying hurricane wakes provide a unique setting in which a dramatically perturbed mixed layer is observable from satellite sea surface temperature. Strong horizontal temperature fronts which border these wakes suggest that two and three dimensional, adiabatic processes play a role. These observations provide the necessary parameters to estimate wake restratification timescales by surface heat fluxes (SF), Ekman buoyancy fluxes (EBF), and mixed layer eddies (MLEs). In the four wakes observed, the timescales for SF and EBF were comparable, while MLEs were much slower. The restratification time for MLEs is reduced for deeper and narrower wakes compared with other mechanisms. Therefore, stronger mixed layer fronts make MLEs competitive with surface heat and wind forcing. Fronts are influenced by winds, waves (Langmuir circulations; LC), MLEs, and symmetric instabilities (SI). The wave averaged (Stokes drift) effects on MLEs are subtle, with aligned (anti-aligned) Stokes and geostrophic flows yielding a slight increase (decrease) in wavenumber and growth rate. Frontal effects on LC are very weak, with the primary result confirming that increased vertical stratification suppresses LC. The effect of Stokes drift on SI is profound. It changes the background flow necessary for SI, and it alters the structure of the SI themselves. Analytic stability criteria show that iii SI exist when the Ertel potential vorticity (PV) is negative. When the Stokes drift is aligned (anti-aligned) with the geostrophic shear, the PV is increased (reduced). This PV criterion is confirmed in more realistic settings with numerical linear stability, and with nonlinear large eddy simulations (LES). Therefore, in the presence of waves, the criterion Ri PV are far more energetic than fronts that exhibit only LC.
Singular solitons and other solutions to a couple of nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
Mustafa Inc; Esma Uluta(s); Anjan Biswas
2013-01-01
This paper addresses the extended (G′/G)-expansion method and applies it to a couple of nonlinear wave equations.These equations are modified the Benjamin-Bona-Mahoney equation and the Boussinesq equation.This extended method reveals several solutions to these equations.Additionally,the singular soliton solutions are revealed,for these two equations,with the aid of the ansatz method.
Modelling of nonlinear shoaling based on stochastic evolution equations
DEFF Research Database (Denmark)
Kofoed-Hansen, Henrik; Rasmussen, Jørgen Hvenekær
1998-01-01
A one-dimensional stochastic model is derived to simulate the transformation of wave spectra in shallow water including generation of bound sub- and super-harmonics, near-resonant triad wave interaction and wave breaking. Boussinesq type equations with improved linear dispersion characteristics a...
Institute of Scientific and Technical Information of China (English)
TANG Jin-yun; TANG Jie; WANG Yuan
2007-01-01
A new analytical model was developed to predict the gravity wave drag (GWD) induced by an isolated 3-dimensional mountain, over which a stratified, nonrotating non-Boussinesq sheared flow is impinged. The model is confined to small amplitude motion and assumes the ambient velocity varying slowly with height. The modified Taylor-Goldstein equation with variable coefficients is solved with a Wentzel-KramersBrillouin (WKB) approximation, formally valid at high Richardson numbers. With this WKB solution, generic formulae of second order accuracy, for the GWD and surface pressure perturbation (both for hydrostatic and non-hydrostatic flow) are presented, enabling a rigorous treatment on the effects by vertical variations in wind profiles. In an ideal test to the circular bell-shaped mountain, it was found that when the wind is linearly sheared,that the GWD decreases as the Richardson number decreases. However, the GWD for a forward sheared wind (wind increases with height) decreases always faster than that for the backward sheared wind (wind deceases with height). This difference is evident whenever the model is hydrostatic or not.
Institute of Scientific and Technical Information of China (English)
HE Hailun; SONG Jinbao; Patrick J. Lynett; LI Shuang
2009-01-01
Fractional energy losses of waves due to wave breaking when passing over a submerged bar are studied systematically using a modified numerical code that is based on the high-order Boussinesq-type equations. The model is first tested by the additional experimental data, and the model's capability of simulating the wave transformation over both gentle slope and steep slope is demonstrated. Then, the model's breaking index is replaced and tested. The new breaking index, which is optimized from the several breaking indices, is not sensitive to the spatial grid length and includes the bottom slopes. Numerical tests show that the modified model with the new breaking index is more stable and efficient for the shallow-water wave breaking. Finally, the modified model is used to study the fractional energy losses for the regular waves propagating and breaking over a submerged bar. Our results have revealed that how the nonlinearity and the dispersion of the incident waves as well as the dimensionless bar height (normalized by water depth) dominate the fractional energy losses. It is also found that the bar slope (limited to gentle slopes that less than 1:10) and the dimensionless bar length (normalized by incident wave length) have negligible effects on the fractional energy losses.
Use of the Boussinesq solution in geotechnical and road engineering: influence of plasticity
Sadek, Marwan; Shahrour, Isam
2007-09-01
The Boussinesq solution for the distribution of stresses in a half-space resulting from surface loads is largely used in geotechnical and road engineering. It is based on the assumption of a linear-elastic homogeneous isotropic half-space for the soil media. Since the soil exhibits nonlinear and irreversible behavior, it is of major interest to study the validity of this solution for elastoplastic soils. This paper includes an investigation of this issue using finite element modeling. The study is conducted by comparing the elastic stress distribution to that obtained using elastoplastic finite element analyses. Results show that the plasticity reduces the attenuation of the vertical stresses in the soil mass, which means that the Boussinesq solution underestimates the stresses in an area which contributes to the soil settlement. To cite this article: M. Sadek, I. Shahrour, C. R. Mecanique 335 (2007).
An Explicit B(a)cklund Transformation of Burgers Equation with Applications
Institute of Scientific and Technical Information of China (English)
L(U) Zhuo-Sheng
2005-01-01
In this paper, an explicit B(a)cklund transformation (BT) of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 (2003) 269; Chaos, Solitons & Fractals 17 (2003) 669]. Based on the BT and some newly obtained seed solutions, infinite sequences of exact solutions for the Burgers equation are generated.Further more, this BT of the Burgers equation is applied to solve the variant Boussinesq equations and the approximate equations of long water wave.
Nonlinear unified equations for water waves propagating over uneven bottoms in the nearshore region
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Considering the continuous characteristics for water waves propagating over complex topography in the nearshore region, the unified nonlinear equations, based on the hypothesis for a typical uneven bottom, are presented by employing the Hamiltonian variational principle for water waves. It is verified that the equations include the following special cases: the extension of Airy's nonlinear shallow-water equations, the generalized mild-slope equation, the dispersion relation for the second-order Stokes waves and the higher order Boussinesq-type equations.
Turbulent Flow over a Flat Plate Using a Three-equation Model
Directory of Open Access Journals (Sweden)
Khalid Alammar
2014-02-01
Full Text Available Aim of this study is to evaluate a three-equation turbulence model based on the Reynolds averaged Navier-Stokes equations. Boussinesq hypothesis is invoked for determining the Reynolds stresses. An average turbulent flat plate flow was simulated. Uncertainty was approximated through validation. Results for the mean axial velocity and friction coefficient were within experimental error.
Non-Boussinesq Integral Model for Horizontal Turbulent Buoyant Round Jets
Directory of Open Access Journals (Sweden)
J. Xiao
2009-01-01
Full Text Available Horizontal buoyant jet is a fundamental flow regime for hydrogen safety analysis in power industry. The purpose of this study is to develop a fast non-Boussinesq engineering model the horizontal buoyant round jets. Verification of this integral model is established with available experimental data and comparisons over a large range of density variations with the CFD codes GASFLOW. The model has proved to be an efficient engineering tool for predicting horizontal strongly buoyant round jets.
On the local well-posedness of a Benjamin-Ono-Boussinesq system
Directory of Open Access Journals (Sweden)
Ruying Xue
2005-01-01
Full Text Available Consider a Benjamin-Ono-Boussinesq system ηt+ux+auxxx+(uηx=0,ut+ηx+uux+cηxxx−duxxt=0, where a, c, and d are constants satisfying a=c>0, d>0 or a0. We prove that this system is locally well posed in Sobolev space Hs(ℝ×Hs+1(ℝ, with s>1/4.
Objective Reduction Solutions to Higher-Order Boussinesq System in (2+1)-Dimensions
Institute of Scientific and Technical Information of China (English)
HU Ya-Hong; ZHENG Chun-Long
2009-01-01
With the help of an objective reduction approach (ORA), abundant exact solutions of (2+1)-dimensional higher-order Boussinesq system (including some hyperboloid function solutions, trigonometric function solutions, and a rational function solution) are obtained. It is shown that some novel soliton structures, like single linearity soliton structure, breath soliton structure, single linearity y-periodic solitary wave structure, libration dromion structure, and kink-like multisoliton structure with actual physical meaning exist in the (2+1)-dimensional higher-order Bonssinesq system.
Turbulent mixing and wave radiation in non-Boussinesq internal bores
DEFF Research Database (Denmark)
Borden, Zac; Koblitz, Tilman; Meiburg, Eckart
2012-01-01
ratio, defined as the ratio of the density of the lighter fluid to the heavier fluid, is greater than approximately one half. For smaller density ratios, undular waves generated at the bore's front dominate over the effects of turbulent mixing, and the expanding layer loses energy across the bore. Based...... on our results, we show that if one can predict the amount of energy radiated by bores through undular waves, it is possible to derive an accurate model for the propagation of non-Boussinesq bores. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4745478]...
Nonlinear wave-structure interactions with a high-order Boussinesq model
DEFF Research Database (Denmark)
Fuhrman, David R.; Bingham, Harry; Madsen, Per A.
2005-01-01
This paper describes the extension of a finite difference model based on a recently derived highly accurate Boussinesq formulation to include domains having arbitrary piecewise-rectangular bottom-mounted (surface-piercing) structures. The resulting linearized system is analyzed for stability on a...... system is receptive to dissipation, and these problems can be overcome in practice using high-order filtering techniques. The resulting model is verified through numerical simulations involving classical linear wave diffraction around a semi-infinite breakwater, linear and nonlinear gap diffraction, and...
A double-layer Boussinesq-type model for highly nonlinear and dispersive waves
Chazel, Florent; Benoit, Michel; Ern, Alexandre; Piperno, Serge
2009-01-01
28 pages, 5 figures. Soumis à Proceedings of the Royal Society of London A. We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. (Agnon et al. 199...
Scenarios of Local Tsunamis in the China Seas by Boussinesq Model
Institute of Scientific and Technical Information of China (English)
赵曦; 刘桦; 王本龙
2014-01-01
The Okinawa Trench in the East China Sea and the Manila Trench in the South China Sea are considered to be the regions with high risk of potential tsunamis induced by submarine earthquakes. Tsunami waves will impact the southeast coast of China if tsunamis occur in these areas. In this paper, the horizontal two-dimensional Boussinesq model is used to simulate tsunami generation, propagation, and runup in a domain with complex geometrical boundaries. The temporary varying bottom boundary condition is adopted to describe the initial tsunami waves motivated by the submarine faults. The Indian Ocean tsunami is simulated by the numerical model as a validation case. The time series of water elevation and runup on the beach are compared with the measured data from field survey. The agreements indicate that the Boussinesq model can be used to simulate tsunamis and predict the waveform and runup. Then, the hypothetical tsunamis in the Okinawa Trench and the Manila Trench are simulated by the numerical model. The arrival time and maximum wave height near coastal cities are predicted by the model. It turns out that the leading depression N-wave occurs when the tsunami propagates in the continental shelf from the Okinawa Trench. The scenarios of the tsunami in the Manila Trench demonstrate significant effects on the coastal area around the South China Sea.
Fan, Jishan; Li, Fucai; Nakamura, Gen
2012-01-01
In this paper we establish some regularity criteria for the 3D Boussinesq system with the temperature-dependent viscosity and thermal diffusivity. We also obtain some uniform estimates for the corresponding 2D case when the fluid viscosity coefficient is a positive constant.
Directory of Open Access Journals (Sweden)
F. Løvholt
2013-06-01
Full Text Available Tsunamis induced by rock slides plunging into fjords constitute a severe threat to local coastal communities. The rock slide impact may give rise to highly non-linear waves in the near field, and because the wave lengths are relatively short, frequency dispersion comes into play. Fjord systems are rugged with steep slopes, and modeling non-linear dispersive waves in this environment with simultaneous run-up is demanding. We have run an operational Boussinesq-type TVD (total variation diminishing model using different run-up formulations. Two different tests are considered, inundation on steep slopes and propagation in a trapezoidal channel. In addition, a set of Lagrangian models serves as reference models. Demanding test cases with solitary waves with amplitudes ranging from 0.1 to 0.5 were applied, and slopes were ranging from 10 to 50°. Different run-up formulations yielded clearly different accuracy and stability, and only some provided similar accuracy as the reference models. The test cases revealed that the model was prone to instabilities for large non-linearity and fine resolution. Some of the instabilities were linked with false breaking during the first positive inundation, which was not observed for the reference models. None of the models were able to handle the bore forming during drawdown, however. The instabilities are linked to short-crested undulations on the grid scale, and appear on fine resolution during inundation. As a consequence, convergence was not always obtained. It is reason to believe that the instability may be a general problem for Boussinesq models in fjords.
Directory of Open Access Journals (Sweden)
Emad A.-B. Abdel-Salam
2013-01-01
Full Text Available 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.
Hilberts, A.; Troch, P.; Boll, J.; Brooks, E.
2004-12-01
Specific yield is a parameter used to quantify the available amount of water for drainage. The instantaneous release of water from the aquifer during drainage however is determined by a related parameter, viz drainable porosity. In this work we present an analytical expression for drainable porosity as a function of water table depth and soil hydraulic parameters. This expression allows us to extend the hillslope-storage Boussinesq equation, recently introduced by the authors, to account for some of the effects of the unsaturated zone on dynamic hydrological behavior during free drainage from complex hillslopes. The concept of a storage-dependent drainable porosity is evaluated by comparing simulation results to field data, which were obtained from a site in Troy, Idaho, where a shallow silt loam soil layer is overlying a fragipan. Evaluation of the model is based on measurements of water table height, soil moisture and outflow. The field site is characterized by the influence of macropores, a decreasing saturated hydraulic conductivity with depth, and a small drainable porosity. Modeling of these data involved making the saturated hydraulic conductivity a function of the depth to the water table, and allowing for variable precipitation inputs.
Pratt, J; Müller, W -C; Chapman, S C; Watkins, N W
2016-01-01
We investigate the utility of the convex hull to analyze physical questions related to the dispersion of a group of much more than four Lagrangian tracer particles in a turbulent flow. Validation of standard dispersion behaviors is a necessary preliminary step for use of the convex hull to describe turbulent flows. In simulations of statistically homogeneous and stationary Navier-Stokes turbulence, neutral fluid Boussinesq convection, and MHD Boussinesq convection we show that the convex hull can be used to reasonably capture the dispersive behavior of a large group of tracer particles. We validate dispersion results produced with convex hull analysis against scalings for Lagrangian particle pair dispersion. In addition to this basic validation study, we show that convex hull analysis provides information that particle pair dispersion does not, in the form of a extreme value statistics, surface area, and volume for a cluster of particles. We use the convex hull surface area and volume to examine the degree of...
A hybrid finite-volume finite-difference rotational Boussinesq-type model of surf-zone hydrodynamics
Tatlock, Benjamin
2015-01-01
An investigation into the numerical and physical behaviour of a hybrid finite-volume finite-difference Boussinesq-type model, using a rotational surface roller approach in the surf-zone is presented. The relevant theory for the required development of a numerical model implementing this technique is outlined. The proposed method looks to achieve a more physically realistic description of the hydrodynamics by considering the rotational nature of the highly turbulent flow found during wave br...
Directory of Open Access Journals (Sweden)
P. Watts
2003-01-01
Full Text Available Case studies of landslide tsunamis require integration of marine geology data and interpretations into numerical simulations of tsunami attack. Many landslide tsunami generation and propagation models have been proposed in recent time, further motivated by the 1998 Papua New Guinea event. However, few of these models have proven capable of integrating the best available marine geology data and interpretations into successful case studies that reproduce all available tsunami observations and records. We show that nonlinear and dispersive tsunami propagation models may be necessary for many landslide tsunami case studies. GEOWAVE is a comprehensive tsunami simulation model formed in part by combining the Tsunami Open and Progressive Initial Conditions System (TOPICS with the fully non-linear Boussinesq water wave model FUNWAVE. TOPICS uses curve fits of numerical results from a fully nonlinear potential flow model to provide approximate landslide tsunami sources for tsunami propagation models, based on marine geology data and interpretations. In this work, we validate GEOWAVE with successful case studies of the 1946 Unimak, Alaska, the 1994 Skagway, Alaska, and the 1998 Papua New Guinea events. GEOWAVE simulates accurate runup and inundation at the same time, with no additional user interference or effort, using a slot technique. Wave breaking, if it occurs during shoaling or runup, is also accounted for with a dissipative breaking model acting on the wave front. The success of our case studies depends on the combination of accurate tsunami sources and an advanced tsunami propagation and inundation model.
NUMERICAL SIMULATION OF SOLITARY WAVE RUN-UP AND OVERTOPPING USING BOUSSINESQ-TYPE MODEL
Institute of Scientific and Technical Information of China (English)
TSUNG Wen-Shuo; HSIAO Shih-Chun; LIN Ting-Chieh
2012-01-01
In this article,the use of a high-order Boussinesq-type model and sets of laboratory experiments in a large scale flume of breaking solitary waves climbing up slopes with two inclinations are presented to study the shoreline behavior of breaking and non-breaking solitary waves on plane slopes.The scale effect on run-up height is briefly discussed.The model simulation capability is well validated against the available laboratory data and present experiments.Then,serial numerical tests are conducted to study the shoreline motion correlated with the effects of beach slope and wave nonlinearity for breaking and non-breaking waves.The empirical formula proposed by Hsiao et al.for predicting the maximum run-up height of a breaking solitary wave on plane slopes with a wide range of slope inclinations is confirmed to be cautious.Furthermore,solitary waves impacting and overtopping an impermeable sloping seawall at various water depths are investigated.Laboratory data of run-up height,shoreline motion,free surface elevation and overtopping discharge are presented.Comparisons of run-up,run-down,shoreline trajectory and wave overtopping discharge are made.A fairly good agreement is seen between numerical results and experimental data.It elucidates that the present depth-integrated model can be used as an efficient tool for predicting a wide spectrum of coastal problems.
Nonlinear evolution equations associated with the chiral-field spectral problem
Energy Technology Data Exchange (ETDEWEB)
Bruschi, M.; Ragnisco, O. (Istituto Nazionale di Fisica Nucleare, Roma (Italy); Dipt. di Fisica, Univ. Rome (Italy))
1985-08-11
In this paper we derive and investigate the class of nonlinear evolution equations (NEEs) associated with the linear problem psisub(x) = lambdaApsi. It turns out that many physically interesting NEEs pertain to this class: for instance, the chiral-field equation, the nonlinear Klein-Gordon equations, the Heisenberg and Papanicolau spin chain models, the modified Boussinesq equation, the Wadati-Konno-Ichikawa equations, etc. We display also the Baecklund transformations for such a class and exploit them to derive in a special case the one-soliton solution.
Exact traveling wave solutions for system of nonlinear evolution equations.
Khan, Kamruzzaman; Akbar, M Ali; Arnous, Ahmed H
2016-01-01
In this work, recently deduced generalized Kudryashov method is applied to the variant Boussinesq equations, and the (2 + 1)-dimensional breaking soliton equations. As a result a range of qualitative explicit exact traveling wave solutions are deduced for these equations, which motivates us to develop, in the near future, a new approach to obtain unsteady solutions of autonomous nonlinear evolution equations those arise in mathematical physics and engineering fields. It is uncomplicated to extend this method to higher-order nonlinear evolution equations in mathematical physics. And it should be possible to apply the same method to nonlinear evolution equations having more general forms of nonlinearities by utilizing the traveling wave hypothesis. PMID:27347461
Singular vectors and conservation laws of quantum KdV type equations
International Nuclear Information System (INIS)
We give a direct proof of the relation between vacuum singular vectors and conservation laws for the quantum KdV equation or equivalently for Φ(1,3)-perturbed conformal field theories. For each degree at which a classical conservation law exists, we find a quantum conserved quantity for a specific value of the central charge. Various generalizations (N=1, 2 supersymmetric, Boussinesq) of this result are presented. (orig.)
Hamiltonian formulation of SL(3) Ur-KdV equation
Chung, B K; Nam, S; Nam, Soonkeon
1993-01-01
We give a unified view of the relation between the $SL(2)$ KdV, the mKdV, and the Ur-KdV equations through the Fr\\'{e}chet derivatives and their inverses. For this we introduce a new procedure of obtaining the Ur-KdV equation, where we require that it has no non-local operators. We extend this method to the $SL(3)$ KdV equation, i.e., Boussinesq(Bsq) equation and obtain the hamiltonian structure of Ur-Bsq equationin a simple form. In particular, we explicitly construct the hamiltonian operator of the Ur-Bsq system which defines the poisson structure of the system, through the Fr\\'{e}chet derivative and its inverse.
Indian Academy of Sciences (India)
Junchao Chen; Biao Li
2012-03-01
In this paper, an extended multiple (′/)-expansion method is proposed to seek exact solutions of nonlinear evolution equations. The validity and advantages of the proposed method is illustrated by its applications to the Sharma–Tasso–Olver equation, the sixth-order Ramani equation, the generalized shallow water wave equation, the Caudrey–Dodd–Gibbon–Sawada–Kotera equation, the sixth-order Boussinesq equation and the Hirota–Satsuma equations. As a result, various complexiton solutions consisting of hyperbolic functions, trigonometric functions, rational functions and their mixture with parameters are obtained. When some parameters are taken as special values, the known double solitary-like wave solutions are derived from the double hyperbolic function solution. In addition, this method can also be used to deal with some high-dimensional and variable coefﬁcients’ nonlinear evolution equations.
Alboussiere, Thierry
2016-01-01
The linear stability threshold of the Rayleigh-Benard configuration is analyzed with compressible effects taken into account. It is assumed that the fluid obeys a Newtonian rheology and Fourier's law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (1916) first obtained analytically the critical value 27pi^4/4 for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This manuscript describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter D and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient a. Different equati...
Kuraz, Michal
2016-06-01
This paper presents pseudo-deterministic catchment runoff model based on the Richards equation model [1] - the governing equation for the subsurface flow. The subsurface flow in a catchment is described here by two-dimensional variably saturated flow (unsaturated and saturated). The governing equation is the Richards equation with a slight modification of the time derivative term as considered e.g. by Neuman [2]. The nonlinear nature of this problem appears in unsaturated zone only, however the delineation of the saturated zone boundary is a nonlinear computationally expensive issue. The simple one-dimensional Boussinesq equation was used here as a rough estimator of the saturated zone boundary. With this estimate the dd-adaptivity algorithm (see Kuraz et al. [4, 5, 6]) could always start with an optimal subdomain split, so it is now possible to avoid solutions of huge systems of linear equations in the initial iteration level of our Richards equation based runoff model.
International Nuclear Information System (INIS)
We investigate properties of convective solutions of the Boussinesq thermal convection in a moderately rotating spherical shell allowing the respective rotation of the inner and outer spheres due to the viscous torque of the fluid. The ratio of the inner and outer radii of the spheres, the Prandtl number, and the Taylor number are fixed to 0.4, 1, and 5002, respectively. The Rayleigh number is varied from 2.6 × 104 to 3.4 × 104. In this parameter range, the behaviours of obtained asymptotic convective solutions are almost similar to those in the system whose inner and outer spheres are restricted to rotate with the same constant angular velocity, although the difference is found in the transition process to chaotic solutions. The convective solution changes from an equatorially symmetric quasi-periodic one to an equatorially symmetric chaotic one, and further to an equatorially asymmetric chaotic one, as the Rayleigh number is increased. This is in contrast to the transition in the system whose inner and outer spheres are assumed to rotate with the same constant angular velocity, where the convective solution changes from an equatorially symmetric quasi-periodic one, to an equatorially asymmetric quasi-periodic one, and to equatorially asymmetric chaotic one. The inner sphere rotates in the retrograde direction on average in the parameter range; however, it sometimes undergoes the prograde rotation when the convective solution becomes chaotic
Institute of Scientific and Technical Information of China (English)
无
2004-01-01
At the beginning of 16th century, mathematicians found it easy to solve equations of the first degree(linear equations, involving x) and of the second degree(quadratic equatiorts, involving x2). Equations of the third degree(cubic equations, involving x3)defeated them.
Song, Y. Tony; Colberg, Frank
2011-01-01
Observational surveys have shown significant oceanic bottom water warming, but they are too spatially and temporally sporadic to quantify the deep ocean contribution to the present-day sea level rise (SLR). In this study, altimetry sea surface height (SSH), Gravity Recovery and Climate Experiment (GRACE) ocean mass, and in situ upper ocean (0-700 m) steric height have been assessed for their seasonal variability and trend maps. It is shown that neither the global mean nor the regional trends of altimetry SLR can be explained by the upper ocean steric height plus the GRACE ocean mass. A non-Boussinesq ocean general circulation model (OGCM), allowing the sea level to rise as a direct response to the heat added into the ocean, is then used to diagnose the deep ocean steric height. Constrained by sea surface temperature data and the top of atmosphere (TOA) radiation measurements, the model reproduces the observed upper ocean heat content well. Combining the modeled deep ocean steric height with observational upper ocean data gives the full depth steric height. Adding a GRACE-estimated mass trend, the data-model combination explains not only the altimetry global mean SLR but also its regional trends fairly well. The deep ocean warming is mostly prevalent in the Atlantic and Indian oceans, and along the Antarctic Circumpolar Current, suggesting a strong relation to the oceanic circulation and dynamics. Its comparison with available bottom water measurements shows reasonably good agreement, indicating that deep ocean warming below 700 m might have contributed 1.1 mm/yr to the global mean SLR or one-third of the altimeter-observed rate of 3.11 +/- 0.6 mm/yr over 1993-2008.
Yoshida, M; Yoshida, Masaki; Kageyama, Akira
2004-01-01
A new numerical finite difference code has been developed to solve a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. A kind of the overset (Chimera) grid named ``Yin-Yang grid'' is used for the spatial discretization. The grid naturally avoids the pole problems which are inevitable in the latitude-longitude grids. The code is applied to numerical simulations of mantle convection with uniform and variable viscosity. The validity of the Yin-Yang grid for the mantle convection simulation is confirmed.
Convective Wave Breaking in the KdV Equation
Brun, Mats K
2016-01-01
The KdV equation is a model equation for waves at the surface of an inviscid incompressible fluid, and it is well known that the equation describes the evolution of unidirectional waves of small amplitude and long wavelength fairly accurately if the waves fall into the Boussinesq regime. The KdV equation allows a balance of nonlinear steepening effects and dispersive spreading which leads to the formation of steady wave profiles in the form of solitary waves and cnoidal waves. While these wave profiles are solutions of the KdV equation for any amplitude, it is shown here that there for both the solitary and the cnoidal waves, there are critical amplitudes for which the horizontal component of the particle velocity matches the phase velocity of the wave. Solitary or cnoidal solutions of the KdV equation which surpass these amplitudes feature incipient wave breaking as the particle velocity exceeds the phase velocity near the crest of the wave, and the model breaks down due to violation of the kinematic surface...
Moiseiwitsch, B L
2005-01-01
Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco
Directory of Open Access Journals (Sweden)
Lloyd K. Williams
1987-01-01
Full Text Available In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.
A new type numerical model foraction balance equation in simulating nearshore waves
Institute of Scientific and Technical Information of China (English)
无
2001-01-01
Several current used wave numerical models are briefly described, the computing techniques of the source terms, numerical wave generation and boundary conditions in the action balance equation model are discussed. Not only the quadruplet wave-wave interactions, but also the triad wave-wave interactions are included in the model, so that nearshore waves could be simulated reasonably. The model is compared with the Boussinesq equation and the mild slope equation. The model is applied to calculating the distribu-tions of wave height and wave period field in the Haian Bay area and to simulating the influences of the unsteady current and water level variation on the wave field. Finally, the de-veloping tendency of the model is discussed.
Self-Consistent Sources for Integrable Equations Via Deformations of Binary Darboux Transformations
Chvartatskyi, Oleksandr; Dimakis, Aristophanes; Müller-Hoissen, Folkert
2016-08-01
We reveal the origin and structure of self-consistent source extensions of integrable equations from the perspective of binary Darboux transformations. They arise via a deformation of the potential that is central in this method. As examples, we obtain in particular matrix versions of self-consistent source extensions of the KdV, Boussinesq, sine-Gordon, nonlinear Schrödinger, KP, Davey-Stewartson, two-dimensional Toda lattice and discrete KP equation. We also recover a (2+1)-dimensional version of the Yajima-Oikawa system from a deformation of the pKP hierarchy. By construction, these systems are accompanied by a hetero binary Darboux transformation, which generates solutions of such a system from a solution of the source-free system and additionally solutions of an associated linear system and its adjoint. The essence of all this is encoded in universal equations in the framework of bidifferential calculus.
Bernoulli equation and flow over a mountain
Sun, Wen-Yih; Sun, Oliver M.
2015-12-01
The Bernoulli equation is applied to an air parcel which originates at a low level at the inflow region, climbs adiabatically over a mountain with an increase in velocity, then descends on the lee side and forms a strong downslope wind. The parcel departs from hydrostatic equilibrium during its vertical motion. The air parcel can be noticeably cooler than the temperature calculated from adiabatic lapse rate, which allows part of enthalpy to be converted to kinetic energy and produces a stronger wind at mountain peak and a severe downslope wind on the lee side. It was found that the hydrostatic assumption tends to suppress the conversion from enthalpy to kinetic energy. It is also shown that the Froude number defined in the atmosphere is equal to the ratio of kinetic energy to the potential energy, same as in Boussinesq fluid. But in the atmosphere, the Froude number cannot be used to determine whether a parcel can move over a mountain or not, unless the vertical motion is weak and the system is near hydrostatic equilibrium. Numerical simulations confirm that except in highly turbulent areas, the potential temperature and Bernoulli function are almost conserved along the streamline, as well as the change of kinetic energy comes from the change of enthalpy instead of potential energy.
Institute of Scientific and Technical Information of China (English)
YUAN Ye-li; QIAO Fang-li; YIN Xun-qiang; HAN Lei; LU Ming
2012-01-01
Based on their differences in physical characteristics and time-space scales,the ocean motions have been divided into four types in the present paper:turbulence,wave-like motion,eddy-like motion and circulation.Applying the three-fold Reynolds averages to the governing equations with Boussinesq approximation,with the averages defined on the former three sub-systems,we derive the governing equation sets of the four sub-systems and refer to their sum as “the ocean dynamic system”.In these equation sets,the interactions among different motions appear in two forms:the first one includes advection transport and shear instability generation of larger scale motions,and the second one is the mixing induced by smaller scale motions in the form of transport flux residue.The governing equation sets are the basis of analytical/numerical descriptions of various ocean processes.
Directory of Open Access Journals (Sweden)
Jing Yin
2015-07-01
Full Text Available A total variation diminishing-weighted average flux (TVD-WAF-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL. The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
Directory of Open Access Journals (Sweden)
A. A. Hemeda
2013-01-01
Full Text Available An extension of the so-called new iterative method (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iteration method (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.
Cao, Chongsheng
2010-01-01
The three--dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom, boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.
Non-Boussinesq turbulent buoyant jet of a low-density gas leaks into high-density ambient
El-Amin, Mohamed
2010-12-01
In this article, we study the problem of low-density gas jet injected into high-density ambient numerically which is important in applications such as fuel injection and leaks. It is assumed that the local rate of entrainment is consisted of two components; one is the component of entrainment due to jet momentum while the other is the component of entrainment due to buoyancy. The integral models of the mass, momentum and concentration fluxes are obtained and transformed to a set of ordinary differential equations using some similarity transformations. The resulting system is solved to determine the centerline quantities which are used to get the mean axial velocity, mean concentration and mean density of the jet. Therefore, the centerline and mean quantities are used together with the governing equation to determine some important turbulent quantities such as, cross-stream velocity, Reynolds stress, velocity- concentration correlation, turbulent eddy viscosity and turbulent eddy diffusivity. Throughout this paper the developed model is verified by comparing the present results with experimental results and jet/plume theory from the literature. © 2010 Elsevier Inc. All rights reserved.
Institute of Scientific and Technical Information of China (English)
BELCAID Aicha; LE PALEC Georges; DRAOUI Abdeslam
2015-01-01
This paper investigates a numerical and experimental study about buoyant wall turbulent jet in a static homogeneous environment. A light fluid of fresh water is injected horizontally and tangentially to a plane wall into homogenous salt water ambient. This later is given with different values of salinity and the initial fractional density is small, so the applicability of the Boussinesq approximation is valid. Since the domain temperature is assumed to be constant, the density of the mixture is a function of the salt concentration only. Mathematical model is based on the finite volume method and reports on an application of standardk-ε turbulence model for steady flow with densimetric Froude numbers of 1-75 and Reynolds numbers of 2 000-6 000. The basic features of the model are the conservation of mass, momentum and concentration. The boundaries of jet body, the radius and cling length are determined. It is found that the jet spreading and behavior depend on the ratio between initial buoyancy flux and momentum, i.e., initial Froude number, and on the influence of wall boundary which corresponds to Coanda effect. Laboratory experiments were conducted with photographic observations of jet trajectories and numerical results are described and compared with the experiments. A good agreement with numerical and experimental results has been achieved.
EXISTENCE TIME OF SOLUTION OF THE (1+2)D KNOBLOCH EQUATION WITH INITIAL-BOUNDARY VALUE PROBLEM
Institute of Scientific and Technical Information of China (English)
2000-01-01
The equation of pattern formation induced by buoyancy or by surface-tension gradient in finite systems confined between horizontal poor heat conductors is introduced by Knobloch[1990] where u is the planform function,μ is the scaled Rayleigh number,K=1 and α represents the effects of a heat transfer finite Biot number.The cofficients β,δ and γ do not vanish when the boundary conditions at top and bottom are not identical (β≠0,δ≠0) or non Boussinesq effects are taken into account (γ ≠ 0).In this paper,the Knobloch equation with α ＞ 0 is considered,the globai existence in L2-space and the finite existence time of solution in V2-space have been obtained respectively.
One-Dimensional Optimal System and Similarity Reductions of Wu—Zhang Equation
Xiong, Na; Li, Yu-Qi; Chen, Jun-Chao; Chen, Yong
2016-07-01
The one-dimensional optimal system for the Lie symmetry group of the (2+1)-dimensional Wu—Zhang equation is constructed by the general and systematic approach. Based on the optimal system, the complete and inequivalent symmetry reduction systems are presented in the form of table. It is noteworthy that a new Painlevé integrable equation with constant coefficient is in the table besides the classic Boussinesq equation and the steady case of the Wu-Zhang equation. Supported by the Global Change Research Program of China under Grant No. 2015CB953904, National Natural Science Foundation of China under Grant Nos. 11375090, 11275072 and 11435005, Research Fund for the Doctoral Program of Higher Education of China under Grant No. 20120076110024, the Network Information Physics Calculation of Basic Research Innovation Research Group of China under Grant No. 61321064, Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things under Grant No. ZF1213, and the Zhejiang Provincial Natural Science Foundation of China under Grant No. LY14A010005
Difference equations by differential equation methods
Hydon, Peter E
2014-01-01
Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.
Random diophantine equations, I
Brüdern, Jörg; Dietmann, Rainer
2012-01-01
We consider additive diophantine equations of degree $k$ in $s$ variables and establish that whenever $s\\ge 3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.
Energy Technology Data Exchange (ETDEWEB)
Boivin, S. [Quebec Univ., Chicoutimi, PQ (Canada); Cayre, F. [Laval Univ., Groupe Interdisciplinaire de Recherche en Elements Finis, Quebec City, PQ (Canada); Dept. Laboratoire National d' Hydraulique, Groupe Recherches et Etudes en Thermohydraulique, 78 - Chatou (France); Herard, J.M. [Dept. Laboratoire National d' Hydraulique, Groupe Recherches et Etudes en Thermohydraulique, 78 - Chatou (France)
2000-09-01
A method to solve the Navier-Stokes equations for incompressible viscous flows and the convection and diffusion of a scalar is proposed in the present paper. This method is based upon a fractional time step scheme and the finite volume method on unstructured meshes. A recently proposed diffusion scheme with interesting theoretical and numerical properties is tested and integrated into the Navier-Stokes solver. Predictions of Poiseuille flows, backward-facing step flows and lid-driven cavity flows are then performed to validate the method. We finally demonstrate the versatility of the method by predicting buoyancy force driven flows of a Boussinesq fluid (natural convection of air in a square cavity with Rayleigh numbers of 10{sup 3} and 10{sup 6}). (authors)
Kinetic energy equations for the average-passage equation system
Johnson, Richard W.; Adamczyk, John J.
1989-01-01
Important kinetic energy equations derived from the average-passage equation sets are documented, with a view to their interrelationships. These kinetic equations may be used for closing the average-passage equations. The turbulent kinetic energy transport equation used is formed by subtracting the mean kinetic energy equation from the averaged total instantaneous kinetic energy equation. The aperiodic kinetic energy equation, averaged steady kinetic energy equation, averaged unsteady kinetic energy equation, and periodic kinetic energy equation, are also treated.
Directory of Open Access Journals (Sweden)
Don Liu
2013-01-01
Full Text Available A system of coupled nonlinear partial differential equations with convective and dispersive terms was modified from Boussinesq-type equations. Through a special formulation, a system of nonlinear partial differential equations was solved alternately and explicitly in time without linearizing the nonlinearity. Coupled compact schemes of sixth order accuracy in space were developed to obtain numerical solutions. Within couple compact schemes, variables and their first and second derivatives were solved altogether. The sixth order accuracy in space is achieved with a memory-saving arrangement of state variables so that the linear system is banded instead of blocked. This facilitates solving very large systems. The efficiency, simplicity, and accuracy make this coupled compact method viable as variational and weighted residual methods. Results were compared with exact solutions which were obtained via devised forcing terms. Error analyses were carried out, and the sixth order convergence in space and second order convergence in time were demonstrated. Long time integration was also studied to show stability and error convergence rates.
The Modified Magnetohydrodynamical Equations
Institute of Scientific and Technical Information of China (English)
EvangelosChaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similar fashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is done by replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vector potential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vector analysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHD equations.
Indian Academy of Sciences (India)
George F R Ellis
2007-07-01
The Raychaudhuri equation is central to the understanding of gravitational attraction in astrophysics and cosmology, and in particular underlies the famous singularity theorems of general relativity theory. This paper reviews the derivation of the equation, and its significance in cosmology.
Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Beginning partial differential equations
O'Neil, Peter V
2014-01-01
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible,combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is or
Reducible functional differential equations
Directory of Open Access Journals (Sweden)
S. M. Shah
1985-01-01
Full Text Available This is the first part of a survey on analytic solutions of functional differential equations (FDE. Some classes of FDE that can be reduced to ordinary differential equations are considered since they often provide an insight into the structure of analytic solutions to equations with more general argument deviations. Reducible FDE also find important applications in the study of stability of differential-difference equations and arise in a number of biological models.
New unified evolution equation
Lim, Jyh-Liong; Li, Hsiang-nan
1998-01-01
We propose a new unified evolution equation for parton distribution functions appropriate for both large and small Bjorken variables $x$, which is an improved version of the Ciafaloni-Catani-Fiorani-Marchesini equation. In this new equation the cancellation of soft divergences between virtual and real gluon emissions is explicit without introducing infrared cutoffs, next-to-leading contributions to the Sudakov resummation can be included systematically. It is shown that the new equation reduc...
Diophantine equations and identities
Directory of Open Access Journals (Sweden)
Malvina Baica
1985-01-01
Full Text Available The general diophantine equations of the second and third degree are far from being totally solved. The equations considered in this paper are i x2−my2=±1 ii x3+my3+m2z3−3mxyz=1iii Some fifth degree diopantine equations
Institute of Scientific and Technical Information of China (English)
谢桂英; 吴勇旗
2009-01-01
利用Hirota方法及Riemann theta函数得到了广义(n+1)维Boussinesq方程的新的周期解,在极限情况下,该周期解退化为孤子解.另外,利用计算机技术和Mathematica绘制了解的三维曲面图.
The Modified Magnetohydrodynamical Equations
Institute of Scientific and Technical Information of China (English)
Evangelos Chaliasos
2003-01-01
After finding the really self-consistent electromagnetic equations for a plasma, we proceed in a similarfashion to find how the magnetohydrodynamical equations have to be modified accordingly. Substantially this is doneby replacing the "Lorentz" force equation by the correct (in our case) force equation. Formally we have to use the vectorpotential instead of the magnetic field intensity. The appearance of the formulae presented is the one of classical vectoranalysis. We thus find a set of eight equations in eight unknowns, as previously known concerning the traditional MHDequations.
Fractional Differential Equations
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Jianping Zhao
2012-01-01
Full Text Available An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powerful mathematical tool for solving fractional differential equations.
Singular stochastic differential equations
Cherny, Alexander S
2005-01-01
The authors introduce, in this research monograph on stochastic differential equations, a class of points termed isolated singular points. Stochastic differential equations possessing such points (called singular stochastic differential equations here) arise often in theory and in applications. However, known conditions for the existence and uniqueness of a solution typically fail for such equations. The book concentrates on the study of the existence, the uniqueness, and, what is most important, on the qualitative behaviour of solutions of singular stochastic differential equations. This is done by providing a qualitative classification of isolated singular points, into 48 possible types.
Lanczos's equation to replace Dirac's equation ?
Gsponer, A; Gsponer, Andre; Hurni, Jean-Pierre
1994-01-01
Lanczos's quaternionic interpretation of Dirac's equation provides a unified description for all elementary particles of spin 0, 1/2, 1, and 3/2. The Lagrangian formulation given by Einstein and Mayer in 1933 predicts two main classes of solutions. (1) Point like partons which come in two families, quarks and leptons. The correct fractional or integral electric and baryonic charges, and zero mass for the neutrino and the u-quark, are set by eigenvalue equations. The electro-weak interaction of the partons is the same as with the Standard model, with the same two free parameters: e and sin^2 theta. There is no need for a Higgs symmetry breaking mechanism. (2) Extended hadrons for which there is no simple eigenvalue equation for the mass. The strong interaction is essentially non-local. The pion mass and pion-nucleon coupling constant determine to first order the nucleon size, mass and anomalous magnetic moment.
International Nuclear Information System (INIS)
We classify (1+3)-dimensional Pauli equations for a spin-(1/2) particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the 11 classes of vector-potentials of the electro-magnetic field A(t,x(vector sign))=(A0(t,x(vector sign)), A(vector sign)(t,x(vector sign))) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is its equivalence to the system of two uncoupled Schroedinger equations. In addition, the magnetic field has to be independent of spatial variables. We prove that coordinate systems and the vector-potentials of the electro-magnetic field providing the separability of the corresponding Pauli equations coincide with those for the Schroedinger equations. Furthermore, an efficient algorithm for constructing all coordinate systems providing the separability of Pauli equation with a fixed vector-potential of the electro-magnetic field is developed. Finally, we describe all vector-potentials A(t,x(vector sign)) that (a) provide the separability of Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c) describe non-zero magnetic field
Nonlinear Rossby waves near the equator with complete Coriolis force%近赤道完整Coriolis力作用下的非线性Rossby波
Institute of Scientific and Technical Information of China (English)
杨红丽; 刘福梅; 王丹妮; 杨联贵
2016-01-01
Nonlinear Rossby Waves near the equator in a potential vorticity equation which includes both the vertical and horizontal components of Coriolis force are studied.The wave evolution is described by the inhomo-geneous Boussinesq equation or the modified Korteweg-de Vries equation depending on the different perturbation methods.From the evolution equations,the effect of the horizontal components of Coriolis force on the nonlinear Rossby waves is evident.As expected,the equations derived also include,as special cases,those obtained before.%从既含有Coriolis力垂直分量又含有水平分量的位涡方程出发,采用不同的摄动方法推导了近赤道非线性Rossby波的演化方程,得到非线性Rossby波振幅演化满足非齐次Boussinesq方程或改进的Korteweg-de Vries方程.从演化方程可以看出Coriolis力水平分量对非线性Rossby波的影响,并且本文取特殊情况时包括了已有的一些结果.
Differential equations for dummies
Holzner, Steven
2008-01-01
The fun and easy way to understand and solve complex equations Many of the fundamental laws of physics, chemistry, biology, and economics can be formulated as differential equations. This plain-English guide explores the many applications of this mathematical tool and shows how differential equations can help us understand the world around us. Differential Equations For Dummies is the perfect companion for a college differential equations course and is an ideal supplemental resource for other calculus classes as well as science and engineering courses. It offers step-by-step techniques, practical tips, numerous exercises, and clear, concise examples to help readers improve their differential equation-solving skills and boost their test scores.
Elliptic partial differential equations
Volpert, Vitaly
If we had to formulate in one sentence what this book is about it might be "How partial differential equations can help to understand heat explosion, tumor growth or evolution of biological species". These and many other applications are described by reaction-diffusion equations. The theory of reaction-diffusion equations appeared in the first half of the last century. In the present time, it is widely used in population dynamics, chemical physics, biomedical modelling. The purpose of this book is to present the mathematical theory of reaction-diffusion equations in the context of their numerous applications. We will go from the general mathematical theory to specific equations and then to their applications. Mathematical anaylsis of reaction-diffusion equations will be based on the theory of Fredholm operators presented in the first volume. Existence, stability and bifurcations of solutions will be studied for bounded domains and in the case of travelling waves. The classical theory of reaction-diffusion equ...
Directory of Open Access Journals (Sweden)
Wei Khim Ng
2009-02-01
Full Text Available We construct nonlinear extensions of Dirac's relativistic electron equation that preserve its other desirable properties such as locality, separability, conservation of probability and Poincaré invariance. We determine the constraints that the nonlinear term must obey and classify the resultant non-polynomial nonlinearities in a double expansion in the degree of nonlinearity and number of derivatives. We give explicit examples of such nonlinear equations, studying their discrete symmetries and other properties. Motivated by some previously suggested applications we then consider nonlinear terms that simultaneously violate Lorentz covariance and again study various explicit examples. We contrast our equations and construction procedure with others in the literature and also show that our equations are not gauge equivalent to the linear Dirac equation. Finally we outline various physical applications for these equations.
Fractional Chemotaxis Diffusion Equations
Langlands, T A M
2010-01-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.
Solving Ordinary Differential Equations
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Fundamental Equation of Economics
Wayne, James J.
2013-01-01
Recent experience of the great recession of 2008 has renewed one of the oldest debates in economics: whether economics could ever become a scientific discipline like physics. This paper proves that economics is truly a branch of physics by establishing for the first time a fundamental equation of economics (FEOE), which is similar to many fundamental equations governing other subfields of physics, for example, Maxwell’s Equations for electromagnetism. From recently established physics laws of...
Differential equations I essentials
REA, Editors of
2012-01-01
REA's Essentials provide quick and easy access to critical information in a variety of different fields, ranging from the most basic to the most advanced. As its name implies, these concise, comprehensive study guides summarize the essentials of the field covered. Essentials are helpful when preparing for exams, doing homework and will remain a lasting reference source for students, teachers, and professionals. Differential Equations I covers first- and second-order equations, series solutions, higher-order linear equations, and the Laplace transform.
Zhalij, Alexander
2002-01-01
We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle interacting with the electro-magnetic field, that are solvable by the method of separation of variables. As a result, we obtain the eleven classes of vector-potentials of the electro-magnetic field A(t,x) providing separability of the corresponding Pauli equations. It is established, in particular, that the necessary condition for the Pauli equation to be separable into second-order matrix ordinary differential equations is...
International Nuclear Information System (INIS)
A new evolution equation is proposed for the gluon density relevant (GLR) for the region of small xB. It generalizes the GLR equation and allows deeper penetration in dense parton systems than the GLR equation does. This generalization consists of taking shadowing effects more comprehensively into account by including multi gluon correlations, and allowing for an arbitrary initial gluon distribution in a hadron. We solve the new equation for fixed αs. It is found that the effects of multi gluon correlations on the deep-inelastic structure function are small. (author) 15 refs, 5 figs, 2 tabs
Wetterich, C
2016-01-01
We propose a gauge invariant flow equation for Yang-Mills theories and quantum gravity that only involves one macroscopic gauge field or metric. It is based on a projection on physical and gauge fluctuations, corresponding to a particular gauge fixing. The freedom in the precise choice of the macroscopic field can be exploited in order to keep the flow equation simple.
Zahari, N. M.; Sapar, S. H.; Mohd Atan, K. A.
2013-04-01
This paper discusses an integral solution (a, b, c) of the Diophantine equations x3n+y3n = 2z2n for n ≥ 2 and it is found that the integral solution of these equation are of the form a = b = t2, c = t3 for any integers t.
Some classical Diophantine equations
Directory of Open Access Journals (Sweden)
Nikita Bokarev
2014-09-01
Full Text Available An attempt to find common solutions complete some Diophantine equations of the second degree with three variables, traced some patterns, suggest a common approach, which being elementary, however, lead to a solution of such equations. Using arithmetic functions allowed to write down the solutions in a single formula with no restrictions on the parameters used.
Ramirez, Erandy; Liddle, Andrew
2004-01-01
We generalize the flow equations approach to inflationary model building to the Randall–Sundrum Type II braneworld scenario. As the flow equations are quite insensitive to the expansion dynamics, we find results similar to, though not identical to, those found in the standard cosmology.
Hazewinkel, M.
1995-01-01
Dedication: I dedicate this paper to Prof. P.C. Baayen, at the occasion of his retirement on 20 December 1994. The beautiful equation which forms the subject matter of this paper was invented by Wouthuysen after he retired. The four complex variable Wouthuysen equation arises from an original space-
Linear Equations: Equivalence = Success
Baratta, Wendy
2011-01-01
The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…
Directory of Open Access Journals (Sweden)
Hannelore Breckner
2000-01-01
Full Text Available We consider a stochastic equation of Navier-Stokes type containing a noise part given by a stochastic integral with respect to a Wiener process. The purpose of this paper is to approximate the solution of this nonlinear equation by the Galerkin method. We prove the convergence in mean square.
The relativistic Pauli equation
Delphenich, David
2012-01-01
After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charged spinning particle in an external electromagnetic field then implies a second order equation in the matrix-valued wave functions that is of Klein-Gordon type and represents the relativistic analogue of the Pauli equation. We conclude by presenting the Lagrangian form for the relativistic Pauli equation.
Applied partial differential equations
Logan, J David
2004-01-01
This primer on elementary partial differential equations presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. What makes this book unique is that it is a brief treatment, yet it covers all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. Mathematical ideas are motivated from physical problems, and the exposition is presented in a concise style accessible to science and engineering students; emphasis is on motivation, concepts, methods, and interpretation, rather than formal theory. This second edition contains new and additional exercises, and it includes a new chapter on the applications of PDEs to biology: age structured models, pattern formation; epidemic wave fronts, and advection-diffusion processes. The student who reads through this book and solves many of t...
Differential equations problem solver
Arterburn, David R
2012-01-01
REA's Problem Solvers is a series of useful, practical, and informative study guides. Each title in the series is complete step-by-step solution guide. The Differential Equations Problem Solver enables students to solve difficult problems by showing them step-by-step solutions to Differential Equations problems. The Problem Solvers cover material ranging from the elementary to the advanced and make excellent review books and textbook companions. They're perfect for undergraduate and graduate studies.The Differential Equations Problem Solver is the perfect resource for any class, any exam, and
Ordinary differential equations
Miller, Richard K
1982-01-01
Ordinary Differential Equations is an outgrowth of courses taught for a number of years at Iowa State University in the mathematics and the electrical engineering departments. It is intended as a text for a first graduate course in differential equations for students in mathematics, engineering, and the sciences. Although differential equations is an old, traditional, and well-established subject, the diverse backgrounds and interests of the students in a typical modern-day course cause problems in the selection and method of presentation of material. In order to compensate for this diversity,
Pierret, Frédéric
2016-02-01
We derived the equations of Celestial Mechanics governing the variation of the orbital elements under a stochastic perturbation, thereby generalizing the classical Gauss equations. Explicit formulas are given for the semimajor axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle, and the mean anomaly, which are expressed in term of the angular momentum vector H per unit of mass and the energy E per unit of mass. Together, these formulas are called the stochastic Gauss equations, and they are illustrated numerically on an example from satellite dynamics.
Beginning partial differential equations
O'Neil, Peter V
2011-01-01
A rigorous, yet accessible, introduction to partial differential equations-updated in a valuable new edition Beginning Partial Differential Equations, Second Edition provides a comprehensive introduction to partial differential equations (PDEs) with a special focus on the significance of characteristics, solutions by Fourier series, integrals and transforms, properties and physical interpretations of solutions, and a transition to the modern function space approach to PDEs. With its breadth of coverage, this new edition continues to present a broad introduction to the field, while also addres
Hyperbolic partial differential equations
Witten, Matthew
1986-01-01
Hyperbolic Partial Differential Equations III is a refereed journal issue that explores the applications, theory, and/or applied methods related to hyperbolic partial differential equations, or problems arising out of hyperbolic partial differential equations, in any area of research. This journal issue is interested in all types of articles in terms of review, mini-monograph, standard study, or short communication. Some studies presented in this journal include discretization of ideal fluid dynamics in the Eulerian representation; a Riemann problem in gas dynamics with bifurcation; periodic M
Uncertain differential equations
Yao, Kai
2016-01-01
This book introduces readers to the basic concepts of and latest findings in the area of differential equations with uncertain factors. It covers the analytic method and numerical method for solving uncertain differential equations, as well as their applications in the field of finance. Furthermore, the book provides a number of new potential research directions for uncertain differential equation. It will be of interest to researchers, engineers and students in the fields of mathematics, information science, operations research, industrial engineering, computer science, artificial intelligence, automation, economics, and management science.
Modern introduction to differential equations
Ricardo, Henry J
2009-01-01
A Modern Introduction to Differential Equations, Second Edition, provides an introduction to the basic concepts of differential equations. The book begins by introducing the basic concepts of differential equations, focusing on the analytical, graphical, and numerical aspects of first-order equations, including slope fields and phase lines. The discussions then cover methods of solving second-order homogeneous and nonhomogeneous linear equations with constant coefficients; systems of linear differential equations; the Laplace transform and its applications to the solution of differential equat
A Comparison of IRT Equating and Beta 4 Equating.
Kim, Dong-In; Brennan, Robert; Kolen, Michael
Four equating methods were compared using four equating criteria: first-order equity (FOE), second-order equity (SOE), conditional mean squared error (CMSE) difference, and the equipercentile equating property. The four methods were: (1) three parameter logistic (3PL) model true score equating; (2) 3PL observed score equating; (3) beta 4 true…
Kinetic equations: computation
Pareschi, Lorenzo
2013-01-01
Kinetic equations bridge the gap between a microscopic description and a macroscopic description of the physical reality. Due to the high dimensionality the construction of numerical methods represents a challenge and requires a careful balance between accuracy and computational complexity.
Applied partial differential equations
Logan, J David
2015-01-01
This text presents the standard material usually covered in a one-semester, undergraduate course on boundary value problems and PDEs. Emphasis is placed on motivation, concepts, methods, and interpretation, rather than on formal theory. The concise treatment of the subject is maintained in this third edition covering all the major ideas: the wave equation, the diffusion equation, the Laplace equation, and the advection equation on bounded and unbounded domains. Methods include eigenfunction expansions, integral transforms, and characteristics. In this third edition, text remains intimately tied to applications in heat transfer, wave motion, biological systems, and a variety other topics in pure and applied science. The text offers flexibility to instructors who, for example, may wish to insert topics from biology or numerical methods at any time in the course. The exposition is presented in a friendly, easy-to-read, style, with mathematical ideas motivated from physical problems. Many exercises and worked e...
Diophantine Equations and Computation
Davis, Martin
Unless otherwise stated, we’ll work with the natural numbers: N = \\{0,1,2,3, dots\\}. Consider a Diophantine equation F(a1,a2,...,an,x1,x2,...,xm) = 0 with parameters a1,a2,...,an and unknowns x1,x2,...,xm For such a given equation, it is usual to ask: For which values of the parameters does the equation have a solution in the unknowns? In other words, find the set: \\{ mid exists x_1,ldots,x_m [F(a_1,ldots,x_1,ldots)=0] \\} Inverting this, we think of the equation F = 0 furnishing a definition of this set, and we distinguish three classes: a set is called Diophantine if it has such a definition in which F is a polynomial with integer coefficients. We write \\cal D for the class of Diophantine sets.
Nonlinear differential equations
Energy Technology Data Exchange (ETDEWEB)
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Frédéric, Pierret
2014-01-01
The equations of celestial mechanics that govern the variation of the orbital elements are completely derived for stochastic perturbation which generalized the classic perturbation equations which are used since Gauss, starting from Newton's equation and it's solution. The six most understandable orbital element, the semi-major axis, the eccentricity, the inclination, the longitude of the ascending node, the pericenter angle and the mean motion are express in term of the angular momentum vector $\\textbf{H}$ per unit of mass and the energy $E$ per unit of mass. We differentiate those expressions using It\\^o's theory of differential equations due to the stochastic nature of the perturbing force. The result is applied to the two-body problem perturbed by a stochastic dust cloud and also perturbed by a stochastic dynamical oblateness of the central body.
Nonlinear differential equations
International Nuclear Information System (INIS)
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics
Garkavenko A. S.
2011-01-01
The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.
Institute of Scientific and Technical Information of China (English)
A.I.Arbab
2013-01-01
A unified complex model of Maxwell's equations is presented.The wave nature of the electromagnetic field vector is related to the temporal and spatial distributions and the circulation of charge and current densities.A new vacuum solution is obtained,and a new transformation under which Maxwell's equations are invariant is proposed.This transformation extends ordinary gauge transformation to include charge-current as well as scalar-vector potential.An electric dipole moment is found to be related to the magnetic charges,and Dirac's quantization is found to determine an uncertainty relation expressing the indeterminacy of electric and magnetic charges.We generalize Maxwell's equations to include longitudinal waves.A formal analogy between this formulation and Dirac's equation is also discussed.
Tsintsadze, Nodar L.; Tsintsadze, Levan N.
2008-01-01
A general derivation of the charging equation of a dust grain is presented, and indicated where and when it can be used. A problem of linear fluctuations of charges on the surface of the dust grain is discussed.
Relativistic Guiding Center Equations
Energy Technology Data Exchange (ETDEWEB)
White, R. B. [PPPL; Gobbin, M. [Euratom-ENEA Association
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
Hedin Equations for Superconductors
Linscheid, A.; Essenberger, F.
2015-01-01
We generalize Hedin equations to a system of superconducting electrons coupled with a system of phonons. The electrons are described by an electronic Pauli Hamiltonian which includes the Coulomb interaction among electrons and an external vector and scalar potential. We derive the continuity equation in the presence of the superconducting condensate and point out how to cast vertex corrections in the form of a non-local effective interaction that can be used to describe both fluctuations of s...
Functional Equations and Fourier Analysis
Yang, Dilian
2010-01-01
By exploring the relations among functional equations, harmonic analysis and representation theory, we give a unified and very accessible approach to solve three important functional equations -- the d'Alembert equation, the Wilson equation, and the d'Alembert long equation, on compact groups.
Tailleux, Remi
2009-01-01
There exist two central measures of turbulent mixing in turbulent stratified fluids, both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy (APE); 2) the turbulent rate of change Wr,turbulent of background potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to D(APE)=Wr,turbulent holding exactly for a Boussinesq fluid with a linear equation of state. It was recently pointed out, however, that this equality no longer holds for a thermally-stratified compressible fluid, the ratio \\xi=Wr,turbulent/D(APE) being then lower than unity and sometimes even negative for water/seawater. In this paper, the behavior of the ratio \\xi is examined for different stratifications having the same buoyancy frequency N(z), but different vertical profiles of the parameter \\Upsilon = \\alpha P/(\\rho C_p), where \\alpha is the thermal expansion, P the hydrostatic pressure, \\rho the...
Integral equations and computation problems
International Nuclear Information System (INIS)
Volterra's Integral Equations and Fredholm's Integral Equations of the second kind are discussed. Computational problems are found in the derivations and the computations. The theorem of the solution of the Fredholm's Integral Equation is discussed in detail. (author)
Scaling Equation for Invariant Measure
Institute of Scientific and Technical Information of China (English)
LIU Shi-Kuo; FU Zun-Tao; LIU Shi-Da; REN Kui
2003-01-01
An iterated function system (IFS) is constructed. It is shown that the invariant measure of IFS satisfies the same equation as scaling equation for wavelet transform (WT). Obviously, IFS and scaling equation of WT both have contraction mapping principle.
Introduction to partial differential equations
Greenspan, Donald
2000-01-01
Designed for use in a one-semester course by seniors and beginning graduate students, this rigorous presentation explores practical methods of solving differential equations, plus the unifying theory underlying the mathematical superstructure. Topics include basic concepts, Fourier series, second-order partial differential equations, wave equation, potential equation, heat equation, approximate solution of partial differential equations, and more. Exercises appear at the ends of most chapters. 1961 edition.
Unified derivation of evolution equations
Li, Hsiang-nan
1998-01-01
We derive the evolution equations of parton distribution functions appropriate in different kinematic regions in a unified and simple way using the resummation technique. They include the Dokshitzer-Gribov-Lipatov-Altarelli-Parisi equation for large momentum transfer $Q$, the Balitskii-Fadin-Kuraev-Lipatov equation for a small Bjorken variable $x$, and the Ciafaloni-Catani-Fiorani-Marchesini equation which embodies the above two equations. The relation among these equations is explored, and p...
Quadratic Diophantine equations
Andreescu, Titu
2015-01-01
This monograph treats the classical theory of quadratic Diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory. The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis.
Stochastic porous media equations
Barbu, Viorel; Röckner, Michael
2016-01-01
Focusing on stochastic porous media equations, this book places an emphasis on existence theorems, asymptotic behavior and ergodic properties of the associated transition semigroup. Stochastic perturbations of the porous media equation have reviously been considered by physicists, but rigorous mathematical existence results have only recently been found. The porous media equation models a number of different physical phenomena, including the flow of an ideal gas and the diffusion of a compressible fluid through porous media, and also thermal propagation in plasma and plasma radiation. Another important application is to a model of the standard self-organized criticality process, called the "sand-pile model" or the "Bak-Tang-Wiesenfeld model". The book will be of interest to PhD students and researchers in mathematics, physics and biology.
Generalization of Hopf Functional Equation
Institute of Scientific and Technical Information of China (English)
无
2002-01-01
This paper generalizes the Hopf functional equation in order to apply it to a wider class of not necessarily incompressible fluid flows. We start by defining characteristic functionals of the velocity field, the density field and the temperature field of a compressible field. Using the continuity equation, the Navier-Stokes equations and the equation of energy we derive a functional equation governing the motion of an ideal gas flow and a van der Waals gas flow, and then give some general methods of deriving a functional equation governing the motion of any compressible fluid flow. These functional equations can be considered as the generalization of the Hopf functional equation.
Amorim, R G G; Silva, Edilberto O
2015-01-01
Symplectic unitary representations for the Poincar\\'{e} group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Klein-Gordon and Dirac equations are derived in phase space. As an application, we study the Dirac equation with electromagnetic interaction in phase space.
Gas Dynamics Equations: Computation
Chen, Gui-Qiang G
2012-01-01
Shock waves, vorticity waves, and entropy waves are fundamental discontinuity waves in nature and arise in supersonic or transonic gas flow, or from a very sudden release (explosion) of chemical, nuclear, electrical, radiation, or mechanical energy in a limited space. Tracking these discontinuities and their interactions, especially when and where new waves arise and interact in the motion of gases, is one of the main motivations for numerical computation for the gas dynamics equations. In this paper, we discuss some historic and recent developments, as well as mathematical challenges, in designing and formulating efficient numerical methods and algorithms to compute weak entropy solutions for the Euler equations for gas dynamics.
Generalized estimating equations
Hardin, James W
2002-01-01
Although powerful and flexible, the method of generalized linear models (GLM) is limited in its ability to accurately deal with longitudinal and clustered data. Developed specifically to accommodate these data types, the method of Generalized Estimating Equations (GEE) extends the GLM algorithm to accommodate the correlated data encountered in health research, social science, biology, and other related fields.Generalized Estimating Equations provides the first complete treatment of GEE methodology in all of its variations. After introducing the subject and reviewing GLM, the authors examine th
The relativistic Pauli equation
Delphenich, David
2012-01-01
After discussing the way that C2 and the algebra of complex 2x2 matrices can be used for the representation of both non-relativistic rotations and Lorentz transformations, we show that Dirac bispinors can be more advantageously represented as 2x2 complex matrices. One can then give the Dirac equation a form for such matrix-valued wave functions that no longer necessitates the introduction of gamma matrices or a choice for their representation. The minimally-coupled Dirac equation for a charge...
Systematic Equation Formulation
DEFF Research Database (Denmark)
Lindberg, Erik
2007-01-01
A tutorial giving a very simple introduction to the set-up of the equations used as a model for an electrical/electronic circuit. The aim is to find a method which is as simple and general as possible with respect to implementation in a computer program. The “Modified Nodal Approach”, MNA, and th......, and the “Controlled Source Approach”, CSA, for systematic equation formulation are investigated. It is suggested that the kernel of the P Spice program based on MNA is reprogrammed....
Institute of Scientific and Technical Information of China (English)
Ding Yi
2009-01-01
In this article, the author derives a functional equation η(s)=［(π/4)s-1/2√2/πг(1-s)sin(πs/2)]η(1-s) of the analytic function η(s) which is defined by η(s)=1-s-3-s-5-s+7-s…for complex variable s with Re s>1, and is defined by analytic continuation for other values of s. The author proves (1) by Ramanujan identity (see [1], [3]). Her method provides a new derivation of the functional equation of Riemann zeta function by using Poisson summation formula.
Cira, Octavian; Smarandache, Florentin
2016-01-01
In this book a multitude of Diophantine equations and their partial or complete solutions are presented. How should we solve, for example, the equation {\\eta}({\\pi}(x)) = {\\pi}({\\eta}(x)), where {\\eta} is the Smarandache function and {\\pi} is Riemann function of counting the number of primes up to x, in the set of natural numbers? If an analytical method is not available, an idea would be to recall the empirical search for solutions. We establish a domain of searching for the solutions and th...
Theory of differential equations
Gel'fand, I M
1967-01-01
Generalized Functions, Volume 3: Theory of Differential Equations focuses on the application of generalized functions to problems of the theory of partial differential equations.This book discusses the problems of determining uniqueness and correctness classes for solutions of the Cauchy problem for systems with constant coefficients and eigenfunction expansions for self-adjoint differential operators. The topics covered include the bounded operators in spaces of type W, Cauchy problem in a topological vector space, and theorem of the Phragmén-Lindelöf type. The correctness classes for the Cau
Markley, F. Landis
1995-01-01
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Equations of mathematical physics
Tikhonov, A N
2011-01-01
Mathematical physics plays an important role in the study of many physical processes - hydrodynamics, elasticity, and electrodynamics, to name just a few. Because of the enormous range and variety of problems dealt with by mathematical physics, this thorough advanced-undergraduate or graduate-level text considers only those problems leading to partial differential equations. The authors - two well-known Russian mathematicians - have focused on typical physical processes and the principal types of equations deailing with them. Special attention is paid throughout to mathematical formulation, ri
On difference Riccati equations and second order linear difference equations
Ishizaki, Katsuya
2011-01-01
In this paper, we treat difference Riccati equations and second order linear difference equations in the complex plane. We give surveys of basic properties of these equations which are analogues in the differential case. We are concerned with the growth and value distributions of transcendental meromorphic solutions of these equations. Some examples are given.
Test equating methods and practices
Kolen, Michael J
1995-01-01
In recent years, many researchers in the psychology and statistical communities have paid increasing attention to test equating as issues of using multiple test forms have arisen and in response to criticisms of traditional testing techniques This book provides a practically oriented introduction to test equating which both discusses the most frequently used equating methodologies and covers many of the practical issues involved The main themes are - the purpose of equating - distinguishing between equating and related methodologies - the importance of test equating to test development and quality control - the differences between equating properties, equating designs, and equating methods - equating error, and the underlying statistical assumptions for equating The authors are acknowledged experts in the field, and the book is based on numerous courses and seminars they have presented As a result, educators, psychometricians, professionals in measurement, statisticians, and students coming to the subject for...
Energy Technology Data Exchange (ETDEWEB)
Richard C. Martineau; Ray A. Berry; Aurélia Esteve; Kurt D. Hamman; Dana A. Knoll; Ryosuke Park; William Taitano
2009-01-01
This report illustrates a comparative study to analyze the physical differences between numerical simulations obtained with both the conservation and incompressible forms of the Navier-Stokes equations for natural convection flows in simple geometries. The purpose of this study is to quantify how the incompressible flow assumption (which is based upon constant density advection, divergence-free flow, and the Boussinesq gravitational body force approximation) differs from the conservation form (which only assumes that the fluid is a continuum) when solving flows driven by gravity acting upon density variations resulting from local temperature gradients. Driving this study is the common use of the incompressible flow assumption in fluid flow simulations for nuclear power applications in natural convection flows subjected to a high heat flux (large temperature differences). A series of simulations were conducted on two-dimensional, differentially-heated rectangular geometries and modeled with both hydrodynamic formulations. From these simulations, the selected characterization parameters of maximum Nusselt number, average Nusselt number, and normalized pressure reduction were calculated. Comparisons of these parameters were made with available benchmark solutions for air with the ideal gas assumption at both low and high heat fluxes. Additionally, we generated body force, velocity, and divergence of velocity distributions to provide a basis for further analysis. The simulations and analysis were then extended to include helium at the Very High Temperature gas-cooled Reactor (VHTR) normal operating conditions. Our results show that the consequences of incorporating the incompressible flow assumption in high heat flux situations may lead to unrepresentative results. The results question the use of the incompressible flow assumption for simulating fluid flow in an operating nuclear reactor, where large temperature variations are present. The results show that the use of
Variation principle of piezothermoelastic bodies, canonical equation and homogeneous equation
Institute of Scientific and Technical Information of China (English)
LIU Yan-hong; ZHANG Hui-ming
2007-01-01
Combining the symplectic variations theory, the homogeneous control equation and isoparametric element homogeneous formulations for piezothermoelastic hybrid laminates problems were deduced. Firstly, based on the generalized Hamilton variation principle, the non-homogeneous Hamilton canonical equation for piezothermoelastic bodies was derived. Then the symplectic relationship of variations in the thermal equilibrium formulations and gradient equations was considered, and the non-homogeneous canonical equation was transformed to homogeneous control equation for solving independently the coupling problem of piezothermoelastic bodies by the incensement of dimensions of the canonical equation. For the convenience of deriving Hamilton isoparametric element formulations with four nodes, one can consider the temperature gradient equation as constitutive relation and reconstruct new variation principle. The homogeneous equation simplifies greatly the solution programs which are often performed to solve nonhomogeneous equation and second order differential equation on the thermal equilibrium and gradient relationship.
Directory of Open Access Journals (Sweden)
Hatem Mejjaoli
2008-12-01
Full Text Available We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Modelling by Differential Equations
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
Do Differential Equations Swing?
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Structural Equation Model Trees
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
Kasari, Hikoya; Yamaguchi, Yoshio
2001-01-01
Contrary to the conventional belief, it was shown that the Breit equation has the eigenvalues for bound states of two oppositely charged Dirac particles interacting through the (static) Coulomb potential. All eigenvalues reduced to those of the Sch\\"odinger case in the non-relativistic limit.
Chi, Do Minh
1999-01-01
We research the natural causality of the Universe. We find that the equation of causality provides very good results on physics. That is our first endeavour and success in describing a quantitative expression of the law of causality. Hence, our theoretical point suggests ideas to build other laws including the law of the Universe's evolution.
Directory of Open Access Journals (Sweden)
Garkavenko A. S.
2011-08-01
Full Text Available The rate equations of the exciton laser in the system of interacting excitons have been obtained and the inverted population conditions and generation have been derived. The possibility of creating radically new gamma-ray laser has been shown.
Nonlocal electrical diffusion equation
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0solar panels, electrochemical phenomena and the description of anomalous complex processes.
Calculus & ordinary differential equations
Pearson, David
1995-01-01
Professor Pearson's book starts with an introduction to the area and an explanation of the most commonly used functions. It then moves on through differentiation, special functions, derivatives, integrals and onto full differential equations. As with other books in the series the emphasis is on using worked examples and tutorial-based problem solving to gain the confidence of students.
Equational binary decision diagrams
Groote, J.F.; Pol, J.C. van de
2000-01-01
We incorporate equations in binary decision diagrams (BDD). The resulting objects are called EQ-BDDs. A straightforward notion of ordered EQ-BDDs (EQ-OBDD) is defined, and it is proved that each EQ-BDD is logically equivalent to an EQ-OBDD. Moreover, on EQ-OBDDs satisfiability and tautology checkin
Lie Symmetries of Ishimori Equation
Institute of Scientific and Technical Information of China (English)
SONG Xu-Xia
2013-01-01
The Ishimori equation is one of the most important (2+1)-dimensional integrable models,which is an integrable generalization of (1+1)-dimensional classical continuous Heisenberg ferromagnetic spin equations.Based on importance of Lie symmetries in analysis of differential equations,in this paper,we derive Lie symmetries for the Ishimori equation by Hirota's direct method.
Anticipated backward stochastic differential equations
Peng, Shige; Yang, Zhe
2009-01-01
In this paper we discuss new types of differential equations which we call anticipated backward stochastic differential equations (anticipated BSDEs). In these equations the generator includes not only the values of solutions of the present but also the future. We show that these anticipated BSDEs have unique solutions, a comparison theorem for their solutions, and a duality between them and stochastic differential delay equations.
Elements of partial differential equations
Sneddon, Ian N
2006-01-01
Geared toward students of applied rather than pure mathematics, this volume introduces elements of partial differential equations. Its focus is primarily upon finding solutions to particular equations rather than general theory.Topics include ordinary differential equations in more than two variables, partial differential equations of the first and second orders, Laplace's equation, the wave equation, and the diffusion equation. A helpful Appendix offers information on systems of surfaces, and solutions to the odd-numbered problems appear at the end of the book. Readers pursuing independent st
Differential Equations with Linear Algebra
Boelkins, Matthew R; Potter, Merle C
2009-01-01
Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, t
Stochastic differential equations and applications
Friedman, Avner
2006-01-01
This text develops the theory of systems of stochastic differential equations, and it presents applications in probability, partial differential equations, and stochastic control problems. Originally published in two volumes, it combines a book of basic theory and selected topics with a book of applications.The first part explores Markov processes and Brownian motion; the stochastic integral and stochastic differential equations; elliptic and parabolic partial differential equations and their relations to stochastic differential equations; the Cameron-Martin-Girsanov theorem; and asymptotic es
Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows
Nieves, David J.
This thesis is a collection of studies concerning an asymptotically reduced equation set derived from the Boussinesq approximation describing rotationally constrained geophysical flow. The first investigation is concerned with a statistical identification of coherent and long-lived structures in rotationally constrained Rayleigh-Benard convection. Presently, physical laboratory limitations challenge experimentalists while spatio-temporal resolution requirements challenges numericists performing direct numerical simulations of the Boussinesq equations. These challenges prevent an exhaustive analysis of the flow morphology in the rapid rotating limit. In this study the flow morphologies obtained from simulations of the reduced equations are investigated from a statistical perspective. Auto- and cross-correlations are computed from temporal and spatial signals that synthesize experimental data that may be obtained in laboratory experiments via thermistor measurements or particle image velocimetry. The statistics used can be employed in laboratory experiments to identify regime transitions in flow morphology, capture radial profiles of coherent structures, and extract transport properties belonging to these structures. These results provide a foundation for comparison and a measure for understanding the extent to which rotationally constrained regime has been accessed by laboratory experiments and direct numerical simulations. A related study comparing the influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is also investigated and briefly summarized for the case of stress-free mechanical boundary conditions. It is shown that the difference between these thermal boundary conditions on the interior geostrophically balanced convection is asymptotically weak. Through a simple rescaling of thermal variables, the leading order reduced system is shown to be equivalent for both thermal
Classical Diophantine equations
1993-01-01
The author had initiated a revision and translation of "Classical Diophantine Equations" prior to his death. Given the rapid advances in transcendence theory and diophantine approximation over recent years, one might fear that the present work, originally published in Russian in 1982, is mostly superseded. That is not so. A certain amount of updating had been prepared by the author himself before his untimely death. Some further revision was prepared by close colleagues. The first seven chapters provide a detailed, virtually exhaustive, discussion of the theory of lower bounds for linear forms in the logarithms of algebraic numbers and its applications to obtaining upper bounds for solutions to the eponymous classical diophantine equations. The detail may seem stark--- the author fears that the reader may react much as does the tourist on first seeing the centre Pompidou; notwithstanding that, Sprind zuk maintainsa pleasant and chatty approach, full of wise and interesting remarks. His emphases well warrant, ...
Differential equations with Mathematica
Abell, Martha L
2004-01-01
The Third Edition of the Differential Equations with Mathematica integrates new applications from a variety of fields,especially biology, physics, and engineering. The new handbook is also completely compatible with recent versions of Mathematica and is a perfect introduction for Mathematica beginners.* Focuses on the most often used features of Mathematica for the beginning Mathematica user* New applications from a variety of fields, including engineering, biology, and physics* All applications were completed using recent versions of Mathematica
Directory of Open Access Journals (Sweden)
D. Diederen
2015-06-01
Full Text Available We present a new equation describing the hydrodynamics in infinitely long tidal channels (i.e., no reflection under the influence of oceanic forcing. The proposed equation is a simple relationship between partial derivatives of water level and velocity. It is formally derived for a progressive wave in a frictionless, prismatic, tidal channel with a horizontal bed. Assessment of a large number of numerical simulations, where an open boundary condition is posed at a certain distance landward, suggests that it can also be considered accurate in the more natural case of converging estuaries with nonlinear friction and a bed slope. The equation follows from the open boundary condition and is therefore a part of the problem formulation for an infinite tidal channel. This finding provides a practical tool for evaluating tidal wave dynamics, by reconstructing the temporal variation of the velocity based on local observations of the water level, providing a fully local open boundary condition and allowing for local friction calibration.
Directory of Open Access Journals (Sweden)
M. Paul Gough
2008-07-01
Full Text Available LandauerÃ¢Â€Â™s principle is applied to information in the universe. Once stars began forming there was a constant information energy density as the increasing proportion of matter at high stellar temperatures exactly compensated for the expanding universe. The information equation of state was close to the dark energy value, w = -1, for a wide range of redshifts, 10 > z > 0.8, over one half of cosmic time. A reasonable universe information bit content of only 1087 bits is sufficient for information energy to account for all dark energy. A time varying equation of state with a direct link between dark energy and matter, and linked to star formation in particular, is clearly relevant to the cosmic coincidence problem. In answering the Ã¢Â€Â˜Why now?Ã¢Â€Â™ question we wonder Ã¢Â€Â˜What next?Ã¢Â€Â™ as we expect the information equation of state to tend towards w = 0 in the future.c
New application to Riccati equation
Taogetusang; Sirendaoerji; Li, Shu-Min
2010-08-01
To seek new infinite sequence of exact solutions to nonlinear evolution equations, this paper gives the formula of nonlinear superposition of the solutions and Bäcklund transformation of Riccati equation. Based on the tanh-function expansion method and homogenous balance method, new infinite sequence of exact solutions to Zakharov-Kuznetsov equation, Karamoto-Sivashinsky equation and the set of (2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov equations are obtained with the aid of symbolic computation system Mathematica. The method is of significance to construct infinite sequence exact solutions to other nonlinear evolution equations.
Telegrapher's equation for light derived from the transport equation
Hoenders, Bernhard J.; Graaff, R.
2005-01-01
Shortcomings of diffusion theory when applied to turbid media such as biological tissue makes the development of more accurate equations desirable. Several authors developed telegrapher's equations in the well known P-1 approximation. The method used in this paper is different: it is based on the asymptotic evaluation of the solutions of the equation of radiative transport with respect to place and time for all values of the albedo. Various coefficients for the telegrapher's equations were de...
Converting fractional differential equations into partial differential equations
He Ji-Huan; Li Zheng-Biao
2012-01-01
A transform is suggested in this paper to convert fractional differential equations with the modified Riemann-Liouville derivative into partial differential equations, and it is concluded that the fractional order in fractional differential equations is equivalent to the fractal dimension.
The compressible adjoint equations in geodynamics: equations and numerical assessment
Ghelichkhan, Siavash; Bunge, Hans-Peter
2016-04-01
The adjoint method is a powerful means to obtain gradient information in a mantle convection model relative to past flow structure. While the adjoint equations in geodynamics have been derived for the conservation equations of mantle flow in their incompressible form, the applicability of this approximation to Earth is limited, because density increases by almost a factor of two from the surface to the Core Mantle Boundary. Here we introduce the compressible adjoint equations for the conservation equations in the anelastic-liquid approximation. Our derivation applies an operator formulation in Hilbert spaces, to connect to recent work in seismology (Fichtner et al (2006)) and geodynamics (Horbach et al (2014)), where the approach was used to derive the adjoint equations for the wave equation and incompressible mantle flow. We present numerical tests of the newly derived equations based on twin experiments, focusing on three simulations. A first, termed Compressible, assumes the compressible forward and adjoint equations, and represents the consistent means of including compressibility effects. A second, termed Mixed, applies the compressible forward equation, but ignores compressibility effects in the adjoint equations, where the incompressible equations are used instead. A third simulation, termed Incompressible, neglects compressibility effects entirely in the forward and adjoint equations relative to the reference twin. The compressible and mixed formulations successfully restore earlier mantle flow structure, while the incompressible formulation yields noticeable artifacts. Our results suggest the use of a compressible formulation, when applying the adjoint method to seismically derived mantle heterogeneity structure.
Ordinary differential equations
Cox, William
1995-01-01
Building on introductory calculus courses, this text provides a sound foundation in the underlying principles of ordinary differential equations. Important concepts, including uniqueness and existence theorems, are worked through in detail and the student is encouraged to develop much of the routine material themselves, thus helping to ensure a solid understanding of the fundamentals required.The wide use of exercises, problems and self-assessment questions helps to promote a deeper understanding of the material and it is developed in such a way that it lays the groundwork for further
Differential Equations as Actions
DEFF Research Database (Denmark)
Ronkko, Mauno; Ravn, Anders P.
1997-01-01
We extend a conventional action system with a primitive action consisting of a differential equation and an evolution invariant. The semantics is given by a predicate transformer. The weakest liberal precondition is chosen, because it is not always desirable that steps corresponding to differential...... actions shall terminate. It is shown that the proposed differential action has a semantics which corresponds to a discrete approximation when the discrete step size goes to zero. The extension gives action systems the power to model real-time clocks and continuous evolutions within hybrid systems....
Dimensional Equations of Entropy
Sparavigna, Amelia Carolina
2015-01-01
Entropy is a quantity which is of great importance in physics and chemistry. The concept comes out of thermodynamics, proposed by Rudolf Clausius in his analysis of Carnot cycle and linked by Ludwig Boltzmann to the number of specific ways in which a physical system may be arranged. Any physics classroom, in its task of learning physics, has therefore to face this crucial concept. As we will show in this paper, the lectures can be enriched by discussing dimensional equations linked to the entropy of some physical systems.
Partial differential equations
Sloan, D; Süli, E
2001-01-01
/homepage/sac/cam/na2000/index.html7-Volume Set now available at special set price ! Over the second half of the 20th century the subject area loosely referred to as numerical analysis of partial differential equations (PDEs) has undergone unprecedented development. At its practical end, the vigorous growth and steady diversification of the field were stimulated by the demand for accurate and reliable tools for computational modelling in physical sciences and engineering, and by the rapid development of computer hardware and architecture. At the more theoretical end, the analytical insight in
Generalized estimating equations
Hardin, James W
2013-01-01
Generalized Estimating Equations, Second Edition updates the best-selling previous edition, which has been the standard text on the subject since it was published a decade ago. Combining theory and application, the text provides readers with a comprehensive discussion of GEE and related models. Numerous examples are employed throughout the text, along with the software code used to create, run, and evaluate the models being examined. Stata is used as the primary software for running and displaying modeling output; associated R code is also given to allow R users to replicat
Lopez, Cesar
2014-01-01
MATLAB is a high-level language and environment for numerical computation, visualization, and programming. Using MATLAB, you can analyze data, develop algorithms, and create models and applications. The language, tools, and built-in math functions enable you to explore multiple approaches and reach a solution faster than with spreadsheets or traditional programming languages, such as C/C++ or Java. MATLAB Differential Equations introduces you to the MATLAB language with practical hands-on instructions and results, allowing you to quickly achieve your goals. In addition to giving an introduct
Conservational PDF Equations of Turbulence
Shih, Tsan-Hsing; Liu, Nan-Suey
2010-01-01
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
On Certain Dual Integral Equations
Directory of Open Access Journals (Sweden)
R. S. Pathak
1974-01-01
Full Text Available Dual integral equations involving H-Functions have been solved by using the theory of Mellin transforms. The proof is analogous to that of Busbridge on solutions of dual integral equations involving Bessel functions.
Program Transformation by Solving Equations
Institute of Scientific and Technical Information of China (English)
朱鸿
1991-01-01
Based on the theory of orthogonal program expansion[8-10],the paper proposes a method to transform programs by solving program equations.By the method,transformation goals are expressed in program equations,and achieved by solving these equations.Although such equations are usually too complicated to be solved directly,the orthogonal expansion of programs makes it possible to reduce such equations into systems of equations only containing simple constructors of programs.Then,the solutions of such equations can be derived by a system of solving and simplifying rules,and algebraic laws of programs.The paper discusses the methods to simplify and solve equations and gives some examples.
Kepler's Differential Equations
Holder, Martin
2011-01-01
Although the differential calculus was invented by Newton, Kepler established his famous laws 70 years earlier by using the same idea, namely to find a path in a nonuniform field of force by small steps. It is generally not known that Kepler demonstrated the elliptic orbit to be composed of intelligeable differential pieces, in modern language, to result from a differential equation. Kepler was first to attribute planetary orbits to a force from the sun, rather than giving them a predetermined geometric shape. Even though neither the force was known nor its relation to motion, he could determine the differential equations of motion from observation. This is one of the most important achievements in the history of physics. In contrast to Newton's Principia and Galilei's Dialogo Kepler's text is not easy to read, for various reasons. Therefore, in the present article, his results -- most of them well known -- are first presented in modern language. Then, in order to justify the claim, the full text of some rele...
International Nuclear Information System (INIS)
This monograph treats most of the usual material to be found in texts on the Dirac equation such as the basic formalism of quantum mechanics, representations of Dirac matrices, covariant realization of the Dirac equation, interpretation of negative energies, Foldy-Wouthuysen transformation, Klein's paradox, spherically symmetric interactions and a treatment of the relativistic hydrogen atom, etc., and also provides excellent additional treatments of a variety of other relevant topics. The monograph contains an extensive treatment of the Lorentz and Poincare groups and their representations. The author discusses in depth Lie algebaic and projective representations, covering groups, and Mackey's theory and Wigner's realization of induced representations. A careful classification of external fields with respect to their behavior under Poincare transformations is supplemented by a basic account of self-adjointness and spectral properties of Dirac operators. A state-of-the-art treatment of relativistic scattering theory based on a time-dependent approach originally due to Enss is presented. An excellent introduction to quantum electrodynamics in external fields is provided. Various appendices containing further details, notes on each chapter commenting on the history involved and referring to original research papers and further developments in the literature, and a bibliography covering all relevant monographs and over 500 articles on the subject, complete this text. This book should satisfy the needs of a wide audience, ranging from graduate students in theoretical physics and mathematics to researchers interested in mathematical physics
Growth Equation with Conservation Law
Lauritsen, Kent Baekgaard
1995-01-01
A growth equation with a generalized conservation law characterized by an integral kernel is introduced. The equation contains the Kardar-Parisi-Zhang, Sun-Guo-Grant, and Molecular-Beam Epitaxy growth equations as special cases and allows for a unified investigation of growth equations. From a dynamic renormalization-group analysis critical exponents and universality classes are determined for growth models with a conservation law.
``Riemann equations'' in bidifferential calculus
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
Hyperbolic Methods for Einstein's Equations
Reula Oscar
1998-01-01
I review evolutionary aspects of general relativity, in particular those related to the hyperbolic character of the field equations and to the applications or consequences that this property entails. I look at several approaches to obtaining symmetric hyperbolic systems of equations out of Einstein's equations by either removing some gauge freedoms from them, or by considering certain linear combinations of a subset of them.
Successfully Transitioning to Linear Equations
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
An Extented Wave Action Equation
Institute of Scientific and Technical Information of China (English)
左其华
2003-01-01
Based on the Navier-Stokes equation, an average wave energy equation and a generalized wave action conservation equation are presented in this paper. The turbulence effects on water particle velocity ui and wave surface elavation ξ as well as energy dissipation are included. Some simplified forms are also given.
Solving Nonlinear Coupled Differential Equations
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
The Schroedinger equation and spin
International Nuclear Information System (INIS)
Galilei invariance of the Schroedinger equation requires linearization of the operator by the introduction of anticommuting matrices as coefficients of the linear form. In an external field this leads directly to the Pauli equation, the non-relativistic limit of Dirac's equation. An overview of the complete argument that defines spin as a non-relativistic concept is presented. 9 refs
Solution of Finite Element Equations
DEFF Research Database (Denmark)
Krenk, Steen
An important step in solving any problem by the finite element method is the solution of the global equations. Numerical solution of linear equations is a subject covered in most courses in numerical analysis. However, the equations encountered in most finite element applications have some special...
Chaliasos, Evangelos
2006-01-01
As we know, from the Einstein equations the vanishing of the four-divergence of the energy-momentum tensor follows. This is the case because the four-divergence of the Einstein tensor vanishes identically. Inversely, we find that from the vanishing of the four-divergence of the energy-momentum tensor not only the Einstein equations follow. Besides, the so-named anti-Einstein equations follow. These equations must be considered as complementary to the Einstein equations. And while from the Ein...
A generalized advection dispersion equation
Indian Academy of Sciences (India)
Abdon Atangana
2014-02-01
This paper examines a possible effect of uncertainties, variability or heterogeneity of any dynamic system when being included in its evolution rule; the notion is illustrated with the advection dispersion equation, which describes the groundwater pollution model. An uncertain derivative is defined; some properties of the operator are presented. The operator is used to generalize the advection dispersion equation. The generalized equation differs from the standard equation in four properties. The generalized equation is solved via the variational iteration technique. Some illustrative figures are presented.
Equation with the many fathers
DEFF Research Database (Denmark)
Kragh, Helge
1984-01-01
of the usual Schrödinger equation. In most of the early versions the Klein-Gordon equation was connected with the general theory of relativity. Klein and some other physicists attempted to express quantum mechanics within a five-dimensional unified theory, embracing general relativity as well......In this essay I discuss the origin and early development of the first relativistic wave equation, known as the Klein-Gordon equation. In 1926 several physicists, among them Klein, Fock, Schrödinger, and de Broglie, announced this equation as a candidate for a relativistic generalization...
Discovering evolution equations with applications
McKibben, Mark
2011-01-01
Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochast
Generalized Klein-Kramers equations
Fa, Kwok Sau
2012-12-01
A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000), 10.1021/jp993491m]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.
Scaling of differential equations
Langtangen, Hans Petter
2016-01-01
The book serves both as a reference for various scaled models with corresponding dimensionless numbers, and as a resource for learning the art of scaling. A special feature of the book is the emphasis on how to create software for scaled models, based on existing software for unscaled models. Scaling (or non-dimensionalization) is a mathematical technique that greatly simplifies the setting of input parameters in numerical simulations. Moreover, scaling enhances the understanding of how different physical processes interact in a differential equation model. Compared to the existing literature, where the topic of scaling is frequently encountered, but very often in only a brief and shallow setting, the present book gives much more thorough explanations of how to reason about finding the right scales. This process is highly problem dependent, and therefore the book features a lot of worked examples, from very simple ODEs to systems of PDEs, especially from fluid mechanics. The text is easily accessible and exam...
Cardona, Carlos
2016-01-01
Recently the CHY approach has been extended to one loop level using elliptic functions and modular forms over a Jacobian variety. Due to the difficulty in manipulating these kind of functions, we propose an alternative prescription that is totally algebraic. This new proposal is based on an elliptic algebraic curve embedded in a $\\mathbb{C}P^2$ space. We show that for the simplest integrand, namely the ${\\rm n-gon}$, our proposal indeed reproduces the expected result. By using the recently formulated $\\Lambda-$algorithm, we found a novel recurrence relation expansion in terms of tree level off-shell amplitudes. Our results connect nicely with recent results on the one-loop formulation of the scattering equations. In addition, this new proposal can be easily stretched out to hyperelliptic curves in order to compute higher genus.
$\\Lambda$ Scattering Equations
Gomez, Humberto
2016-01-01
The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter $\\Lambda$ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting $\\Lambda$ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the $\\Lambda$ algorithm.
Institute of Scientific and Technical Information of China (English)
黄虎; 丁平兴; 吕秀红
2001-01-01
The Hamiltonian formalism for surface waves and the mild-slope approximation were empolyed in handling the case of slowly varying three-dimensional currents and an uneven bottom, thus leading to an extended mild-slope equation. The bottom topography consists of two components: the slowly varying component whose horizontal length scale is longer than the surface wave length, and the fast varying component with the amplitude being smaller than that of the surface wave. The frequency of the fast varying depth component is, however, comparable to that of the surface waves. The extended mild- slope equation is more widely applicable and contains as special cases famous mild-slope equations below: the classical mild-slope equation of Berkhoff , Kirby' s mild-slope equation with current, and Dingemans' s mild-slope equation for rippled bed. The extended shallow water equations for ambient currents and rapidly varying topography are also obtained.
The Riccati Differential Equation and a Diffusion-Type Equation
Suazo, Erwin; Vega-Guzman, Jose M
2008-01-01
We construct an explicit solution of the Cauchy initial value problem for certain diffusion-type equation with variable coefficients on the entire real line. The corresponding Green function (heat kernel) is given in terms of elementary functions and certain integrals involving a characteristic function, which should be found as an analytic or numerical solution of the second order linear differential equation with time-dependent coefficients. Some special and limiting cases are outlined. Solution of the corresponding nonhomogeneous equation is also found.
Energy Technology Data Exchange (ETDEWEB)
Menikoff, Ralph [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
Differential equations methods and applications
Said-Houari, Belkacem
2015-01-01
This book presents a variety of techniques for solving ordinary differential equations analytically and features a wealth of examples. Focusing on the modeling of real-world phenomena, it begins with a basic introduction to differential equations, followed by linear and nonlinear first order equations and a detailed treatment of the second order linear equations. After presenting solution methods for the Laplace transform and power series, it lastly presents systems of equations and offers an introduction to the stability theory. To help readers practice the theory covered, two types of exercises are provided: those that illustrate the general theory, and others designed to expand on the text material. Detailed solutions to all the exercises are included. The book is excellently suited for use as a textbook for an undergraduate class (of all disciplines) in ordinary differential equations. .
Energy Conservation Equations of Motion
Vinokurov, Nikolay A
2015-01-01
A conventional derivation of motion equations in mechanics and field equations in field theory is based on the principle of least action with a proper Lagrangian. With a time-independent Lagrangian, a function of coordinates and velocities that is called energy is constant. This paper presents an alternative approach, namely derivation of a general form of equations of motion that keep the system energy, expressed as a function of generalized coordinates and corresponding velocities, constant. These are Lagrange equations with addition of gyroscopic forces. The important fact, that the energy is defined as the function on the tangent bundle of configuration manifold, is used explicitly for the derivation. The Lagrangian is derived from a known energy function. A development of generalized Hamilton and Lagrange equations without the use of variational principles is proposed. The use of new technique is applied to derivation of some equations.
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Directory of Open Access Journals (Sweden)
A. Jaruga
2014-11-01
Full Text Available This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case; and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Directory of Open Access Journals (Sweden)
A. Jaruga
2015-04-01
Full Text Available This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case; and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 1.0: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.
2015-04-01
This paper accompanies the first release of libmpdata++, a C++ library implementing the multi-dimensional positive-definite advection transport algorithm (MPDATA) on regular structured grid. The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; a shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
libmpdata++ 0.1: a library of parallel MPDATA solvers for systems of generalised transport equations
Jaruga, A.; Arabas, S.; Jarecka, D.; Pawlowska, H.; Smolarkiewicz, P. K.; Waruszewski, M.
2014-11-01
This paper accompanies first release of libmpdata++, a C++ library implementing the Multidimensional Positive-Definite Advection Transport Algorithm (MPDATA). The library offers basic numerical solvers for systems of generalised transport equations. The solvers are forward-in-time, conservative and non-linearly stable. The libmpdata++ library covers the basic second-order-accurate formulation of MPDATA, its third-order variant, the infinite-gauge option for variable-sign fields and a flux-corrected transport extension to guarantee non-oscillatory solutions. The library is equipped with a non-symmetric variational elliptic solver for implicit evaluation of pressure gradient terms. All solvers offer parallelisation through domain decomposition using shared-memory parallelisation. The paper describes the library programming interface, and serves as a user guide. Supported options are illustrated with benchmarks discussed in the MPDATA literature. Benchmark descriptions include code snippets as well as quantitative representations of simulation results. Examples of applications include: homogeneous transport in one, two and three dimensions in Cartesian and spherical domains; shallow-water system compared with analytical solution (originally derived for a 2-D case); and a buoyant convection problem in an incompressible Boussinesq fluid with interfacial instability. All the examples are implemented out of the library tree. Regardless of the differences in the problem dimensionality, right-hand-side terms, boundary conditions and parallelisation approach, all the examples use the same unmodified library, which is a key goal of libmpdata++ design. The design, based on the principle of separation of concerns, prioritises the user and developer productivity. The libmpdata++ library is implemented in C++, making use of the Blitz++ multi-dimensional array containers, and is released as free/libre and open-source software.
Quaternion Dirac Equation and Supersymmetry
Rawat, Seema; Negi, O. P. S.
2007-01-01
Quaternion Dirac equation has been analyzed and its supersymetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, non zero mass, scalar pote...
Introduction to ordinary differential equations
Rabenstein, Albert L
1966-01-01
Introduction to Ordinary Differential Equations is a 12-chapter text that describes useful elementary methods of finding solutions using ordinary differential equations. This book starts with an introduction to the properties and complex variable of linear differential equations. Considerable chapters covered topics that are of particular interest in applications, including Laplace transforms, eigenvalue problems, special functions, Fourier series, and boundary-value problems of mathematical physics. Other chapters are devoted to some topics that are not directly concerned with finding solutio
THE ERMAKOV EQUATION: A COMMENTARY
P.G.L. Leach; Andriopoulos, K.
2008-01-01
We present a short history of the Ermakov Equation with an emphasis on its discovery by theWest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008), ...
Luo, Da-Wei; Pyshkin, P. V.; Yu, Ting; Lin, Hai-Qing; You, J. Q.; Wu, Lian-Ao
2016-01-01
We provide an alternative approach to relativistic dynamics based on the Feshbach projection technique. Instead of directly studying the Dirac equation, we derive a two-component equation for the upper spinor. This approach allows one to investigate the underlying physics in a different perspective. For particles with small mass such as the neutrino, the leading order equation has a Hermitian effective Hamiltonian, implying there is no leakage between the upper and lower spinors. In the weak ...
Spinor wave equation of photon
Wu, Xiang-Yao; Liu, Xiao-Jing; Zhang, Si-Qi; Wang, Jing; Li, Hong; Fan, Xi-Hui; Li, Jing-Wu
2012-01-01
In this paper, we give the spinor wave equations of free and unfree photon, which are the differential equation of space-time one order. For the free photon, the spinor wave equations are covariant, and the spinors $\\psi$ are corresponding to the the reducibility representations $D^{10}+D^{01}$ and $D^{10}+D^{01}+D^{1/2 1/2}$ of the proper Lorentz group.
Correct Linearization of Einstein's Equations
Directory of Open Access Journals (Sweden)
Rabounski D.
2006-06-01
Full Text Available Regularly Einstein's equations can be reduced to a wave form (linearly dependent from the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel's symbols. As shown here, the origin of the problem is that one uses the general covariant theory of measurement. Here the wave form of Einstein's equations is obtained in the terms of Zelmanov's chronometric invariants (physically observable projections on the observer's time line and spatial section. The obtained equations depend on solely the second derivatives even if gravitation, the space rotation and Christoffel's symbols. The correct linearization proves: the Einstein equations are completely compatible with weak waves of the metric.
Electronic representation of wave equation
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Quaternion Dirac Equation and Supersymmetry
Rawat, Seema; Negi, O. P. S.
2009-08-01
Quaternion Dirac equation has been analyzed and its supersymmetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, nonzero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.
Quaternion Dirac Equation and Supersymmetry
Rawat, S; Rawat, Seema
2007-01-01
Quaternion Dirac equation has been analyzed and its supersymetrization has been discussed consistently. It has been shown that the quaternion Dirac equation automatically describes the spin structure with its spin up and spin down components of two component quaternion Dirac spinors associated with positive and negative energies. It has also been shown that the supersymmetrization of quaternion Dirac equation works well for different cases associated with zero mass, non zero mass, scalar potential and generalized electromagnetic potentials. Accordingly we have discussed the splitting of supersymmetrized Dirac equation in terms of electric and magnetic fields.
A modified electromagnetic wave equation
International Nuclear Information System (INIS)
The aim of this paper is to find an alternative to the usual electromagnetic wave equation: that is, we want to find a different equation with the same solutions. The final goal is to solve electromagnetic problems with iterative methods. The curl curl operator that appears in the electromagnetic wave equation is difficult to invert numerically, and this cannot be done iteratively. The addition of a higher order term that emphasizes the diagonal terms in the operator may help the solution of the problem, and the new equation should be solvable by an iterative algorithm. The additional mode is suppressed by suitable boundary conditions. (author) 5 figs., 9 refs
Tippe Top Equations and Equations for the Related Mechanical Systems
Directory of Open Access Journals (Sweden)
Nils Rutstam
2012-04-01
Full Text Available The equations of motion for the rolling and gliding Tippe Top (TT are nonintegrable and difficult to analyze. The only existing arguments about TT inversion are based on analysis of stability of asymptotic solutions and the LaSalle type theorem. They do not explain the dynamics of inversion. To approach this problem we review and analyze here the equations of motion for the rolling and gliding TT in three equivalent forms, each one providing different bits of information about motion of TT. They lead to the main equation for the TT, which describes well the oscillatory character of motion of the symmetry axis 3ˆ during the inversion. We show also that the equations of motion of TT give rise to equations of motion for two other simpler mechanical systems: the gliding heavy symmetric top and the gliding eccentric cylinder. These systems can be of aid in understanding the dynamics of the inverting TT.
Solutions of relativistic radial quasipotential equations
Energy Technology Data Exchange (ETDEWEB)
Minh, V.X.; Kadyshevskii, V.G.; Zhidkov, E.P.
1985-11-01
A systematic approach to the investigation of relativistic radial quasipotential equations is developed. The quasipotential equations can be interpreted either as linear equations in finite differences of fourth and second orders, respectively, or as differential equations of infinite order.
Anomalous Fractional Diffusion Equation for Transport Phenomena
Institute of Scientific and Technical Information of China (English)
QiuhuaZENG; HouqiangLI; 等
1999-01-01
We derive the standard diffusion equation from the continuity equation and by discussing the defectiveness of earlier proposed equations,we get the generalized fractional diffusion equation for anomalous diffusion.
Students' Understanding of Quadratic Equations
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
Enclosing Solutions of Integral Equations
DEFF Research Database (Denmark)
Madsen, Kaj; NA NA NA Caprani, Ole; Stauning, Ole
1996-01-01
We present a method for enclosing the solution of an integral equation. It is assumed that a solution exists and that the corresponding integral operator T is a contraction near y. When solving the integral equation by iteration we obtain a result which is normally different from y because...
Partial Completion of Equational Theories
Institute of Scientific and Technical Information of China (English)
孙永强; 林凯; 陆朝俊
2000-01-01
In this paper, the notion of partial completion of equational theories is proposed, which is a procedure to construct a confluent term rewriting system from an equational theory without requirement of termination condition. A partial completion algorithm is presented with a brief description of its application in a program development system.
Uncertainty of empirical correlation equations
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Numerical Solution of Parabolic Equations
DEFF Research Database (Denmark)
Østerby, Ole
These lecture notes are designed for a one-semester course on finite-difference methods for parabolic equations. These equations which traditionally are used for describing diffusion and heat-conduction problems in Geology, Physics, and Chemistry have recently found applications in Finance Theory...
Institute of Scientific and Technical Information of China (English)
M. Ko(c)ak; B. G(o)nül
2007-01-01
The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schr(o)dinger-like one. Earlier results are discussed in a unified framework, and some solutions of a large class of potentials are given.
Differential equations a concise course
Bear, H S
2011-01-01
Concise introduction for undergraduates includes, among other topics, a survey of first order equations, discussions of complex-valued solutions, linear differential operators, inverse operators and variation of parameters method, the Laplace transform, Picard's existence theorem, and an exploration of various interpretations of systems of equations. Numerous clearly stated theorems and proofs, examples, and problems followed by solutions.
On asymptotics for difference equations
Rafei, M.
2012-01-01
In this thesis a class of nonlinear oscillator equations is studied. Asymptotic approximations of first integrals for nonlinear difference equations are constructed by using the recently developed perturbation method based on invariance vectors. The asymptotic approximations of the solutions of the
Kocak, M.; Gonul, B.
2007-01-01
The solutions, in terms of orthogonal polynomials, of Dirac equation with analytically solvable potentials are investigated within a novel formalism by transforming the relativistic equation into a Schrodinger like one. Earlier results are discussed in a unified framework and certain solutions of a large class of potentials are given.
Loewner equations and dispersionless hierarchies
Energy Technology Data Exchange (ETDEWEB)
Takebe, Takashi [Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Bunkyo-ku, Tokyo, 112-8610 (Japan); Teo, Lee-Peng [Faculty of Information Technology, Multimedia University, Jalan Multimedia, Cyberjaya, 63100, Selangor Darul Ehsan (Malaysia); Zabrodin, Anton [Institute of Biochemical Physics, Kosygina str. 4, 119991 Moscow, Russia and ITEP, Bol. Cheremushkinskaya str. 25, 117259 Moscow (Russian Federation)
2006-09-15
Using the Hirota representation of dispersionless dKP and dToda hierarchies, we show that the chordal Loewner equations and radial Loewner equations respectively serve as consistency conditions for one-variable reductions of these integrable hierarchies. We also clarify the geometric meaning of this result by relating it to the eigenvalue distribution of normal random matrices in the large N limit.
Singularity: Raychaudhuri equation once again
Indian Academy of Sciences (India)
Naresh Dadhich
2007-07-01
I first recount Raychaudhuri's deep involvement with the singularity problem in general relativity. I then argue that precisely the same situation has arisen today in loop quantum cosmology as obtained when Raychaudhuri discovered his celebrated equation. We thus need a new analogue of the Raychaudhuri equation in quantum gravity.
International Nuclear Information System (INIS)
The Boltzmann-Uhlenbeck (BUU) equation, which is the time evolution of the wigner function of the single particle Green's function, is dervied by using the closed-time Green's function approach. The quantum mechanical approximation in derving the BUU equation is discussed
Conservation Laws of Differential Equations in Finance
Institute of Scientific and Technical Information of China (English)
QIN Mao-Chang; MEI Feng-Xiang; SHANG Mei
2005-01-01
Conservation laws of some differential equations in fiance are studied in this paper. This method does not involve the use or existence of a variational principle. As an alternative, linearize the given equation and find adjoint equation of the linearized equation, the conservation laws can be constructed directly from the symmetries and adjoint symmetries of the associated linearized equation and its adjoint equation.
Transport Equations for Oscillating Neutrinos
Zhang, Yunfan
2013-01-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse su...
A Generalized Cubic Functional Equation
Institute of Scientific and Technical Information of China (English)
P. K. SAHOO
2005-01-01
In this paper, we determine the general solution of the functional equation f1 (2x + y) +f2(2x - y) ＝ f3(x + y) + f4(x - y) + f5(x) without assuming any regularity condition on the unknown functions f1,f2,f3, f4,f5: R → R. The general solution of this equation is obtained by finding the general solution of the functional equations f(2x + y) + f(2x - y) = g(x + y) + g(x - y) + h(x) and f(2x + y) - f(2x - y) ＝ g(x + y) - g(x - y). The method used for solving these functional equations is elementary but exploits an important result due to Hosszu. The solution of this functional equation can also be determined in certain type of groups using two important results due to Székelyhidi.
Nominal Logic with Equations Only
Clouston, Ranald
2011-01-01
Many formal systems, particularly in computer science, may be captured by equations modulated by side conditions asserting the "freshness of names"; these can be reasoned about with Nominal Equational Logic (NEL). Like most logics of this sort NEL employs this notion of freshness as a first class logical connective. However, this can become inconvenient when attempting to translate results from standard equational logic to the nominal setting. This paper presents proof rules for a logic whose only connectives are equations, which we call Nominal Equation-only Logic (NEoL). We prove that NEoL is just as expressive as NEL. We then give a simple description of equality in the empty NEoL-theory, then extend that result to describe freshness in the empty NEL-theory.
Determining dynamical equations is hard
Cubitt, Toby S; Wolf, Michael M
2010-01-01
The behaviour of any physical system is governed by its underlying dynamical equations--the differential equations describing how the system evolves with time--and much of physics is ultimately concerned with discovering these dynamical equations and understanding their consequences. At the end of the day, any such dynamical law is identified by making measurements at different times, and computing the dynamical equation consistent with the acquired data. In this work, we show that, remarkably, this process is a provably computationally intractable problem (technically, it is NP-hard). That is, even for a moderately complex system, no matter how accurately we have specified the data, discovering its dynamical equations can take an infeasibly long time (unless P=NP). As such, we find a complexity-theoretic solution to both the quantum and the classical embedding problems; the classical version is a long-standing open problem, dating from 1937, which we finally lay to rest.
Some Variations on Maxwell's Equations
Ascoli, G A; Ascoli, Giorgio A.; Goldin, Gerald A.
2006-01-01
In the first sections of this article, we discuss two variations on Maxwell's equations that have been introduced in earlier work---a class of nonlinear Maxwell theories with well-defined Galilean limits (and correspondingly generalized Yang-Mills equations), and a linear modification motivated by the coupling of the electromagnetic potential with a certain nonlinear Schroedinger equation. In the final section, revisiting an old idea of Lorentz, we write Maxwell's equations for a theory in which the electrostatic force of repulsion between like charges differs fundamentally in magnitude from the electrostatic force of attraction between unlike charges. We elaborate on Lorentz' description by means of electric and magnetic field strengths, whose governing equations separate into two fully relativistic Maxwell systems---one describing ordinary electromagnetism, and the other describing a universally attractive or repulsive long-range force. If such a force cannot be ruled out {\\it a priori\\/} by known physical ...
Stochastic differential equations, backward SDEs, partial differential equations
Pardoux, Etienne
2014-01-01
This research monograph presents results to researchers in stochastic calculus, forward and backward stochastic differential equations, connections between diffusion processes and second order partial differential equations (PDEs), and financial mathematics. It pays special attention to the relations between SDEs/BSDEs and second order PDEs under minimal regularity assumptions, and also extends those results to equations with multivalued coefficients. The authors present in particular the theory of reflected SDEs in the above mentioned framework and include exercises at the end of each chapter. Stochastic calculus and stochastic differential equations (SDEs) were first introduced by K. Itô in the 1940s, in order to construct the path of diffusion processes (which are continuous time Markov processes with continuous trajectories taking their values in a finite dimensional vector space or manifold), which had been studied from a more analytic point of view by Kolmogorov in the 1930s. Since then, this topic has...
Higher derivative gravity: Field equation as the equation of state
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Higher derivative gravity: field equation as the equation of state
Dey, Ramit; Mohd, Arif
2016-01-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. Extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Wave equations for pulse propagation
Shore, B. W.
1987-06-01
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity.
Integral equation methods for electromagnetics
Volakis, John
2012-01-01
This text/reference is a detailed look at the development and use of integral equation methods for electromagnetic analysis, specifically for antennas and radar scattering. Developers and practitioners will appreciate the broad-based approach to understanding and utilizing integral equation methods and the unique coverage of historical developments that led to the current state-of-the-art. In contrast to existing books, Integral Equation Methods for Electromagnetics lays the groundwork in the initial chapters so students and basic users can solve simple problems and work their way up to the mo
Galois theory of difference equations
Put, Marius
1997-01-01
This book lays the algebraic foundations of a Galois theory of linear difference equations and shows its relationship to the analytic problem of finding meromorphic functions asymptotic to formal solutions of difference equations. Classically, this latter question was attacked by Birkhoff and Tritzinsky and the present work corrects and greatly generalizes their contributions. In addition results are presented concerning the inverse problem in Galois theory, effective computation of Galois groups, algebraic properties of sequences, phenomena in positive characteristics, and q-difference equations. The book is aimed at advanced graduate researchers and researchers.
THE ERMAKOV EQUATION: A COMMENTARY
Directory of Open Access Journals (Sweden)
P. G. L. Leach
2008-08-01
Full Text Available We present a short history of the Ermakov Equation with an emphasis on its discovery by theWest and the subsequent boost to research into invariants for nonlinear systems although recognizing some of the significant developments in the East. We present the modern context of the Ermakov Equation in the algebraic and singularity theory of ordinary differential equations and applications to more divers fields. The reader is referred to the previous article (Appl. Anal. Discrete Math., 2 (2008, 123–145 for an English translation of Ermakov’s original paper.
Reflection algebra and functional equations
Energy Technology Data Exchange (ETDEWEB)
Galleas, W., E-mail: w.galleas@uu.nl; Lamers, J., E-mail: j.lamers@uu.nl
2014-09-15
In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations satisfied by the partition function.
Soliton equations and Hamiltonian systems
Dickey, L A
2002-01-01
The theory of soliton equations and integrable systems has developed rapidly during the last 30 years with numerous applications in mechanics and physics. For a long time, books in this field have not been written but the flood of papers was overwhelming: many hundreds, maybe thousands of them. All this output followed one single work by Gardner, Green, Kruskal, and Mizura on the Korteweg-de Vries equation (KdV), which had seemed to be merely an unassuming equation of mathematical physics describing waves in shallow water. Besides its obvious practical use, this theory is attractive also becau
Equational theories of tropical sernirings
DEFF Research Database (Denmark)
Aceto, Luca; Esik, Zoltan; Ingolfsdottir, Anna
2003-01-01
of these commutative idempotent weak semirings, the paper offers characterizations of the equations that hold in them, decidability results for their equational theories, explicit descriptions of the free algebras in the varieties they generate, and relative axiomatization results. Udgivelsesdato: APR 11......This paper studies the equational theories of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime...
Direct 'delay' reductions of the Toda equation
International Nuclear Information System (INIS)
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painleve equations. The Lax pair associated with this equation is obtained, also by reduction. (fast track communication)
Direct "Delay" Reductions of the Toda Equation
Joshi, Nalini
2008-01-01
A new direct method of obtaining reductions of the Toda equation is described. We find a canonical and complete class of all possible reductions under certain assumptions. The resulting equations are ordinary differential-difference equations, sometimes referred to as delay-differential equations. The representative equation of this class is hypothesized to be a new version of one of the classical Painlev\\'e equations. The Lax pair associated to this equation is obtained, also by reduction.
Integral Transform Approach to Generalized Tricomi Equations
Yagdjian, Karen
2014-01-01
We present some integral transform that allows to obtain solutions of the generalized Tricomi equation from solutions of a simpler equation. We used in [13,14],[41]-[46] the particular version of this transform in order to investigate in a unified way several equations such as the linear and semilinear Tricomi equations, Gellerstedt equation, the wave equation in Einstein-de Sitter spacetime, the wave and the Klein-Gordon equations in the de Sitter and anti-de Sitter spacetimes.
Geophysical interpretation using integral equations
Eskola, L
1992-01-01
Along with the general development of numerical methods in pure and applied to apply integral equations to geophysical modelling has sciences, the ability improved considerably within the last thirty years or so. This is due to the successful derivation of integral equations that are applicable to the modelling of complex structures, and efficient numerical algorithms for their solution. A significant stimulus for this development has been the advent of fast digital computers. The purpose of this book is to give an idea of the principles by which boundary-value problems describing geophysical models can be converted into integral equations. The end results are the integral formulas and integral equations that form the theoretical framework for practical applications. The details of mathematical analysis have been kept to a minimum. Numerical algorithms are discussed only in connection with some illustrative examples involving well-documented numerical modelling results. The reader is assu med to have a back...
Solving Differential Equations in R
Although R is still predominantly applied for statistical analysis and graphical representation, it is rapidly becoming more suitable for mathematical computing. One of the fields where considerable progress has been made recently is the solution of differential equations. Here w...
IKT for quantum hydrodynamic equations
Tessarotto, Massimo; Ellero, Marco; Nicolini, Piero
2007-11-01
A striking feature of standard quantum mechanics (SQM) is its analogy with classical fluid dynamics. In fact, it is well-known that the Schr"odinger equation is equivalent to a closed set of partial differential equations for suitable real-valued functions of position and time (denoted as quantum fluid fields) [Madelung, 1928]. In particular, the corresponding quantum hydrodynamic equations (QHE) can be viewed as the equations of a classical compressible and non-viscous fluid, endowed with potential velocity and quantized velocity circulation. In this reference, an interesting theoretical problem, in its own right, is the construction of an inverse kinetic theory (IKT) for such a type of fluids. In this note we intend to investigate consequences of the IKT recently formulated for QHE [M.Tessarotto et al., Phys. Rev. A 75, 012105 (2007)]. In particular a basic issue is related to the definition of the quantum fluid fields.
Spin equation and its solutions
Bagrov, V G; Baldiotti, M C; Levin, A D
2005-01-01
The aim of the present article is to study in detail the so-called spin equation (SE) and present both the methods of generating new solution and a new set of exact solutions. We recall that the SE with a real external field can be treated as a reduction of the Pauli equation to the (0+1)-dimensional case. Two-level systems can be described by an SE with a particular form of the external field. In this article, we also consider associated equations that are equivalent or (in one way or another) related to the SE. We describe the general solution of the SE and solve the inverse problem for this equation. We construct the evolution operator for the SE and consider methods of generating new sets of exact solutions. Finally, we find a new set of exact solutions of the SE.
Diophantine approximations and Diophantine equations
Schmidt, Wolfgang M
1991-01-01
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum
Invariant foliations for parabolic equations
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
It is proved for parabolic equations that under certain conditions the weak (un-)stable manifolds possess invariant foliations, called strongly (un-)stable foliations. The relevant results on center manifolds are generalized to weak hyperbolic manifolds.
Overdetermined Systems of Linear Equations.
Williams, Gareth
1990-01-01
Explored is an overdetermined system of linear equations to find an appropriate least squares solution. A geometrical interpretation of this solution is given. Included is a least squares point discussion. (KR)
Correct Linearization of Einstein's Equations
Directory of Open Access Journals (Sweden)
Rabounski D.
2006-04-01
Full Text Available Routinely, Einstein’s equations are be reduced to a wave form (linearly independent of the second derivatives of the space metric in the absence of gravitation, the space rotation and Christoffel’s symbols. As shown herein, the origin of the problem is the use of the general covariant theory of measurement. Herein the wave form of Einstein’s equations is obtained in terms of Zelmanov’s chronometric invariants (physically observable projections on the observer’s time line and spatial section. The equations so obtained depend solely upon the second derivatives, even for gravitation, the space rotation and Christoffel’s symbols. The correct linearization proves that the Einstein equations are completely compatible with weak waves of the metric.
Hidden Statistics of Schroedinger Equation
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
A New Unified Evolution Equation
Lim, Jyh-Liong
1998-01-01
WE propose a new unified evolution equation for parton distribution functions appropriate for both large and small Bjorken x. Compared with the Ciafaloni- Catani-Fiorani-Marchesini equation, the cancellation of soft poles between virtual and real gluon emissions is made explicitly without introducing infrared cutoffs, next-to-leading contributions to the Sudakov resummation can be included systematically, and the scales of the running coupling constants are determined unambiguously.
On basic equation of statistical physics
Institute of Scientific and Technical Information of China (English)
邢修三
1996-01-01
Considering that thermodynamic irreversibility, the principle of entropy increase and hydrodynamic equations cannot be derived rigorously and in a unified way from the Liouville equations, the anomalous Langevin equation in Liouville space or its equivalent generalized Liouville equation is proposed as a basic equation of statistical physics. This equation reflects the fact that the law of motion of statistical thermodynamics is stochastic, but not deterministic. From that the nonequilibrium entropy, the principle of entropy increase, the theorem of minimum entropy production and the BBGKY diffusion equation hierarchy have been derived. The hydrodynamic equations, such as the generalized Navier-Stokes equation and the mass drift-diffusion equation, etc. have been derived from the BBGKY diffusion equation hierarchy. This equation has the same equilibrium solution as that of the Liouville equation. All these are unified and rigorous without adding any extra assumption. But it is difficult to prove that th
Computational partial differential equations using Matlab
Li, Jichun
2008-01-01
Brief Overview of Partial Differential Equations The parabolic equations The wave equations The elliptic equations Differential equations in broader areasA quick review of numerical methods for PDEsFinite Difference Methods for Parabolic Equations Introduction Theoretical issues: stability, consistence, and convergence 1-D parabolic equations2-D and 3-D parabolic equationsNumerical examples with MATLAB codesFinite Difference Methods for Hyperbolic Equations IntroductionSome basic difference schemes Dissipation and dispersion errors Extensions to conservation lawsThe second-order hyperbolic PDE
Equationally Compact Acts : Coproducts / Peeter Normak
Normak, Peeter
1998-01-01
In this article equational compactness of acts and its generalizations are discussed. As equational compactness does not carry over to coproducts a slight generalization of c-equational campactness is introduced. It is proved that a coproduct of acts is c-equationally compact if and only if all components are c-equationally campact
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
ON THE EQUIVALENCE OF THE ABEL EQUATION
Institute of Scientific and Technical Information of China (English)
无
2006-01-01
This article uses the reflecting function of Mironenko to study some complicated differential equations which are equivalent to the Abel equation. The results are applied to discuss the behavior of solutions of these complicated differential equations.
Non-linear constitutive equations for gravitoelectromagnetism
Duplij, Steven; Di Grezia, Elisabetta; Esposito, Giampiero; Kotvytskiy, Albert
2013-01-01
This paper studies non-linear constitutive equations for gravitoelectromagnetism. Eventually, the problem is solved of finding, for a given particular solution of the gravity-Maxwell equations, the exact form of the corresponding non-linear constitutive equations.
Multi-Time Equations, Classical and Quantum
Petrat, Sören
2013-01-01
Multi-time equations are evolution equations involving several time variables, one for each particle. Such equations have been considered for the purpose of making theories manifestly Lorentz invariant. We compare their status and significance in classical and quantum physics.
First-order partial differential equations
Rhee, Hyun-Ku; Amundson, Neal R
2001-01-01
This first volume of a highly regarded two-volume text is fully usable on its own. After going over some of the preliminaries, the authors discuss mathematical models that yield first-order partial differential equations; motivations, classifications, and some methods of solution; linear and semilinear equations; chromatographic equations with finite rate expressions; homogeneous and nonhomogeneous quasilinear equations; formation and propagation of shocks; conservation equations, weak solutions, and shock layers; nonlinear equations; and variational problems. Exercises appear at the end of mo
How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation
Heras, Jose A.
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
How to obtain the covariant form of Maxwell's equations from the continuity equation
Energy Technology Data Exchange (ETDEWEB)
Heras, Jose A [Departamento de Ciencias Basicas, Universidad Autonoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo No. 180, Col. Reynosa, 02200, Mexico D. F. (Mexico); Departamento de Fisica y Matematicas, Universidad Iberoamericana, Prolongacion Paseo de la Reforma 880, Mexico D. F. 01210 (Mexico)
2009-07-15
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
Introductory course on differential equations
Gorain, Ganesh C
2014-01-01
Introductory Course on DIFFERENTIAL EQUATIONS provides an excellent exposition of the fundamentals of ordinary and partial differential equations and is ideally suited for a first course of undergraduate students of mathematics, physics and engineering. The aim of this book is to present the elementary theories of differential equations in the forms suitable for use of those students whose main interest in the subject are based on simple mathematical ideas. KEY FEATURES: Discusses the subject in a systematic manner without sacrificing mathematical rigour. A variety of exercises drill the students in problem solving in view of the mathematical theories explained in the book. Worked out examples illustrated according to the theories developed in the book with possible alternatives. Exhaustive collection of problems and the simplicity of presentation differentiate this book from several others. Material contained will help teachers as well as aspiring students of different competitive examinations.
Differential Equations for Morphological Amoebas
Welk, Martin; Breuß, Michael; Vogel, Oliver
This paper is concerned with amoeba median filtering, a structure-adaptive morphological image filter. It has been introduced by Lerallut et al. in a discrete formulation. Experimental evidence shows that iterated amoeba median filtering leads to segmentation-like results that are similar to those obtained by self-snakes, an image filter based on a partial differential equation. We investigate this correspondence by analysing a space-continuous formulation of iterated median filtering. We prove that in the limit of vanishing radius of the structuring elements, iterated amoeba median filtering indeed approximates a partial differential equation related to self-snakes and the well-known (mean) curvature motion equation. We present experiments with discrete iterated amoeba median filtering that confirm qualitative and quantitative predictions of our analysis.
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
The respiratory system in equations
Maury, Bertrand
2013-01-01
The book proposes an introduction to the mathematical modeling of the respiratory system. A detailed introduction on the physiological aspects makes it accessible to a large audience without any prior knowledge on the lung. Different levels of description are proposed, from the lumped models with a small number of parameters (Ordinary Differential Equations), up to infinite dimensional models based on Partial Differential Equations. Besides these two types of differential equations, two chapters are dedicated to resistive networks, and to the way they can be used to investigate the dependence of the resistance of the lung upon geometrical characteristics. The theoretical analysis of the various models is provided, together with state-of-the-art techniques to compute approximate solutions, allowing comparisons with experimental measurements. The book contains several exercises, most of which are accessible to advanced undergraduate students.
Stability Analysis of Ecomorphodynamic Equations
Bärenbold, Fabian; Perona, Paolo
2014-01-01
Although riparian vegetation is present in or along many water courses of the world, its active role resulting from the interaction with flow and sediment processes has only recently become an active field of research. Especially, the role of vegetation in the process of river pattern formation has been explored and demonstrated mostly experimentally and numerically until now. In the present work, we shed light on this subject by performing a linear stability analysis on a simple model for riverbed vegetation dynamics coupled with the set of classical river morphodynamic equations. The vegetation model only accounts for logistic growth, local positive feedback through seeding and resprouting, and mortality by means of uprooting through flow shear stress. Due to the simplicity of the model, we can transform the set of equations into an eigenvalue problem and assess the stability of the linearized equations when slightly perturbated away from a spatially homogeneous solution. If we couple vegetation dynamics wi...
Random equations in nilpotent groups
Gilman, Robert; Romankov, Vitalii
2011-01-01
In this paper we study satisfiability of random equations in an infinite finitely generated nilpotent group G. We show that the set SAT(G,k) of all equations in k > 1 variables over G which are satisfiable in G has an intermediate asymptotic density in the space of all equations in k variables over G. When G is a free abelian group of finite rank, we compute this density precisely; otherwise we give some non-trivial upper and lower bounds. For k = 1 the set SAT(G,k) is negligible. Usually the asymptotic densities of interesting sets in groups are either zero or one. The results of this paper provide new examples of algebraically significant sets of intermediate asymptotic density.
Students' understanding of quadratic equations
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Basic linear partial differential equations
Treves, Francois
2006-01-01
Focusing on the archetypes of linear partial differential equations, this text for upper-level undergraduates and graduate students features most of the basic classical results. The methods, however, are decidedly nontraditional: in practically every instance, they tend toward a high level of abstraction. This approach recalls classical material to contemporary analysts in a language they can understand, as well as exploiting the field's wealth of examples as an introduction to modern theories.The four-part treatment covers the basic examples of linear partial differential equations and their
Hamiltonian systems as selfdual equations
Institute of Scientific and Technical Information of China (English)
2008-01-01
Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action func-tionals obtained by a generalization of Bogomolnyi's trick of 'completing squares'. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the correspond- ing Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.
Stability theory of differential equations
Bellman, Richard
2008-01-01
Suitable for advanced undergraduates and graduate students, this was the first English-language text to offer detailed coverage of boundedness, stability, and asymptotic behavior of linear and nonlinear differential equations. It remains a classic guide, featuring material from original research papers, including the author's own studies.The linear equation with constant and almost-constant coefficients receives in-depth attention that includes aspects of matrix theory. No previous acquaintance with the theory is necessary, since author Richard Bellman derives the results in matrix theory from
Nielsen number and differential equations
Directory of Open Access Journals (Sweden)
Andres Jan
2005-01-01
Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via Poincaré's translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial -structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.
Fundamentals of equations of state
Eliezer, Shalom; Hora, Heinrich
2002-01-01
The equation of state was originally developed for ideal gases, and proved central to the development of early molecular and atomic physics. Increasingly sophisticated equations of state have been developed to take into account molecular interactions, quantization, relativistic effects, etc. Extreme conditions of matter are encountered both in nature and in the laboratory, for example in the centres of stars, in relativistic collisions of heavy nuclei, in inertial confinement fusion (where a temperature of 10 9 K and a pressure exceeding a billion atmospheres can be achieved). A sound knowledg
Applied analysis and differential equations
Cârj, Ovidiu
2007-01-01
This volume contains refereed research articles written by experts in the field of applied analysis, differential equations and related topics. Well-known leading mathematicians worldwide and prominent young scientists cover a diverse range of topics, including the most exciting recent developments. A broad range of topics of recent interest are treated: existence, uniqueness, viability, asymptotic stability, viscosity solutions, controllability and numerical analysis for ODE, PDE and stochastic equations. The scope of the book is wide, ranging from pure mathematics to various applied fields such as classical mechanics, biomedicine, and population dynamics.
Group analysis of differential equations
Ovsiannikov, L V
1982-01-01
Group Analysis of Differential Equations provides a systematic exposition of the theory of Lie groups and Lie algebras and its application to creating algorithms for solving the problems of the group analysis of differential equations.This text is organized into eight chapters. Chapters I to III describe the one-parameter group with its tangential field of vectors. The nonstandard treatment of the Banach Lie groups is reviewed in Chapter IV, including a discussion of the complete theory of Lie group transformations. Chapters V and VI cover the construction of partial solution classes for the g
Differential equations and mathematical biology
Jones, DS; Sleeman, BD
2009-01-01
""… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions.""-SIAM Review, Sept. 2010, Vol. 52, No. 3""… Where this text stands out i
Partial differential equations an introduction
Colton, David
2004-01-01
Intended for a college senior or first-year graduate-level course in partial differential equations, this text offers students in mathematics, engineering, and the applied sciences a solid foundation for advanced studies in mathematics. Classical topics presented in a modern context include coverage of integral equations and basic scattering theory. This complete and accessible treatment includes a variety of examples of inverse problems arising from improperly posed applications. Exercises at the ends of chapters, many with answers, offer a clear progression in developing an understanding of
Integral equations on time scales
Georgiev, Svetlin G
2016-01-01
This book offers the reader an overview of recent developments of integral equations on time scales. It also contains elegant analytical and numerical methods. This book is primarily intended for senior undergraduate students and beginning graduate students of engineering and science courses. The students in mathematical and physical sciences will find many sections of direct relevance. The book contains nine chapters and each chapter is pedagogically organized. This book is specially designed for those who wish to understand integral equations on time scales without having extensive mathematical background.
Radar equations for modern radar
Barton, David K
2012-01-01
Based on the classic Radar Range-Performance Analysis from 1980, this practical volume extends that work to ensure applicability of radar equations to the design and analysis of modern radars. This unique book helps you identify what information on the radar and its environment is needed to predict detection range. Moreover, it provides equations and data to improve the accuracy of range calculations. You find detailed information on propagation effects, methods of range calculation in environments that include clutter, jamming and thermal noise, as well as loss factors that reduce radar perfo
On a nonhomogeneous Burgers' equation
Institute of Scientific and Technical Information of China (English)
DING; Xiaqi(
2001-01-01
［1］Hopf, E., The partial differential equation ut + uux = μuxx, Comm. Pure Appl. Math., 1950, 3: 201-230.［2］Ding, X. Q. , Luo, P. Z. , Generalized expansions in Hilbert space, Acta Mathematica Scientia, 1999, 19(3): 241 250.［3］Titchmarsh, E., Introduction to the Theory of Fourier Integrals, 2nd ed., Oxford: Oxford University Press, 1948.［4］Ladyzhenskaya, O. A., Solonnikov, V. A., Ural' ceva, N. N., Linear and Quasilinear Equations of Parabolic Type,Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, 1968.
E. M. E. Zayed; K. A. E. Alurrfi
2014-01-01
We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.
Directory of Open Access Journals (Sweden)
E. M. E. Zayed
2014-01-01
Full Text Available We apply the generalized projective Riccati equations method to find the exact traveling wave solutions of some nonlinear evolution equations with any-order nonlinear terms, namely, the nonlinear Pochhammer-Chree equation, the nonlinear Burgers equation and the generalized, nonlinear Zakharov-Kuznetsov equation. This method presents wider applicability for handling many other nonlinear evolution equations in mathematical physics.
Algebraic solution of master equations
R. Rangel; L. Carvalho
2003-01-01
We present a simple analytical method to solve master equations for finite temperatures and any initial conditions, which consists in the expansion of the density operator into normal modes. These modes and the expansion coefficients are obtained algebraically by using ladder superoperators. This algebraic technique is successful in cases in which the Liouville superoperator is quadratic in the creation and annihilation operators.
International Nuclear Information System (INIS)
In 1977, Dave Young published an equation-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram
Equational axioms of test algebra
Hollenberg, M.
2008-01-01
We present a complete axiomatization of test algebra ([24,18,29]), the two-sorted algebraic variant of Propositional Dynamic Logic (PDL,[21,7]). The axiomatization consists of adding a finite number of equations to any axiomatization of Kleene algebra ([15,26,17,4]) and algebraic translations of the
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus. PMID:27586766
Stability of Functional Differential Equations
Lemm, Jeffrey M
1986-01-01
This book provides an introduction to the structure and stability properties of solutions of functional differential equations. Numerous examples of applications (such as feedback systrems with aftereffect, two-reflector antennae, nuclear reactors, mathematical models in immunology, viscoelastic bodies, aeroautoelastic phenomena and so on) are considered in detail. The development is illustrated by numerous figures and tables.
Homographic scheme for Riccati equation
Dubois, François
2011-01-01
In this paper we present a numerical scheme for the resolution of matrix Riccati equation, usualy used in control problems. The scheme is unconditionnaly stable and the solution is definite positive at each time step of the resolution. We prove the convergence in the scalar case and present several numerical experiments for classical test cases.
Renaissance Learning Equating Study. Report
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Quaternionic Monge-Ampere equations
Alesker, Semyon
2002-01-01
The main result of this paper is the existence and uniqueness of solution of the Dirichlet problem for quaternionic Monge-Ampere equations in quaternionic strictly pseudoconvex bounded domains in H^n. We continue the study of the theory of plurisubharmonic functions of quaternionic variables started by the author at [2].
Pendulum Motion and Differential Equations
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Solution and transcritical bifurcation of Burgers equation
Institute of Scientific and Technical Information of China (English)
Tang Jia-Shi; Zhao Ming-Hua; Han Feng; Zhang Liang
2011-01-01
Burgers equation is reduced into a first-order ordinary differential equation by using travelling wave transformation and it has typical bifurcation characteristics. We can obtain many exact solutions of the Burgers equation, discuss its transcritical bifurcation and control dynamical behaviours by extending the stable region. The transcritical bifurcation exists in the (2 + 1)-dimensional Burgers equation.
Stochastic dynamic equations on general time scales
Martin Bohner; Olexandr M. Stanzhytskyi; Anastasiia O. Bratochkina
2013-01-01
In this article, we construct stochastic integral and stochastic differential equations on general time scales. We call these equations stochastic dynamic equations. We provide the existence and uniqueness theorem for solutions of stochastic dynamic equations. The crucial tool of our construction is a result about a connection between the time scales Lebesgue integral and the Lebesgue integral in the common sense.
A Bayesian Nonparametric Approach to Test Equating
Karabatsos, George; Walker, Stephen G.
2009-01-01
A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are…
Exact Vacuum Solutions to the Einstein Equation
Institute of Scientific and Technical Information of China (English)
无
2007-01-01
In this paper, the author presents a framework for getting a series of exact vacuum solutions to the Einstein equation. This procedure of resolution is based on a canonical form of the metric. According to this procedure, the Einstein equation can be reduced to some 2-dimensional Laplace-like equations or rotation and divergence equations,which are much convenient for the resolution.
Functional Equations and Inequalities with Applications
Kannappan, Palaniappan
2009-01-01
Presents a comprehensive study of the classical topic of functional equations. This monograph explores different aspects of functional equations and their applications to related topics, such as differential equations, integral equations, the Laplace transformation, the calculus of finite differences, and many other basic tools in analysis.
On Backward Stochstic Partial Differential Equations.
2001-01-01
We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations. This is a type of equations which appear as adjoint equations in the maximum principle approach to optimal control of systems described by stochastic partial differential equations.
Algebraic entropy for differential-delay equations
Viallet, Claude M.
2014-01-01
We extend the definition of algebraic entropy to a class of differential-delay equations. The vanishing of the entropy, as a structural property of an equation, signals its integrability. We suggest a simple way to produce differential-delay equations with vanishing entropy from known integrable differential-difference equations.
The AGL equation from the dipole picture
Gay-Ducati, M B
1999-01-01
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at small x in the DLA limit.
The AGL equation from the dipole picture
International Nuclear Information System (INIS)
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to a unitarized gluon distribution in the small x regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this paper that the AGL equation and, consequently, the GLR equation, can also be obtained from the dipole picture in the double logarithmic limit, using an evolution equation, recently proposed, which includes all multiple pomeron exchanges in the leading logarithmic approximation. Our conclusion is that the AGL equation is a good candidate for a unitarized evolution equation at small x in the DLA limit
Explicit Integration of Friedmann's Equation with Nonlinear Equations of State
Chen, Shouxin; Yang, Yisong
2015-01-01
This paper is a continuation of our earlier study on the integrability of the Friedmann equations in the light of the Chebyshev theorem. Our main focus will be on a series of important, yet not previously touched, problems when the equation of state for the perfect-fluid universe is nonlinear. These include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born--Infeld, and two-fluid models. We show that some of these may be integrated using Chebyshev's result while other are out of reach by the theorem but may be integrated explicitly by other methods. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution. For example, in the Chaplygin gas universe, it is seen that, as far as there is a tiny presence of nonlinear matter, linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics ...
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
Four nonlinear acoustical wave equations that apply to both perfect gasses and arbitrary fluids with a quadratic equation of state are studied. Shock and rarefaction wave solutions to the equations are studied. In order to assess the accuracy of the wave equations, their solutions are compared...... to solutions of the basic equations from which the wave equations are derived. A straightforward weakly nonlinear equation is the most accurate for shock modeling. A higher order wave equation is the most accurate for modeling of smooth disturbances. Investigations of the linear stability properties...... of solutions to the wave equations, reveal that the solutions may become unstable. Such instabilities are not found in the basic equations. Interacting shocks and standing shocks are investigated....
The AGL Equation from a Dipole Picture
Gay-Ducati, M B
1999-01-01
The AGL equation includes all multiple pomeron exchanges in the double logarithmic approximation (DLA) limit, leading to an unitarized gluon distribution in the small $x$ regime. This equation was originally obtained using the Glauber-Mueller approach. We demonstrate in this contribution that the AGL equation can also be obtained from the dipole picture. Our conclusion is that the AGL equation is a good candidate for an unitarized evolution equation at small $x$ in the DLA limit.
Dual Isomonodromic Problems and Whitham Equations
Takasaki, Kanehisa
1997-01-01
The author's recent results on an asymptotic description of the Schlesinger equation are generalized to the JMMS equation. As in the case of the Schlesinger equation, the JMMS equation is reformulated to include a small parameter $\\epsilon$. By the method of multiscale analysis, the isomonodromic problem is approximated by slow modulations of an isospectral problem. A modulation equation of this slow dynamics is proposed, and shown to possess a number of properties similar to the Seiberg- Wit...
Techniques for solving Boolean equation systems
Keinänen, Misa
2006-01-01
Boolean equation systems are ordered sequences of Boolean equations decorated with least and greatest fixpoint operators. Boolean equation systems provide a useful framework for formal verification because various specification and verification problems, for instance, μ-calculus model checking can be represented as the problem of solving Boolean equation systems. The general problem of solving a Boolean equation system is a computationally hard task, and no polynomial time solution technique ...
The Pauli equation in scale relativity
Celerier, Marie-Noelle; Nottale, Laurent
2006-01-01
In standard quantum mechanics, it is not possible to directly extend the Schrodinger equation to spinors, so the Pauli equation must be derived from the Dirac equation by taking its non-relativistic limit. Hence, it predicts the existence of an intrinsic magnetic moment for the electron and gives its correct value. In the scale relativity framework, the Schrodinger, Klein-Gordon and Dirac equations have been derived from first principles as geodesics equations of a non-differentiable and cont...
Integrable (2k)-Dimensional Hitchin Equations
Ward, R S
2016-01-01
This letter describes a completely-integrable system of Yang-Mills-Higgs equations which generalizes the Hitchin equations on a Riemann surface to arbitrary k-dimensional complex manifolds. The system arises as a dimensional reduction of a set of integrable Yang-Mills equations in 4k real dimensions. Our integrable system implies other generalizations such as the Simpson equations and the non-abelian Seiberg-Witten equations. Some simple solutions in the k=2 case are described.
Algebrization of Nonautonomous Differential Equations
Directory of Open Access Journals (Sweden)
María Aracelia Alcorta-García
2015-01-01
Full Text Available Given a planar system of nonautonomous ordinary differential equations, dw/dt=F(t,w, conditions are given for the existence of an associative commutative unital algebra A with unit e and a function H:Ω⊂R2×R2→R2 on an open set Ω such that F(t,w=H(te,w and the maps H1(τ=H(τ,ξ and H2(ξ=H(τ,ξ are Lorch differentiable with respect to A for all (τ,ξ∈Ω, where τ and ξ represent variables in A. Under these conditions the solutions ξ(τ of the differential equation dξ/dτ=H(τ,ξ over A define solutions (x(t,y(t=ξ(te of the planar system.
Decoherent Histories and Hydrodynamic Equations
Halliwell, J J
1998-01-01
For a system consisting of a large collection of particles, a set of variables that will generally become effectively classical are the local densities (number, momentum, energy). That is, in the context of the decoherent histories approach to quantum theory, it is expected that histories of these variables will be approximately decoherent, and that their probabilites will be strongly peaked about hydrodynamic equations. This possibility is explored for the case of the diffusion of the number density of a dilute concentration of foreign particles in a fluid. It is shown that, for certain physically reasonable initial states, the probabilities for histories of number density are strongly peaked about evolution according to the diffusion equation. Decoherence of these histories is also shown for a class of initial states which includes non-trivial superpositions of number density. Histories of phase space densities are also discussed. The case of histories of number, momentum and energy density for more general...
Power equations in endurance sports.
van Ingen Schenau, G J; Cavanagh, P R
1990-01-01
This paper attempts to clarify the formulation of power equations applicable to a variety of endurance activities. An accurate accounting of the relationship between the metabolic power input and the mechanical power output is still elusive, due to such issues as storage and recovery of strain energy and the differing energy costs of concentric and eccentric muscle actions. Nevertheless, an instantaneous approach is presented which is based upon the application of conventional Newtonian mechanics to a rigid segment model of the body, and does not contain assumptions regarding the exact nature of segmental interactions--such as energy transfer, etc. The application of the equation to running, cycling, speed skating, swimming and rowing is discussed and definitions of power, efficiency, and economy are presented.
Differential Equations of Ideal Memristors
Directory of Open Access Journals (Sweden)
Z. Biolek
2015-06-01
Full Text Available Ideal memristor is a resistor with a memory, which adds dynamics to its behavior. The most usual characteristics describing this dynamics are the constitutive relation (i.e. the relation between flux and charge, or Parameter-vs-state- map (PSM, mostly represented by the memristance-to-charge dependence. One of the so far unheeded tools for memristor description is its differential equation (DEM, composed exclusively of instantaneous values of voltage, current, and their derivatives. The article derives a general form of DEM that holds for any ideal memristor and shows that it is always a nonlinear equation of the first order; the PSM forms are found for memristors which are governed by DEMs of the Bernoulli and the Riccati types; a classification of memristors according to the type of their dynamics with respect to voltage and current is carried out.
Nielsen number and differential equations
Directory of Open Access Journals (Sweden)
Jan Andres
2005-06-01
Full Text Available In reply to a problem of Jean Leray (application of the Nielsen theory to differential equations, two main approaches are presented. The first is via PoincarÃƒÂ©'s translation operator, while the second one is based on the Hammerstein-type solution operator. The applicability of various Nielsen theories is discussed with respect to several sorts of differential equations and inclusions. Links with the Sharkovskii-like theorems (a finite number of periodic solutions imply infinitely many subharmonics are indicated, jointly with some further consequences like the nontrivial RÃŽÂ´-structure of solutions of initial value problems. Some illustrating examples are supplied and open problems are formulated.
The equations icons of knowledge
Bais, Sander
2005-01-01
For thousands of years mankind has tried to understand nature. Exploring the world on all scales with instruments of ever more ingenuity, we have been able to unravel some of the great mysteries that surround us. While collecting an overwhelming multitude of observational facts, we discovered fundamental laws that govern the structure and evolution of physical reality. We know that nature speaks to us in the language of mathematics. In this language most of our basic understanding of the physical world can be expressed in an unambiguous and concise way. The most artificial language turns out to be the most natural of all. The laws of nature correspond to equations. These equations are the icons of knowledge that mark crucial turning points in our thinking about the world we happen to live in. They form the symbolic representation of most of what we know, and as such constitute an important and robust part of our culture.
Nonlocal higher order evolution equations
Rossi, Julio D.
2010-06-01
In this article, we study the asymptotic behaviour of solutions to the nonlocal operator ut(x, t)1/4(-1)n-1 (J*Id -1)n (u(x, t)), x ∈ ℝN, which is the nonlocal analogous to the higher order local evolution equation vt(-1)n-1(Δ)nv. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity. © 2010 Taylor & Francis.
Sensitivity for the Smoluchowski equation
Energy Technology Data Exchange (ETDEWEB)
Bailleul, I F, E-mail: i.bailleul@statslab.cam.ac.uk [Statistical Laboratory, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB (United Kingdom)
2011-06-17
This paper investigates the question of sensitivity of the solutions {mu}{sup {lambda}}{sub t} of the Smoluchowski equation on R{sub +}* with respect to the parameters {lambda} in the interaction kernel K{sup {lambda}}. It is proved that {mu}{sup {lambda}}{sub t} is a C{sup 1} function of (t, {lambda}) with values in a good space of measures under the hypotheses K{sup {lambda}}(x, y) {<=} {psi}(x) {psi}(y), for some sub-linear function {psi}, and {integral}{psi}{sup 4+{epsilon}}(x) {mu}{sub 0}(dx) < {infinity}, and that the derivative is the unique solution of a related equation.
Eigenwavelets of the Wave equation
Kaiser, Gerald
2004-01-01
We study a class of localized solutions of the wave equation, called eigenwavelets, obtained by extending its fundamental solutions to complex spacetime in the sense of hyperfunctions. The imaginary spacetime variables y, which form a timelike vector, act as scale parameters generalizing the scale variable of wavelets in one dimension. They determine the shape of the wavelets in spacetime, making them pulsed beams that can be focused as tightly as desired around a single ray by letting y appr...
Handbook of structural equation modeling
Hoyle, Rick H
2012-01-01
The first comprehensive structural equation modeling (SEM) handbook, this accessible volume presents both the mechanics of SEM and specific SEM strategies and applications. The editor, contributors, and editorial advisory board are leading methodologists who have organized the book to move from simpler material to more statistically complex modeling approaches. Sections cover the foundations of SEM; statistical underpinnings, from assumptions to model modifications; steps in implementation, from data preparation through writing the SEM report; and basic and advanced applications, inclu
Generalized bootstrap for estimating equations
Chatterjee, Snigdhansu; Bose, Arup
2005-01-01
We introduce a generalized bootstrap technique for estimators obtained by solving estimating equations. Some special cases of this generalized bootstrap are the classical bootstrap of Efron, the delete-d jackknife and variations of the Bayesian bootstrap. The use of the proposed technique is discussed in some examples. Distributional consistency of the method is established and an asymptotic representation of the resampling variance estimator is obtained.
Equation of State Project Overview
Energy Technology Data Exchange (ETDEWEB)
Crockett, Scott [Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
2015-09-11
A general overview of the Equation of State (EOS) Project will be presented. The goal is to provide the audience with an introduction of what our more advanced methods entail (DFT, QMD, etc.. ) and how these models are being utilized to better constrain the thermodynamic models. These models substantially reduce our regions of interpolation between the various thermodynamic limits. I will also present a variety example of recent EOS work.
Instantaneous Bethe-Salpeter equation
International Nuclear Information System (INIS)
We present a systematic algebraic and numerical investigation of the instantaneous Beth-Salpeter equation. Emphasis is placed on confining interaction kernels of the Lorentz scalar, time component vector, and full vector-types. We explore the stability of the solutions and Regge behavior for each of these interactions, and conclude that only time component vector confinement leads to normal Regge structure and stable solutions for all quark masses
Operator equations and invariant subspaces
Directory of Open Access Journals (Sweden)
Valentin Matache
1994-05-01
Full Text Available Banach space operators acting on some fixed space X are considered. If two such operators A and B verify the condition A2=B2 and if A has nontrivial hyperinvariant subspaces, then B has nontrivial invariant subspaces. If A and B commute and satisfy a special type of functional equation, and if A is not a scalar multiple of the identity, the author proves that if A has nontrivial invariant subspaces, then so does B.
Differential equations in airplane mechanics
Carleman, M T
1922-01-01
In the following report, we will first draw some conclusions of purely theoretical interest, from the general equations of motion. At the end, we will consider the motion of an airplane, with the engine dead and with the assumption that the angle of attack remains constant. Thus we arrive at a simple result, which can be rendered practically utilizable for determining the trajectory of an airplane descending at a constant steering angle.
Wave equations in higher dimensions
Dong, Shi-Hai
2011-01-01
Higher dimensional theories have attracted much attention because they make it possible to reduce much of physics in a concise, elegant fashion that unifies the two great theories of the 20th century: Quantum Theory and Relativity. This book provides an elementary description of quantum wave equations in higher dimensions at an advanced level so as to put all current mathematical and physical concepts and techniques at the reader’s disposal. A comprehensive description of quantum wave equations in higher dimensions and their broad range of applications in quantum mechanics is provided, which complements the traditional coverage found in the existing quantum mechanics textbooks and gives scientists a fresh outlook on quantum systems in all branches of physics. In Parts I and II the basic properties of the SO(n) group are reviewed and basic theories and techniques related to wave equations in higher dimensions are introduced. Parts III and IV cover important quantum systems in the framework of non-relativisti...
Effective Schroedinger equations on submanifolds
Energy Technology Data Exchange (ETDEWEB)
Wachsmuth, Jakob
2010-02-11
In this thesis the time dependent Schroedinger equation is considered on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When the potential is scaled in the directions normal to C by a small parameter epsilon, the solutions concentrate in an epsilon-neighborhood of the submanifold. An effective Schroedinger equation on the submanifold C is derived and it is shown that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order {epsilon}{sup 3} vertical stroke t vertical stroke at time t. Furthermore, it is proved that, under reasonable conditions, the eigenvalues of the corresponding Hamiltonians below a certain energy coincide upto errors of order {epsilon}{sup 3}. These results holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. The most important consequence of this thesis is that now constraining potentials that change their shape along the submanifold can be treated, which is the typical situation in applications like molecular dynamics and quantum waveguides.
Torsion Effects and LLG Equation
Ferreira, Cristine N; Neto, J A Helayël
2016-01-01
Based on the non-relativistic regime of the Dirac equation coupled to a torsion pseudo-vector, we study the dynamics of magnetization and how it is affected by the presence of torsion. We consider that torsion interacting terms in Dirac equation appear in two ways one of these is thhrough the covariant derivative considering the spin connection and gauge magnetic field and the other is through a non-minimal spin torsion coupling. We show within this framework, that it is possible to obtain the most general Landau, Lifshitz and Gilbert (LLG) equation including the torsion effects, where we refer to torsion as a geometric field playing an important role in the spin coupling process. We show that the torsion terms can give us two important landscapes in the magnetization dynamics: one of them related with damping and the other related with the screw dislocation that give us a global effect like a helix damping sharped. These terms are responsible for changes in the magnetization precession dynamics.
ADVANCED WAVE-EQUATION MIGRATION
Energy Technology Data Exchange (ETDEWEB)
L. HUANG; M. C. FEHLER
2000-12-01
Wave-equation migration methods can more accurately account for complex wave phenomena than ray-tracing-based Kirchhoff methods that are based on the high-frequency asymptotic approximation of waves. With steadily increasing speed of massively parallel computers, wave-equation migration methods are becoming more and more feasible and attractive for imaging complex 3D structures. We present an overview of several efficient and accurate wave-equation-based migration methods that we have recently developed. The methods are implemented in the frequency-space and frequency-wavenumber domains and hence they are called dual-domain methods. In the methods, we make use of different approximate solutions of the scalar-wave equation in heterogeneous media to recursively downward continue wavefields. The approximations used within each extrapolation interval include the Born, quasi-Born, and Rytov approximations. In one of our dual-domain methods, we use an optimized expansion of the square-root operator in the one-way wave equation to minimize the phase error for a given model. This leads to a globally optimized Fourier finite-difference method that is a hybrid split-step Fourier and finite-difference scheme. Migration examples demonstrate that our dual-domain migration methods provide more accurate images than those obtained using the split-step Fourier scheme. The Born-based, quasi-Born-based, and Rytov-based methods are suitable for imaging complex structures whose lateral variations are moderate, such as the Marmousi model. For this model, the computational cost of the Born-based method is almost the same as the split-step Fourier scheme, while other methods takes approximately 15-50% more computational time. The globally optimized Fourier finite-difference method significantly improves the accuracy of the split-step Fourier method for imaging structures having strong lateral velocity variations, such as the SEG/EAGE salt model, at an approximately 30% greater
a Multiple Riccati Equations Rational-Exponent Method and its Application to Whitham-Broer Equation
Liu, Qing; Wang, Zi-Hua; Jia, Dong-Li
2013-03-01
According to two dependent solutions to a generalized Riccati equation together with the equation itself, a multiple Riccati equations rational-exponent method is proposed and applied to Whitham-Broer-Kaup equation. It shows that this method is a more concise and efficient approach and can uniformly derive many types of combined solutions to nonlinear partial differential equations.
Gibbs adsorption and the compressibility equation
International Nuclear Information System (INIS)
A new approach for deriving the equation of state is developed. It is shown that the integral in the compressibility equation is identical to the isotherm for Gibbs adsorption in radial coordinates. The Henry, Langmuir, and Frumkin adsorption isotherms are converted into equations of state. It is shown that using Henry's law gives an expression for the second virial coefficient that is identical to the result from statistical mechanics. Using the Langmuir isotherm leads to a new analytic expression for the hard-sphere equation of state which can be explicit in either pressure or density. The Frumkin isotherm results in a new equation of state for the square-well potential fluid. Conversely, new adsorption isotherms can be derived from equations of state using the compressibility equation. It is shown that the van der Waals equation gives an adsorption isotherm equation that describes both polymolecular adsorption and the unusual adsorption behavior observed for supercritical fluids. copyright 1995 American Institute of Physics
Kinetic equations for an unstable plasma
International Nuclear Information System (INIS)
In this work, we establish the plasma kinetic equations starting from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations. We demonstrate that relations existing between correlation functions may help to justify the truncation of the hierarchy. Then we obtain the kinetic equations of a stable or unstable plasma. They do not reduce to an equation for the one-body distribution function, but generally involve two coupled equations for the one-body distribution function and the spectral density of the fluctuating electric field. We study limiting cases where the Balescu-Lenard equation, the quasi-linear theory, the Pines-Schrieffer equations and the equations of weak turbulence in the random phase approximation are recovered. At last we generalise the H-theorem for the system of equations and we define conditions for irreversible behaviour. (authors)
Transport Properties of the Universal Quantum Equation
Institute of Scientific and Technical Information of China (English)
A.I.Arbab
2012-01-01
The universal quantum equation (UQE) is found to describe the transport properties of the quantum particles.This equation describes a wave equation interacting with constant scalar and vector potentials propagating in spacetime.A new transformation that sends the Schr(o)dinger equation with a potential energy V =-1/2mc2 to Dirac's equation is proposed.The Cattaneo telegraph equation as well as a one-dimensional UQE are compatible with our recently proposed generalized continuity equations.Furthermore,a new wave equation resulted from the invariance of the UQE under the post-Galilean transformations is derived.This equation is found to govern a Klein Gordon's particle interacting with a photon-like vector field (ether) whose magnitude is proportional to the particle's mass.
On Reducing a System of Equations to a Single Equation
DEFF Research Database (Denmark)
Frandsen, G.S.; Shparlinski, I.E.
2004-01-01
For a system of polynomial equations over Q;p; we present an efficient construction of a single polynomial of quite small degree whose zero set over Q;p; coincides with the zero set over Q;p; of the original system. We also show that the polynomial has some other attractive features such as low a...... additive and straight-line complexity.The proof is based on a link established here between the above problem and some recent number theoretic result about zeros of p-adic forms....
From Newton's Equation to Fractional Diffusion and Wave Equations
Directory of Open Access Journals (Sweden)
Vázquez Luis
2011-01-01
Full Text Available Fractional calculus represents a natural instrument to model nonlocal (or long-range dependence phenomena either in space or time. The processes that involve different space and time scales appear in a wide range of contexts, from physics and chemistry to biology and engineering. In many of these problems, the dynamics of the system can be formulated in terms of fractional differential equations which include the nonlocal effects either in space or time. We give a brief, nonexhaustive, panoramic view of the mathematical tools associated with fractional calculus as well as a description of some fields where either it is applied or could be potentially applied.
Partial differential equations of mathematical physics and integral equations
Guenther, Ronald B
1996-01-01
This book was written to help mathematics students and those in the physical sciences learn modern mathematical techniques for setting up and analyzing problems. The mathematics used is rigorous, but not overwhelming, while the authors carefully model physical situations, emphasizing feedback among a beginning model, physical experiments, mathematical predictions, and the subsequent refinement and reevaluation of the physical model itself. Chapter 1 begins with a discussion of various physical problems and equations that play a central role in applications. The following chapters take up the t
Handbook of differential equations stationary partial differential equations
Chipot, Michel
2006-01-01
This handbook is volume III in a series devoted to stationary partial differential quations. Similarly as volumes I and II, it is a collection of self contained state-of-the-art surveys written by well known experts in the field. The topics covered by this handbook include singular and higher order equations, problems near critically, problems with anisotropic nonlinearities, dam problem, T-convergence and Schauder-type estimates. These surveys will be useful for both beginners and experts and speed up the progress of corresponding (rapidly developing and fascinating) areas of mathematics. Ke
Using fundamental equations to describe basic phenomena
DEFF Research Database (Denmark)
Jakobsen, Arne; Rasmussen, Bjarne D.
1999-01-01
When the fundamental thermodynamic balance equations (mass, energy, and momentum) are used to describe the processes in a simple refrigeration system, then one finds that the resulting equation system will have a degree of freedom equal to one. Further investigations reveal that it is the equation...... and subcooling are introduced. Since the degree of freedom was equal to one, using both the superheat and subcooling require that one of the fundamental equations must be omitted from the equation system.The main purpose of the paper is to clarify the relation between the fundamental balance equations...
Partial differential equations for scientists and engineers
Farlow, Stanley J
1993-01-01
Most physical phenomena, whether in the domain of fluid dynamics, electricity, magnetism, mechanics, optics, or heat flow, can be described in general by partial differential equations. Indeed, such equations are crucial to mathematical physics. Although simplifications can be made that reduce these equations to ordinary differential equations, nevertheless the complete description of physical systems resides in the general area of partial differential equations.This highly useful text shows the reader how to formulate a partial differential equation from the physical problem (constructing th
Mathematical physics with partial differential equations
Kirkwood, James
2011-01-01
Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field - the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform
International Workshop on Elliptic and Parabolic Equations
Schrohe, Elmar; Seiler, Jörg; Walker, Christoph
2015-01-01
This volume covers the latest research on elliptic and parabolic equations and originates from the international Workshop on Elliptic and Parabolic Equations, held September 10-12, 2013 at the Leibniz Universität Hannover. It represents a collection of refereed research papers and survey articles written by eminent scientist on advances in different fields of elliptic and parabolic partial differential equations, including singular Riemannian manifolds, spectral analysis on manifolds, nonlinear dispersive equations, Brownian motion and kernel estimates, Euler equations, porous medium type equations, pseudodifferential calculus, free boundary problems, and bifurcation analysis.
EQUATIONS FOR GRAIN SIZE DISTRIBUTION CURVE
Institute of Scientific and Technical Information of China (English)
Prabhata K.SWAMEE; Nimisha SWAMEE
2004-01-01
The grain size distribution of particulate material is of particular interest in the field of sediment transport. The size distribution is described by various equations, however no equation is flexible enough to satisfy the grain size distribution data faithfully. Presented herein are the equations for unimodal and multimodal grain size distribution curves. A graphical method has been given to evaluate the parameters involved in these equations. The size distribution equation can be used to estimate many properties of sediment sample like number of sediment particles, surface area of the particles and hydraulic conductivity. It is hoped that the equations will find many applications in studying sedimentation processes.
Savoye, Philippe
2009-01-01
In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.
PICARD ITERATION FOR NONSMOOTH EQUATIONS
Institute of Scientific and Technical Information of China (English)
Song-bai Sheng; Hui-fu Xu
2001-01-01
This paper presents an analysis of the generalized Newton method, approximate Newton methods, and splitting methods for solving nonsmooth equations from Picard iteration viewpoint. It is proved that the radius of the weak Jacobian (RGJ) of Picard iteration function is equal to its least Lipschitz constant. Linear convergence or superlinear convergence results can be obtained provided that RGJ of the Picard iteration function at a solution point is less than one or equal to zero. As for applications, it is pointed out that the approximate Newton methods, the generalized Newton method for piecewise C1problems and splitting methods can be explained uniformly with the same viewpoint.
Advanced lab on Fresnel equations
Petrova-Mayor, Anna; Gimbal, Scott
2015-11-01
This experimental and theoretical exercise is designed to promote students' understanding of polarization and thin-film coatings for the practical case of a scanning protected-metal coated mirror. We present results obtained with a laboratory scanner and a polarimeter and propose an affordable and student-friendly experimental arrangement for the undergraduate laboratory. This experiment will allow students to apply basic knowledge of the polarization of light and thin-film coatings, develop hands-on skills with the use of phase retarders, apply the Fresnel equations for metallic coating with complex index of refraction, and compute the polarization state of the reflected light.
Ho, Choon-Lin; Hosotani, Yutaka
Starting from the quantum field theory of nonrelativistic matter on a torus interacting with Chern-Simons gauge fields, we derive the Schrödinger equation for an anyon system. The nonintegrable phases of the Wilson line integrals on a torus play an essential role. In addition to generating degenerate vacua, they enter in the definition of a many-body Schrödinger wave function in quantum mechanics, which can be defined as a regular function of the coordinates of anyons. It obeys a non-Abelian representation of the braid group algebra, being related to Einarsson’s wave function by a singular gauge transformation.
BMN correlators by loop equations
International Nuclear Information System (INIS)
In the BMN approach to N=4 SYM a large class of correlators of interest are expressible in terms of expectation values of traces of words in a zero-dimensional gaussian complex matrix model. We develop a loop-equation based, analytic strategy for evaluating such expectation values to any order in the genus expansion. We reproduce the expectation values which were needed for the calculation of the one-loop, genus one correction to the anomalous dimension of BMN-operators and which were earlier obtained by combinatorial means. Furthermore, we present the expectation values needed for the calculation of the one-loop, genus two correction. (author)
Experimental determination of circuit equations
Shulman, Jason; Widjaja, Matthew; Gunaratne, Gemunu H
2013-01-01
Kirchhoff's laws offer a general, straightforward approach to circuit analysis. Unfortunately, use of the laws becomes impractical for all but the simplest of circuits. This work presents a novel method of analyzing direct current resistor circuits. It is based on an approach developed to model complex networks, making it appropriate for use on large, complicated circuits. It is unique in that it is not an analytic method. It is based on experiment, yet the approach produces the same circuit equations obtained by more traditional means.
Directory of Open Access Journals (Sweden)
Sérgio N. Duarte
2001-05-01
importance in the calculation of the drain spacing. Several equations are suggested for the calculation of this spacing. However, there is not a single equation that adapts well to all the situations found in the field. This work was conducted at the Drainage Laboratory belonging to the Rural Engineering Department of the Escola Superior de Agricultura "Luiz de Queiroz"/USP, Brazil, using a reduced hydrologic model filled with sand, with the objective of defining the most appropriate equation in two different situations: when the drains are installed on or above an impermeable layer. Tests of flow (recharge and hydraulic head (level of the water-table under steady state conditions allowed the determination of the hydraulic conductivity using the Ernst formula. In a second stage, measuring the volume of water drained from the soil, the drenable porosity were determined. Later on, measuring the flow and water-table depth under conditions of non permanent state, the measured data were compared with those estimated by the formulas of Boussinesq-Glover, Schilfgaarde, Glover-Dumm, Terzidis, Hammad, Kirkham and Glover. The results indicated that the best spacing formula for drains to be used under conditions where the same are located over the impermeable layer is that of Boussinesq-Glover. On the other hand, when drains are located above the impermeable layer the Schilfgaarde equation followed by the Glover-Dumm equation gave the better results.
On oscillatory solutions of certain difference equations
Directory of Open Access Journals (Sweden)
Grzegorz Grzegorczyk
2006-01-01
Full Text Available Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.
On oscillatory solutions of certain difference equations
Grzegorz Grzegorczyk; Jarosław Werbowski
2006-01-01
Some difference equations with deviating arguments are discussed in the context of the oscillation problem. The aim of this paper is to present the sufficient conditions for oscillation of solutions of the equations discussed.
The Spin-2 Equation on Minkowski Background
Beyer, Florian; Frauendiener, Jörg; Whale, Ben
2014-01-01
The linearised general conformal field equations in their first and second order form are used to study the behaviour of the spin-2 zero-rest-mass equation on Minkowski background in the vicinity of space-like infinity.
Hydrodynamic Equations for Microscopic Phase Densities
Gerasimenko, V. I.; Shtyk, V. O.; Zagorodny, A. G.
2009-01-01
The evolution equations for the generalized microscopic phase densities are introduced. The evolution equations of average values of microscopic phase densities are derived and a solution of the initial-value problem of the obtained hydrodynamic type hierarchy is constructed.
Introduction to linear algebra and differential equations
Dettman, John W
1986-01-01
Excellent introductory text focuses on complex numbers, determinants, orthonormal bases, symmetric and hermitian matrices, first order non-linear equations, linear differential equations, Laplace transforms, Bessel functions, more. Includes 48 black-and-white illustrations. Exercises with solutions. Index.
Exact Solutions to Short Pulse Equation
Institute of Scientific and Technical Information of China (English)
FU Zun-Tao; ZHENG Ming-Hua; LIU Shi-Kuo
2009-01-01
In this paper, dependent and independent variable transformations are introduced to solve the short pulse equation. It is shown that different kinds of solutions can be obtained to the short pulse equation.
Exact Solutions to Degasperis-Procesi Equation
Institute of Scientific and Technical Information of China (English)
ZHANG Lin-Na; FU Zun-Tao; LIU Shi-Kuo
2008-01-01
In this paper,dependent and independent variable transformations are introduced to solve the Degasperis-Procesi equation.It is shown that different kinds of solutions can be obtained to the Degasperis-Procesi equation.
Simultaneous Independent Linear Equations and Goldbach Conjecture
Linggen, Song
2007-01-01
It was verified that if Goldbach conjecture was a fault, the number of simultaneous independent linear equations educed from this assumption would be unreasonably at least one more than the number of unknowns involved in these equations.
A new class of variational equation problems
Institute of Scientific and Technical Information of China (English)
无
2003-01-01
Applying an analysis method to a group of multivariable equations, a new class of variational equations are proved. This method is more concise and more direct than the others. This result can be applied to some stochastic control models.
Functional differential equations of third order
Directory of Open Access Journals (Sweden)
Tuncay Candan
2005-04-01
Full Text Available In this paper, we consider the third-order neutral functional differential equation with distributed deviating arguments. We give sufficient conditions for the oscillatory behavior of this functional differential equation.
OSCILLATION CRITERIA FOR FORCED SUPERLINEAR DIFFERENCE EQUATIONS
Institute of Scientific and Technical Information of China (English)
无
2008-01-01
Using Riccati transformation techniques,some oscillation criteria for the forced second-order superlinear difference equations are established.These criteria are dis- crete analogues of the criteria for differential equations proposed by Yan.
An axiomatic approach to Maxwell's equations
Heras, José A
2016-01-01
This paper suggests an axiomatic approach to Maxwell's equations. The basis of this approach is a theorem formulated for two sets of functions localized in space and time. If each set satisfies a continuity equation then the theorem provides an integral representation for each function. A corollary of this theorem yields Maxwell's equations with magnetic monopoles. It is pointed out that the causality principle and the conservation of electric and magnetic charges are the most fundamental physical axioms underlying these equations. Another application of the corollary yields Maxwell's equations in material media. The theorem is also formulated in the Minkowski space-time and applied to obtain the covariant form of Maxwell's equations with magnetic monopoles and the covariant form of Maxwell's equations in material media. The approach makes use of the infinite-space Green function of the wave equation and is therefore suitable for an advanced course in electrodynamics.
Dirac and Maxwell equations in Split Octonions
Beradze, Revaz
2016-01-01
The split octonionic form of Dirac and Maxwell equations are found. In contrast with the previous attempts these equations are derived from the octonionic analyticity condition and also we use different basis of the 8-dimensional space of split octonions.
Linear superposition solutions to nonlinear wave equations
Institute of Scientific and Technical Information of China (English)
Liu Yu
2012-01-01
The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.
Fractional Complex Transform for Fractional Differential Equations
Li, Zheng-Biao; He, Ji-Huan
2010-01-01
Fractional complex transform is proposed to convert fractional differential equations into ordinary differential equations, so that all analytical methods devoted to advanced calculus can be easily applied to fractional calculus. Two examples are given.
Solving Equations of Multibody Dynamics
Jain, Abhinandan; Lim, Christopher
2007-01-01
Darts++ is a computer program for solving the equations of motion of a multibody system or of a multibody model of a dynamic system. It is intended especially for use in dynamical simulations performed in designing and analyzing, and developing software for the control of, complex mechanical systems. Darts++ is based on the Spatial-Operator- Algebra formulation for multibody dynamics. This software reads a description of a multibody system from a model data file, then constructs and implements an efficient algorithm that solves the dynamical equations of the system. The efficiency and, hence, the computational speed is sufficient to make Darts++ suitable for use in realtime closed-loop simulations. Darts++ features an object-oriented software architecture that enables reconfiguration of system topology at run time; in contrast, in related prior software, system topology is fixed during initialization. Darts++ provides an interface to scripting languages, including Tcl and Python, that enable the user to configure and interact with simulation objects at run time.
Classical equations for quantum systems
Energy Technology Data Exchange (ETDEWEB)
Gell-Mann, M. (Theoretical Astrophysics Group (T-6), Los Alamos National Laboratory, Los Alamos, New Mexico 87545) (United States) (Santa Fe Institute, 1660 Old Pecos Trail, Santa Fe, New Mexico 87501); Hartle, J.B. (Department of Physics, University of California enSanta Barbara, Santa Barbara, (California) 93106)
1993-04-15
The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that [ital decohere], which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models.
Quantization of Equations of Motion
Directory of Open Access Journals (Sweden)
D. Kochan
2007-01-01
Full Text Available The Classical Newton-Lagrange equations of motion represent the fundamental physical law of mechanics. Their traditional Lagrangian and/or Hamiltonian precursors when available are essential in the context of quantization. However, there are situations that lack Lagrangian and/or Hamiltonian settings. This paper discusses a description of classical dynamics and presents some irresponsible speculations about its quantization by introducing a certain canonical two-form ?. By its construction ? embodies kinetic energy and forces acting within the system (not their potential. A new type of variational principle employing differential two-form ? is introduced. Variation is performed over “umbilical surfaces“ instead of system histories. It provides correct Newton-Lagrange equations of motion. The quantization is inspired by the Feynman path integral approach. The quintessence is to rearrange it into an “umbilical world-sheet“ functional integral in accordance with the proposed variational principle. In the case of potential-generated forces, the new approach reduces to the standard quantum mechanics. As an example, Quantum Mechanics with friction is analyzed in detail.
Partial differential equations of mathematical physics
Sobolev, S L
1964-01-01
Partial Differential Equations of Mathematical Physics emphasizes the study of second-order partial differential equations of mathematical physics, which is deemed as the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. The book discusses in detail a wide spectrum of topics related to partial differential equations, such as the theories of sets and of Lebesgue integration, integral equations, Green's function, and the proof of the Fourier method. Theoretical physicists, experimental physicists, mathematicians engaged in pure and applied math
BOUNDARY CONTROL OF MKDV-BURGERS EQUATION
Institute of Scientific and Technical Information of China (English)
TIAN Li-xin; ZHAO Zhi-feng; WANG Jing-feng
2006-01-01
The boundary control of MKdV-Burgers equation was considered by feedback control on the domain [0,1]. The existence of the solution of MKdV-Burgers equation with the feedback control law was proved. On the base, priori estimates for the solution was given. At last, the existence of the weak solution of MKdV-Burgers equation was proved and the global-exponential and asymptotic stability of the solution of MKdV-Burgers equation was given.
Nonlinear SCHRÖDINGER-PAULI Equations
Ng, Wei Khim; Parwani, Rajesh R.
2011-11-01
We obtain novel nonlinear Schrüdinger-Pauli equations through a formal non-relativistic limit of appropriately constructed nonlinear Dirac equations. This procedure automatically provides a physical regularisation of potential singularities brought forward by the nonlinear terms and suggests how to regularise previous equations studied in the literature. The enhancement of contributions coming from the regularised singularities suggests that the obtained equations might be useful for future precision tests of quantum nonlinearity.
Weierstrass solutions for dissipative BBM equation
Mancas, Stefan C.; Spradlin, Greg; Khanal, Harihar
2013-01-01
In this paper the effect of a small dissipation on waves is included to find exact solutions to the modified BBM equation. Using Lyapunov functions and dynamical systems theory, we prove that when viscosity is added to the BBM equation, in certain regions there still exist bounded traveling wave solutions in the form of solitary waves, periodic, and elliptic functions. By using the canonical form of Abel equation, the polynomial Appell invariant make the equation integrable in terms of Weiers...
A Note on Indefinite Stochastic Riccati Equations
Qian, Zhongmin
2012-01-01
An indefinite stochastic Riccati Equation is a matrix-valued, highly nonlinear backward stochastic differential equation together with an algebraic, matrix positive definiteness constraint. We introduce a new approach to solve a class of such equations (including the existence of solutions) driven by one-dimensional Brownian motion. The idea is to replace the original equation by a system of BSDEs (without involving any algebraic constraint) whose existence of solutions automatically enforces the original algebraic constraint to be satisfied.
On a complex differential Riccati equation
Energy Technology Data Exchange (ETDEWEB)
Khmelnytskaya, Kira V; Kravchenko, Vladislav V [Department of Mathematics, CINVESTAV del IPN, Unidad Queretaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, Queretaro, Qro. C.P. 76230 Mexico (Mexico)], E-mail: vkravchenko@qro.cinvestav.mx
2008-02-29
We consider a nonlinear partial differential equation for complex-valued functions which is related to the two-dimensional stationary Schroedinger equation and enjoys many properties similar to those of the ordinary differential Riccati equation such as the famous Euler theorems, the Picard theorem and others. Besides these generalizations of the classical 'one-dimensional' results, we discuss new features of the considered equation including an analogue of the Cauchy integral theorem.
Solutions manual to accompany Ordinary differential equations
Greenberg, Michael D
2014-01-01
Features a balance between theory, proofs, and examples and provides applications across diverse fields of study Ordinary Differential Equations presents a thorough discussion of first-order differential equations and progresses to equations of higher order. The book transitions smoothly from first-order to higher-order equations, allowing readers to develop a complete understanding of the related theory. Featuring diverse and interesting applications from engineering, bioengineering, ecology, and biology, the book anticipates potential difficulties in understanding the various solution steps
Homogenization of ordinary and linear transport equations
Peirone, Roberto
1996-01-01
The homogenization of first order ordinary differential equations in $\\mathbb{R}^N$ and associated linear transport equations are studied. We prove the equivalence between $G$-convergence and strong $G$-convergence for the ordinary equations. We give a sufficient condition, which is also necessary in the autonomous case, for the weak homogenization of the linear transport equations. This condition is satisfied when div$_x f=0$.
Some constant solutions to Zamolodchikov's tetrahedron equations
Hietarinta, Jarmo
1992-01-01
In this letter we present constant solutions to the tetrahedron equations proposed by Zamolodchikov. In general, from a given solution of the Yang-Baxter equation there are two ways to construct solutions to the tetrahedron equation. There are also other kinds of solutions. We present some two-dimensional solutions that were obtained by directly solving the equations using either an upper triangular or Zamolodchikov's ansatz.
Introduction to differential equations with dynamical systems
Campbell, Stephen L
2011-01-01
Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Cam
Onsager equations and time dependent neutron transport
International Nuclear Information System (INIS)
The diffusion of neutrons following an abrupt, localized temperature fluctuation can be conducted in the framework of Onsager-type transport equations. Considering Onsager equations as a generalized Fick's law, time-dependent particle and energy 'generalized diffusion equations' can be obtained. Aim of the present paper is to obtain the time-dependent diffusion Onsager-type equations for the diffusion of neutrons and to apply them to simple trial cases to gain a feeling for their behaviour. (author)
Some Aspects of Extended Kinetic Equation
Directory of Open Access Journals (Sweden)
Dilip Kumar
2015-09-01
Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.
Notes on the infinity Laplace equation
Lindqvist, Peter
2016-01-01
This BCAM SpringerBriefs is a treaty of the Infinity-Laplace Equation, which has inherited many features from the ordinary Laplace Equation, and is based on lectures by the author. The Infinity.Laplace Equation has delightful counterparts to the Dirichlet integral, the mean value property, the Brownian motion, Harnack's inequality, and so on. This "fully non-linear" equation has applications to image processing and to mass transfer problems, and it provides optimal Lipschitz extensions of boundary values.
Symmetry Breaking for Black-Scholes Equations
Institute of Scientific and Technical Information of China (English)
YANG Xuan-Liu; ZHANG Shun-Li; QU Chang-Zheng
2007-01-01
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
Symmetry Breaking for Black-Scholes Equations
Yang, Xuan-Liu; Zhang, Shun-Li; Qu, Chang-Zheng
2007-06-01
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
Symmetry Breaking for Black-Scholes Equations
International Nuclear Information System (INIS)
Black-Scholes equation is used to model stock option pricing. In this paper, optimal systems with one to four parameters of Lie point symmetries for Black-Scholes equation and its extension are obtained. Their symmetry breaking interaction associated with the optimal systems is also studied. As a result, symmetry reductions and corresponding solutions for the resulting equations are obtained.
The Modified Enskog Equation for Mixtures
Beijeren, H. van; Ernst, M.H.
1973-01-01
In a previous paper it was shown that a modified form of the Enskog equation, applied to mixtures of hard spheres, should be considered as the correct extension of the usual Enskog equation to the case of mixtures. The main argument was that the modified Enskog equation leads to linear transport coe
Some Functional Equations Originating from Number Theory
Indian Academy of Sciences (India)
Soon-Mo Jung; Jae-Hyeong Bae
2003-05-01
We will introduce new functional equations (3) and (4) which are strongly related to well-known formulae (1) and (2) of number theory, and investigate the solutions of the equations. Moreover, we will also study some stability problems of those equations.
Errors in equations for galaxy rotation speeds
Nicholson, Kenneth F.
2003-01-01
Shown are the errors and difficulties of the equations used for galaxy rotation speeds in the book "Galactic Dynamics" (Binney and Tremaine). A usable and accurate set of equations is then presented. The new equations allow easy determination of galaxy mass distribution from the rotation profile with no need for dark matter or any knowledge of galaxy surface light.
Differential Galois Theory of Linear Difference Equations
Hardouin, Charlotte; Singer, Michael F.
2008-01-01
We present a Galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations. With this we are able to reprove Hoelder's Theorem that the Gamma function satisfies no polynomial differential equation and are able to give general results that imply, for example, that no differential relationship holds among solutions of certain classes of q-hypergeometric functions.
Difference Galois theory of linear differential equations
Di Vizio, Lucia; Hardouin, Charlotte; Wibmer, Michael
2013-01-01
We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.
The Schrodinger equation and negative energies
Bruce, S
2008-01-01
We present a nonrelativistic wave equation for the electron in (3+1)-dimensions which includes negative-energy eigenstates. We solve this equation for three well-known instances, reobtaining the corresponding Pauli equation (but including negative-energy eigenstates) in each case.
Symmetry properties of fractional diffusion equations
Energy Technology Data Exchange (ETDEWEB)
Gazizov, R K; Kasatkin, A A; Lukashchuk, S Yu [Ufa State Aviation Technical University, Karl Marx strausse 12, Ufa (Russian Federation)], E-mail: gazizov@mail.rb.ru, E-mail: alexei_kasatkin@mail.ru, E-mail: lsu@mail.rb.ru
2009-10-15
In this paper, nonlinear anomalous diffusion equations with time fractional derivatives (Riemann-Liouville and Caputo) of the order of 0-2 are considered. Lie point symmetries of these equations are investigated and compared. Examples of using the obtained symmetries for constructing exact solutions of the equations under consideration are presented.
The Effect of Repeaters on Equating
Kim, HeeKyoung; Kolen, Michael J.
2010-01-01
Test equating might be affected by including in the equating analyses examinees who have taken the test previously. This study evaluated the effect of including such repeaters on Medical College Admission Test (MCAT) equating using a population invariance approach. Three-parameter logistic (3-PL) item response theory (IRT) true score and…
Contact Structures of Partial Differential Equations
Eendebak, P.T.
2007-01-01
We study the geometry of contact structures of partial differential equations. The main classes we study are first order systems of two equations in two independent and two dependent variables and the second order scalar equations in two independent variables. The contact distribution in these two c