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...
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.
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.
Symmetries and conservation laws of lattice Boussinesq equations
International Nuclear Information System (INIS)
Sequences of canonical conservation laws and generalized symmetries for the lattice Boussinesq and the lattice modified Boussinesq systems are successively derived. The interpretation of these symmetries as differential-difference equations leads to corresponding hierarchies of such equations for which conservation laws and Lax pairs are constructed. Finally, using the continuous symmetry reduction approach, an integrable, multidimensionally consistent system of partial differential equations is derived in relation with the lattice modified Boussinesq system. -- Highlights: ► Symmetries and conservation laws for lattice Boussinesq system are constructed. ► Corresponding results for lattice modified Boussinesq system are presented. ► The generating PDE (GPDE) for lattice modified Boussinesq is derived. ► Lax pair and Bäcklund transformation for this GPDE are explicitly given.
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.
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.
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.
Global rough solutions to the cubic nonlinear Boussinesq equation
Farah, Luiz Gustavo; Linares, Felipe
2008-01-01
We prove that the initial value problem (IVP) for the cubic defocusing nonlinear Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^2u)_{xx}=0$ on the real line is globally well-posed in $H^{s}(\\R)$ provided $2/3
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...
On the Cauchy problem for the damped Boussinesq equation
Varlamov, Vladimir
1996-01-01
A classic solution to the Cauchy problem for the damped Boussinesq equation $u_{tt}-2Bu_{txx}=-\\alpha u_{xxxx}+u_{xx}-\\beta(u^2)_{xx}$, $x\\in\\Bbb R^1$, $t>0$, $\\alpha, B=\\text{const}>0$, $\\beta=\\text{const}\\in\\Bbb R^1$, with small initial data is constructed by means of the application of both the spectral and perturbation theories. Large time asymptotics of this solution are obtained. Its main term accounts for two solitons traveling in opposite directions. Each of th...
On spatially periodic solutions of the damped Boussinesq equation
Vladimir V. Varlamov
1997-01-01
A classical solution of the damped Boussinesq equation $$ u_{tt}-2bu_{txx}=-\\alpha u_{xxxx}+u_{xx}+\\beta (u^2)_{xx},\\quad x\\in {\\Bbb R}^1,t>0, $$ with $\\alpha ,b=\\text{const}>0$, $\\beta =\\text{const}\\in{\\Bbb R}^1$, $\\alpha >b^2$, and small initial data is constructed by means of the successive application of the spectral theory and the perturbation one. Its long-time asymptotic representation is obtained which shows that the major term increases linearly with time and the secon...
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...... and practical aspects of a viable and efficient numerical solution. Two Boussinesq-type models have been devised and tested in the course of this project. The first model is customised to the solution of higher-order Boussinesq equations, formulated in terms of the horizontal volume-flux vector. The...
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.; Warburton, T.
2008-01-01
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......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...
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.
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
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...
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.
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.
Single Peak Solitons for the Boussinesq-Like B(2,2 Equation
Directory of Open Access Journals (Sweden)
Lina Zhang
2013-01-01
Full Text Available The nonlinear dispersive Boussinesq-like B(2,2 equation utt+(u2xx−(u2xxxx=0, which exhibits single peak solitons, is investigated. Peakons, cuspons and smooth soliton solutions are obtained by setting the B(2,2 equation under inhomogeneous boundary condition. Asymptotic behavior and numerical simulations are provided for these three types of single peak soliton solutions of the B(2,2 equation.
Single Peak Solitons for the Boussinesq-Like B(2,2) Equation
Lina Zhang; Shumin Li; Aiyong Chen
2013-01-01
The nonlinear dispersive Boussinesq-like B(2,2) equation utt+(u2)xx−(u2)xxxx=0, which exhibits single peak solitons, is investigated. Peakons, cuspons and smooth soliton solutions are obtained by setting the B(2,2) equation under inhomogeneous boundary condition. Asymptotic behavior and numerical simulations are provided for these three types of single peak soliton solutions of the B(2,2) equation.
New superfield extension of Boussinesq and its (x, t) interchanged equation from odd Poisson Bracket
International Nuclear Information System (INIS)
A new superfield extension of the Boussinesq equation and its corresponding (x, t) interchanged variant are deduced from the odd Poisson-Bracket-formalism, which is similar to the antibracket of Batalin and Vilkovisky. In the former case we obtain the equation deduced by Figueroa-O'Farrill et al. from a different approach. In each case we have deduced the bi-Hamiltonian structure and some basic symmetries associated with them. (orig.)
Exact periodic solutions of the sixth-order generalized Boussinesq equation
International Nuclear Information System (INIS)
This paper examines a class of nonlinear sixth-order generalized Boussinesq-like equations (SGBE): utt = uxx + 3(u2)xx + uxxxx + αuxxxxxx, α in R, depending on the positive parameter α. Hirota's bilinear transformation method is applied to the above class of non-integrable equations and exact periodic solutions have been obtained. The results confirmed the well-known nonlinear superposition principle.
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.
Local well-posedness for the Sixth-Order Boussinesq Equation
Farah, Luiz Gustavo
2010-01-01
This work studies the local well-posedness of the initial-value problem for the nonlinear sixth-order Boussinesq equation $u_{tt}=u_{xx}+\\beta u_{xxxx}+u_{xxxxxx}+(u^2)_{xx}$, where $\\beta=\\pm1$. We prove local well-posedness with initial data in non-homogeneous Sobolev spaces $H^s(\\R)$ for negative indices of $s \\in \\R$.
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.
Global attractor for the lattice dynamical system of a nonlinear Boussinesq equation
Directory of Open Access Journals (Sweden)
Ahmed Y. Abdallah
2005-08-01
Full Text Available We will study the lattice dynamical system of a nonlinear Boussinesq equation. Our objective is to explore the existence of the global attractor for the solution semiflow of the introduced lattice system and to investigate its upper semicontinuity with respect to a sequence of finite-dimensional approximate systems. As far as we are aware, our result here is the first concerning the lattice dynamical system corresponding to a differential equation of second order in time variable and fourth order in spatial variable with nonlinearity involving the gradients.
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.
Unstructured nodal DG-FEM solution of high-order Boussinesq-type equations
DEFF Research Database (Denmark)
Engsig-Karup, Allan Peter
2007-01-01
high-order Boussinesq equations. Remarkably, it is demonstrated that the linear eigenspectra of the linearized semi-discrete equation system is bounded and hence the stable time increment is not dictated by the spatial discretization. This is a favorable property for explicit time-integration schemes...... equations constitute a highly complex system of coupled equations which put any numerical method to the test. The main problems that need to be overcome to solve the equations are the treatment of strongly nonlinear convection-type terms and spatially varying coefficient terms; efficient and robust solution...... of the resultant 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...
Exact solutions of (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations
International Nuclear Information System (INIS)
The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space—time translations. Three types of symmetry reduction equations and similar solutions for the (3+1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space—time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system. (general)
Painlevé properties and exact solutions for the high-dimensional Schwartz Boussinesq equation
International Nuclear Information System (INIS)
The usual (1+1)-dimensional Schwartz Boussinesq equation is extended to the (1+1)-dimensional space-time symmetric form and the general (n+1)-dimensional space–time symmetric form. These extensions are Painlevé integrable in the sense that they possess the Painlevé property. The single soliton solutions and the periodic travelling wave solutions for arbitrary dimensional space–time symmetric form are obtained by the Painlevé–Bäcklund transformation. (fluids, plasmas and electric discharges)
Global solutions for the generalized Boussinesq equation in low-order Sobolev spaces
Farah, Luiz Gustavo
2010-01-01
We show that the Cauchy problem for the defocusing generalized Boussinesq equation $u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0$, $k\\geq1$, on the real line is globally well-posed in $H^{s}(\\R)$ for $s>1-({1}/{3k})$. We use the "$I$-method" to define a modification of the energy functional that is "almost conserved" in time. Our result extends the previous one obtained by Farah and Linares (2010 \\textit{J. London Math. Soc.} \\textbf{81} 241-254) when $k=1$.
Global solutions in lower order Sobolev spaces for the generalized Boussinesq equation
Directory of Open Access Journals (Sweden)
Luiz G. Farah
2012-03-01
Full Text Available We show that the Cauchy problem for the defocusing generalized Boussinesq equation $$ u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u_{xx}=0, quad kgeq 1, $$ on the real line is globally well-posed in $H^s(mathbb{R}$ with s>1-(1/(3k. To do this, we use the I-method, introduced by Colliander, Keel, Staffilani, Takaoka and Tao [8,9], to define a modification of the energy functional that is almost conserved in time. Our result extends a previous result obtained by Farah and Linares [16] for the case k=1.
Global solutions in lower order Sobolev spaces for the generalized Boussinesq equation
Farah, Luiz G.; Hongwei Wang
2012-01-01
We show that the Cauchy problem for the defocusing generalized Boussinesq equation $$ u_{tt}-u_{xx}+u_{xxxx}-(|u|^{2k}u)_{xx}=0, quad kgeq 1, $$ on the real line is globally well-posed in $H^s(mathbb{R})$ with s>1-(1/(3k)). To do this, we use the I-method, introduced by Colliander, Keel, Staffilani, Takaoka and Tao [8,9], to define a modification of the energy functional that is almost conserved in time. Our result extends a previous result obtained by Farah and Linares [16] for the...
On Triply Periodic Wave Solutions for （2d-1）-Dimensional Boussinesq Equation
Institute of Scientific and Technical Information of China (English)
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.
Double criticality and the two-way Boussinesq equation in stratified shallow water hydrodynamics
Bridges, Thomas J.; Ratliff, Daniel J.
2016-06-01
Double criticality and its nonlinear implications are considered for stratified N-layer shallow water flows with N = 1, 2, 3. Double criticality arises when the linearization of the steady problem about a uniform flow has a double zero eigenvalue. We find that there are two types of double criticality: non-semisimple (one eigenvector and one generalized eigenvector) and semi-simple (two independent eigenvectors). Using a multiple scales argument, dictated by the type of singularity, it is shown that the weakly nonlinear problem near double criticality is governed by a two-way Boussinesq equation (non-semisimple case) and a coupled Korteweg-de Vries equation (semisimple case). Parameter values and reduced equations are constructed for the examples of two-layer and three-layer stratified shallow water hydrodynamics.
Numerical investigations on the finite time singularity in two-dimensional Boussinesq equations
Yin, Z
2006-01-01
To investigate the finite time singularity in three-dimensional (3D) Euler flows, the simplified model of 3D axisymmetric incompressible fluids (i.e., two-dimensional Boussinesq approximation equations) is studied numerically. The system describes a cap-like hot zone of fluid rising from the bottom, while the edges of the cap lag behind, forming eye-like vortices. The hot liquid is driven by the buoyancy and meanwhile attracted by the vortices, which leads to the singularity-forming mechanism in our simulation. In the previous 2D Boussinesq simulations, the symmetricial initial data is used. However, it is observed that the adoption of symmetry leads to coordinate singularity. Moreover, as demonstrated in this work that the locations of peak values for the vorticity and the temperature gradient becomes far apart as $t$ approaches the predicted blow-up time. This suggests that the symmetry assumption may be unreasonable for searching solution blow-ups. One of the main contributions of this work is to propose a...
Tailleux, Rémi
This paper seeks to illustrate the point that physical inconsistencies between thermodynamics and dynamics usually introduce nonconservative production/destruction terms in the local total energy balance equation in numerical ocean general circulation models (OGCMs). Such terms potentially give rise to undesirable forces and/or diabatic terms in the momentum and thermodynamic equations, respectively, which could explain some of the observed errors in simulated ocean currents and water masses. In this paper, a theoretical framework is developed to provide a practical method to determine such nonconservative terms, which is illustrated in the context of a relatively simple form of the hydrostatic Boussinesq primitive equation used in early versions of OGCMs, for which at least four main potential sources of energy nonconservation are identified; they arise from: (1) the "hanging" kinetic energy dissipation term; (2) assuming potential or conservative temperature to be a conservative quantity; (3) the interaction of the Boussinesq approximation with the parameterizations of turbulent mixing of temperature and salinity; (4) some adiabatic compressibility effects due to the Boussinesq approximation. In practice, OGCMs also possess spurious numerical energy sources and sinks, but they are not explicitly addressed here. Apart from (1), the identified nonconservative energy sources/sinks are not sign definite, allowing for possible widespread cancellation when integrated globally. Locally, however, these terms may be of the same order of magnitude as actual energy conversion terms thought to occur in the oceans. Although the actual impact of these nonconservative energy terms on the overall accuracy and physical realism of the oceans is difficult to ascertain, an important issue is whether they could impact on transient simulations, and on the transition toward different circulation regimes associated with a significant reorganization of the different energy reservoirs
A moist Boussinesq shallow water equations set for testing atmospheric models
International Nuclear Information System (INIS)
The shallow water equations have long been used as an initial test for numerical methods applied to atmospheric models with the test suite of Williamson et al. being used extensively for validating new schemes and assessing their accuracy. However the lack of physics forcing within this simplified framework often requires numerical techniques to be reworked when applied to fully three dimensional models. In this paper a novel two-dimensional shallow water equations system that retains moist processes is derived. This system is derived from three-dimensional Boussinesq approximation of the hydrostatic Euler equations where, unlike the classical shallow water set, we allow the density to vary slightly with temperature. This results in extra (or buoyancy) terms for the momentum equations, through which a two-way moist-physics dynamics feedback is achieved. The temperature and moisture variables are advected as separate tracers with sources that interact with the mean-flow through a simplified yet realistic bulk moist-thermodynamic phase-change model. This moist shallow water system provides a unique tool to assess the usually complex and highly non-linear dynamics–physics interactions in atmospheric models in a simple yet realistic way. The full non-linear shallow water equations are solved numerically on several case studies and the results suggest quite realistic interaction between the dynamics and physics and in particular the generation of cloud and rain. - Highlights: • Novel shallow water equations which retains moist processes are derived from the three-dimensional hydrostatic Boussinesq equations. • The new shallow water set can be seen as a more general one, where the classical equations are a special case of these equations. • This moist shallow water system naturally allows a feedback mechanism from the moist physics increments to the momentum via buoyancy. • Like full models, temperature and moistures are advected as tracers that interact
Dralle, David N.; Boisramé, Gabrielle F. S.; Thompson, Sally E.
2014-11-01
The linearized hillslope Boussinesq equation, introduced by Brutsaert (1994), describes the dynamics of saturated, subsurface flow from hillslopes with shallow, unconfined aquifers. In this paper, we use a new analytical technique to solve the linearized hillslope Boussinesq equation to predict water table dynamics and hillslope discharge to channels. The new solutions extend previous analytical treatments of the linearized hillslope Boussinesq equation to account for the impact of spatiotemporal heterogeneity in water table recharge. The results indicate that the spatial character of recharge may significantly alter both steady state subsurface storage characteristics and the transient hillslope hydrologic response, depending strongly on similarity measures of controls on the subsurface flow dynamics. Additionally, we derive new analytical solutions for the linearized hillslope-storage Boussinesq equation and explore the interaction effects of recharge structure and hillslope morphology on water storage and base flow recession characteristics. A theoretical recession analysis, for example, demonstrates that decreasing the relative amount of downslope recharge has a similar effect as increasing hillslope convergence. In general, the theory suggests that recharge heterogeneity can serve to diminish or enhance the hydrologic impacts of hillslope morphology.
Hybridizable discontinuous Galerkin projection methods for Navier-Stokes and Boussinesq equations
Ueckermann, M. P.; Lermusiaux, P. F. J.
2016-02-01
Schemes for the incompressible Navier-Stokes and Boussinesq equations are formulated and derived combining the novel Hybridizable Discontinuous Galerkin (HDG) method, a projection method, and Implicit-Explicit Runge-Kutta (IMEX-RK) time-integration schemes. We employ an incremental pressure correction and develop the corresponding HDG finite element discretization including consistent edge-space fluxes for the velocity predictor and pressure correction. We then derive the proper forms of the element-local and HDG edge-space final corrections for both velocity and pressure, including the HDG rotational correction. We also find and explain a consistency relation between the HDG stability parameters of the pressure correction and velocity predictor. We discuss and illustrate the effects of the time-splitting error. We then detail how to incorporate the HDG projection method time-split within standard IMEX-RK time-stepping schemes. Our high-order HDG projection schemes are implemented for arbitrary, mixed-element unstructured grids, with both straight-sided and curved meshes. In particular, we provide a quadrature-free integration method for a nodal basis that is consistent with the HDG method. To prevent numerical oscillations, we develop a selective nodal limiting approach. Its applications show that it can stabilize high-order schemes while retaining high-order accuracy in regions where the solution is sufficiently smooth. We perform spatial and temporal convergence studies to evaluate the properties of our integration and selective limiting schemes and to verify that our solvers are properly formulated and implemented. To complete these studies and to illustrate a range of properties for our new schemes, we employ an unsteady tracer advection benchmark, a manufactured solution for the steady diffusion and Stokes equations, and a standard lock-exchange Boussinesq problem.
A moist Boussinesq shallow water equations set for testing atmospheric models
Zerroukat, M.; Allen, T.
2015-06-01
The shallow water equations have long been used as an initial test for numerical methods applied to atmospheric models with the test suite of Williamson et al. [1] being used extensively for validating new schemes and assessing their accuracy. However the lack of physics forcing within this simplified framework often requires numerical techniques to be reworked when applied to fully three dimensional models. In this paper a novel two-dimensional shallow water equations system that retains moist processes is derived. This system is derived from three-dimensional Boussinesq approximation of the hydrostatic Euler equations where, unlike the classical shallow water set, we allow the density to vary slightly with temperature. This results in extra (or buoyancy) terms for the momentum equations, through which a two-way moist-physics dynamics feedback is achieved. The temperature and moisture variables are advected as separate tracers with sources that interact with the mean-flow through a simplified yet realistic bulk moist-thermodynamic phase-change model. This moist shallow water system provides a unique tool to assess the usually complex and highly non-linear dynamics-physics interactions in atmospheric models in a simple yet realistic way. The full non-linear shallow water equations are solved numerically on several case studies and the results suggest quite realistic interaction between the dynamics and physics and in particular the generation of cloud and rain.
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.
The global attractor of the 2D Boussinesq equations with fractional Laplacian in Subcritical case
Huang, Aimin; Huo, Wenru
2015-01-01
We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.
1997-01-01
For the damped Boussinesq equation $u_{tt}-2bu_{txx}= -\\alpha u_{xxxx}+ u_{xx}+\\beta(u^2)_{xx},x\\in(0,\\pi),t > 0;\\alpha,b = const > 0,\\beta = const\\in R^1$ , 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 so...
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.
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...... 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...
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.
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
functions of arbitrary order in space on each element of an unstructured computational domain. A fourth order explicit Runge-Kutta scheme is used to advance the solution in time. Methods for introducing artificial damping to control mild nonlinear instabilities are also discussed. The accuracy 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...
The application of Fast Fourier transforms to the primitive equations of Boussinesq convection
International Nuclear Information System (INIS)
We have described a numerical scheme which is second-order in both space and time. The use of Fast Fourier Transform techniques for the solution of pressure equation guarantees accurate incompressibility at all time and enabled us to consider using iteration for part of this scheme. The iterations converge satisfactorily for values of the timestep of the order of one-half to one-quarter of the space step. Numerical calculations are being undertaken to clarify the range of Reynolds numbers and timestep over which the iteration converges. (orig.)
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
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
Institute of Scientific and Technical Information of China (English)
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.
International Nuclear Information System (INIS)
There exists much good work in the area of usual solitons, but there appears little in the field of compacton solutions. Only a few mathematical tools were employed so far. Recently, Yan [Chaos, Solitons and Fractals 14 (2002) 1151] extended the decomposition method to seek compacton solutions of B(m,n) equation utt=(un)xx+(um)xxxx. In this paper we present a different approach, integral approach, to investigate the compacton solutions of the B(m,n) equation. Not only Yan's results but also many new compacton solutions of the B(m,n) equation are obtained. Our approach is simple and also suitable for studying compacton solutions of some other equations
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.
On the global regularity of axisymmetric Navier-Stokes-Boussinesq system
Abidi, Hamadi; Hmidi, Taoufik; Keraani, Sahbi
2009-01-01
In this paper we prove a global well-posedness result for tridimensional Navier-Stokes-Boussinesq system with axisymmetric initial data. This system couples Navier-Stokes equations with a transport equation governing the density.
Global well-posedness for the Euler-Boussinesq system with axisymmetric data
Hmidi, Taoufik; Rousset, Frederic
2010-01-01
In this paper we prove the global well-posedness for the three-dimensional Euler-Boussinesq system with axisymmetric initial data without swirl. This system couples the Euler equation with a transport-diffusion equation governing the temperature.
Global well-posedness for Euler-Boussinesq system with critical dissipation
Hmidi, Taoufik; Keraani, Sahbi; Rousset, Frederic
2009-01-01
In this paper we study a fractional diffusion Boussinesq model which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion for the temperature. We prove global well-posedness results.
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
Energy Technology Data Exchange (ETDEWEB)
Zhao, Min, E-mail: zhaomin1223@126.com; Zhou, Shengfan, E-mail: zhoushengfan@yahoo.com [Department of Mathematics, Zhejiang Normal University, Jinhua 321004 (China)
2015-09-15
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.
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
Variational Boussinesq model for simulations of coastal waves and tsunamis
Adytia, Didit; Groesen, van, E.; Tan, Soon Keat; Huang, Zhenhua
2009-01-01
In this paper we describe the basic ideas of a so-called Variational Boussinesq Model which is based on the Hamiltonian structure of gravity surface waves. By using a rather simple approach to prescribe the profile of vertical fluid potential in the expression for the kinetic energy, we obtain a set of dynamic equations extended with one additional elliptic equation for the amplitude of the vertical profile. All expressions in the energy contain at most first order derivatives, which makes a ...
Ingersoll, Andrew P.
2005-01-01
Expressions are derived for the potential energy of a fluid whose density depends on three variables: temperature, pressure, and salinity. The thermal expansion coefficient is a function of depth, and the application is to thermobaric convection in the oceans. Energy conservation, with conversion between kinetic and potential energies during adiabatic, inviscid motion, exists for the Boussinesq and anelastic approximations but not for all approximate systems of equations. In the Boussinesq/an...
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.
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....
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...
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...
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
Global well-posedness for a Boussinesq- Navier-Stokes System with critical dissipation
Hmidi, Taoufik; Keraani, Sahbi; Rousset, Frederic
2009-01-01
In this paper we study a fractional diffusion Boussinesq model which couples a Navier-Stokes type equation with fractional diffusion for the velocity and a transport equation for the temperature. We establish global well-posedness results with rough initial data.
Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data
Hmidi, Taoufik; Rousset, Frederic
2009-01-01
In this paper we prove the global well-posedness for a three-dimensional Boussinesq system with axisymmetric initial data. This system couples the Navier-Stokes equation with a transport-diffusion equation governing the temperature. Our result holds uniformly with respect to the heat conductivity coefficient $\\kappa \\geq 0$ which may vanish.
Global existence and uniqueness for a non linear Boussinesq system in dimension two
Sulaiman, Samira
2012-01-01
We study the global well-posedness of a two-dimensional Boussinesq system which couples the incompressible Euler equation for the velocity and a transport equation with fractional diffusion of type $|\\DD|^{\\alpha}$ for the temperature. We prove that for $\\alpha>1,$ there exists a unique global solution for initial data with critical regularities.
Derivation of Boussinesq's shoaling law using a coupled BBM system
Directory of Open Access Journals (Sweden)
H. Kalisch
2013-03-01
Full Text Available This paper is focused on finding rules for waveheight change in a solitary wave as it runs up a slowly increasing bottom. A coupled BBM system is used to describe the solitary waves. Expressions for energy density and energy flux associated with the BBM system are derived, and the principle of energy conservation is used to develop an equation relating the waveheight and undisturbed depth to the initial undisturbed depth and the incident waveheight. In the limit of zero waveheight, Boussinesq's shoaling law is recovered.
Falqui, G; Tondo, G
1999-01-01
We discuss the Boussinesq system with $t_5$ stationary, within a general framework for the analysis of stationary flows of n-Gel'fand-Dickey hierarchies. We show how a careful use of its bihamiltonian structure can be used to provide a set of separation coordinates for the corresponding Hamilton--Jacobi equations.
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.
Note on modulational instability of Boussinesq wavetrains
Directory of Open Access Journals (Sweden)
S. Roy Choudhury
1992-06-01
Full Text Available It is demonstrated that modulational instability may occur in Boussinesq wavetrains for wavenumbers k2>1, in contrast to the result found by Shivamoggi and Debnath. Both Whitham's averaged Lagrangian formulation and the technique of Benjamin and Feir are used.
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...
A Finite-Dimensional Completely Integrable System Associated with Boussinesq Hierarchy
International Nuclear Information System (INIS)
In this paper, a new completely integrable system related to the complex spectral problem -φxx + (i/4)uφx + (i/4)(uφ)x + (1/4)vφ = iλφx and the constrained flows of the Boussinesq equations are generated. According to the viewpoint of Hamiltonian mechanics, the Euler-Lagrange equations and the Legendre transformations, a reasonable Jacobi-Ostrogradsky coordinate system is obtained. Moreover, by means of the constrained conditions between the potential u, v and the eigenfunction φ, the involutive representations of the solutions for the Boussinesq equation hierarchy are given. (electromagnetism, optics, acoustics, heat transfer, classical mechanics, and fluid dynamics)
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.
Dark soliton solutions of (N+1)-dimensional nonlinear evolution equations
Demiray, Seyma Tuluce; Bulut, Hasan
2016-06-01
In this study, we investigate exact solutions of (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation by using generalized Kudryashov method. (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation can be returned to nonlinear ordinary differential equation by suitable transformation. Then, generalized Kudryashov method has been used to seek exact solutions of the (N+1)-dimensional double sinh-Gordon equation and (N+1)-dimensional generalized Boussinesq equation. Also, we obtain dark soliton solutions for these (N+1)-dimensional nonlinear evolution equations. Finally, we denote that this method can be applied to solve other nonlinear evolution equations.
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.
Solution of 2D Boussinesq systems with FreeFem++: The flat bottom case
Sadaka, Georges
2012-01-01
FreeFem++ is an open source platform to solve partial differential equations numerically, based on finite element methods. The FreeFem++ platform has been developed to facilitate teaching and basic research through prototyping. For the moment this platform is restricted to the numerical simulations of problems which admit a variational formulation. We will use FreeFem++ in this work to solve a three-parameter family of Boussinesq type systems in two space dimensions which approximate the three-dimensional Euler equations over an horizontal bottom.
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...
Díaz Díaz, Jesús Ildefonso; Rakotoson, J. M.; Schmidt, P G
2008-01-01
We propose a modification of the classical Navier-Stokes-Boussinesq system of equations, which governs buoyancy-driven flows of viscous, incompressible fluids. This modification is motivated by unresolved issues regarding the global solvability of the classical system in situations where viscous heating cannot be neglected. A simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. In the present ...
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.
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...
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...
A Boussinesq system for two-way propagation of interfacial waves
Nguyen, Hai Yen
2007-01-01
The theory of internal waves between two layers of immiscible fluids is important both for its applications in oceanography and engineering, and as a source of interesting mathematical model equations that exhibit nonlinearity and dispersion. A Boussinesq system for two-way propagation of interfacial waves in a rigid lid configuration is derived. In most cases, the nonlinearity is quadratic. However, when the square of the depth ratio is close to the density ratio, the coefficients of the quadratic nonlinearities become small and cubic nonlinearities must be considered. The propagation as well as the collision of solitary waves and/or fronts is studied numerically.
A New Integrable Couplings of Classical-Boussinesq Hierarchy with Self-Consistent Sources
International Nuclear Information System (INIS)
A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with sl-tilde(4) is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-Boussinesq hierarchy with self-consistent sources by using of loop algebra sl-tilde(4). In this paper, we also point out that there exist some errors in Yu's paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources. (general)
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...
Travelling Wave Solutions of the Schrödinger-Boussinesq System
Reza Abazari; Adem Kılıcman
2012-01-01
We establish exact solutions for the Schrödinger-Boussinesq System $i{u}_{t}+{u}_{xx}-a\\mathrm{uv}=0$ , ${v}_{tt}-{v}_{xx}+{v}_{xxxx}-b{({|u|}^{\\mathrm{2}})}_{xx}=0$ , where $a$ and $b$ are real constants. The ( ${G}^{\\prime }/G$ )-expansion method is used to construct exact periodic and soliton solutions of this equation. Our work is motivated by the fact that the ( ${G}^{\\prime }/G$ )-expansion method provides not only more general forms of solutions but also periodic and solitary waves....
International Nuclear Information System (INIS)
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite β that we solve the perpendicular component of Ohm's law to conserve the physical energy while ensuring the relation ∇ · j = 0
EXP-function method and its application to nonlinear equations
International Nuclear Information System (INIS)
Exp-function method is used to find a unified solution of a nonlinear wave equation. Variant Boussinesq equations are selected to illustrate the effectiveness and simplicity of the method. A generalized solitary solution with free parameters is obtained
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 $\
Integrable Couplings of Classical-Boussinesq Hierarchy and Its Hamiltonian Structure
International Nuclear Information System (INIS)
By using a Lie algebra, an integrable couplings of the classical-Boussinesq hierarchy is obtained. Then, the Hamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity. (general)
Conservation Laws and Self-Consistent Sources for a Super-Classical-Boussinesq Hierarchy
International Nuclear Information System (INIS)
The super-classical-Boussinesq hierarchy with self-consistent sources is considered. Then, infinitely many conservation laws for the integrable super-classical-Boussinesq hierarchy are established. (general)
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.
Incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity
Czech Academy of Sciences Publication Activity Database
Consiglieri, L.; Nečasová, Šárka; Sokolowski, J.
2009-01-01
Roč. 38, č. 4 (2009), s. 1193-1215. ISSN 0324-8569 R&D Projects: GA ČR GA201/05/0005; GA ČR GA201/08/0012 Institutional research plan: CEZ:AV0Z10190503 Keywords : Maxwell-Boussinesq approximation Subject RIV: BA - General Mathematics Impact factor: 0.378, year: 2009
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.
Incompressible Maxwell-Boussinesq approximation: Existence, uniqueness and shape sensitivity
Consiglieri, Luisa; Necasova, Sarka; Sokolowski, Jan
2009-01-01
We prove the existence and uniqueness of weak solutions to the variational formulation of the Maxwell-Boussinesq approximation problem. Some further regularity in $W^{1,2+\\delta}$, $\\delta>0$, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak solutions. As a result, the existence of the strong material derivatives for the weak solutions of the problem is shown. The result can be used to establish the shape differe...
On the Oberbeck-Boussinesq approximation on unbounded domains
Czech Academy of Sciences Publication Activity Database
Feireisl, Eduard; Schonbek, M.E.
Berlin : Springer, 2012 - (Holden, H.; Karlsen, K.), s. 131-168 ISBN 978-3-642-25360-7. - (Abel Symposia. 7) R&D Projects: GA ČR GA201/09/0917 Institutional research plan: CEZ:AV0Z10190503 Keywords : Oberbeck-Boussinesq system * singular limit * unbounded domain Subject RIV: BA - General Mathematics http://link.springer.com/chapter/10.1007/978-3-642-25361-4_7
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.
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
Abundant soliton solutions for the coupled Schrödinger-Boussinesq system via an analytical method
Manafian, Jalil; Aghdaei, Mehdi Fazli
2016-04-01
In this paper, the improved tan(Φ(ξ)/2)-expansion method is proposed to find the exact soliton solutions of the coupled Schrödinger-Boussinesq (SB) system. The exact particular solutions are of five types: hyperbolic function solution (exact soliton wave solution), trigonometric function solution (exact periodic wave solution), rational exponential solution (exact singular kink-type wave solution), logarithmic solution and rational solution (exact singular cupson wave solution). We obtained the further solutions comparing with other methods. The results demonstrate that the new tan(Φ(ξ)/2)-expansion method is more efficient than the Ansatz method applied by Bilige et al. (2013). Recently this method was developed for searching the exact travelling-wave solutions of nonlinear partial differential equations. Abundant exact travelling-wave solutions including solitons, kink, periodic and rational solutions have been found. These solutions might play an important role in Laser and plasma. It is shown that this method, with the help of symbolic computation, provides a straightforward and powerful mathematical tool for solving the nonlinear problems.
Inner harbour wave agitation using boussinesq wave model
Directory of Open Access Journals (Sweden)
Panigrahi Jitendra K.
2015-01-01
Full Text Available Short crested waves play an important role for planning and design of harbours. In this context a numerical simulation is carried out to evaluate wave tranquility inside a real harbour located in east coast of India. The annual offshore wave climate proximity- to harbour site is established using Wave Model (WAM hindcast wave data. The deep water waves are transformed to harbour front using a Near Shore spectral Wave model (NSW. A directional analysis is carried out to determine the probable incident wave directions towards the harbour. Most critical threshold wave height and wave period is chosen for normal operating conditions using exceedence probability analysis. Irregular random waves from various directions are generated confirming to Pierson Moskowitz spectrum at 20m water depth. Wave incident into inner harbor through harbor entrance is performed using Boussinesq Wave model (BW. Wave disturbance experienced inside the harbour and at various berths are analysed. The paper discusses the progresses took place in short wave modeling and it demonstrates application of wave climate for the evaluation of harbor tranquility using various types of wave models.
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.
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
fully nonlinear and highly dispersive waves traveling over a rapidly varying bathymetry. Finally, we cover applications of this Boussinesq model, and we study a number of nonlinear wave phenomena in deep and shallow water. These include (1) Kinematics in highly nonlinear progressive deep-water waves; (2......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......) Kinematics in progressive solitary waves; (3) Reflection of solitary waves from a vertical wall; (4) Reflection and diffraction around a vertical plate; (5) Quartet and quintet interactions and class I and II instabilities; (6) Extreme events from focused directionally spread waveelds; (7) Bragg scattering...
Non-Boussinesq turbulent buoyant jet resulting from hydrogen leakage in air
Energy Technology Data Exchange (ETDEWEB)
El-Amin, M.F. [Department of Mechanical Engineering, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395 (Japan)
2009-09-15
This paper is devoted to introduce a numerical investigation of a vertical axisymmetric non-Boussinesq buoyant jet resulting from hydrogen leakage in air as an example of injecting a low-density gas jet into high-density ambient. As the domain temperature is assumed to be constant and therefore the density of the mixture is a function of the concentration only, the binary gas mixture is assumed to be of a linear mixing type. Also, it is assumed that the rate of entrainment to be a function of the plume centerline velocity and the ratio of the mean plume and ambient densities. On the other hand, the local rate of entrainment may be considered to be consisted from two components; one is the component of entrainment due to jet momentum while the other is the component of entrainment due to buoyancy. Firstly, the integral models of the mass, momentum and concentration fluxes are obtained and transformed to a set of ordinary differential equations using some non-dimensional transformations known as similarity transformations. The given ordinary differential system is integrated numerically and the mean centerline mass fraction, jet width and mean centerline velocity are obtained. In the second step, the mean axial velocity, mean concentration and mean density of the jet are obtained. Finally in the third step of this article, several quantities of interest, including the cross-stream velocity, Reynolds stress, velocity-concentration correlation (radial flux), turbulent eddy viscosity and turbulent eddy diffusivity, are obtained. In addition, the turbulent Schmidt number is estimated and the normalized jet-feed material density and the normalized momentum flux density are correlated. (author)
Consiglieri, L.; Nečasová, Š. (Šárka); Sokolowski, J.
2010-01-01
The Boussinesq approximation to the Fourier–Navier–Stokes (F–N–S) flows under the electromagnetic field is considered. Such a model is the so-called Maxwell–Boussinesq approximation. We propose a new approach to the problem. We prove the existence and uniqueness of weak solutions to the variational formulation of the model. Some further regularity in W1,2+δ, δ>0, is obtained for the weak solutions. The shape sensitivity analysis by the boundary variations technique is performed for the weak s...
International Nuclear Information System (INIS)
The results from direct numerical simulations of turbulent Boussinesq convection are briefly presented. The flow is computed for a cylindrical cell of aspect ratio 1/2 in order to compare with the results from recent experiments. The results span eight decades of Ra from 2x106 to 2x1014 and form the baseline data for a strictly Boussinesq fluid of constant Prandtl number (Pr=0.7). A conclusion is that the Nusselt number varies nearly as the 1/3 power of Ra for about four decades towards the upper end of the Ra range covered. (author)
International Nuclear Information System (INIS)
Based on the closed connections among the homogeneous balance (HB) method, Weiss-Tabor-Carneval (WTC) method and Clarkson-Kruskal (CK) method, we study Baecklund transformation and similarity reductions of the variable coefficients variant Boussinesq system. In the meantime, new exact solutions also are found
Energy Technology Data Exchange (ETDEWEB)
Moussa, M.H.M. [Department of Mathematic, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo (Egypt)], E-mail: m_h_m_moussa@yahoo.com; El Shikh, Rehab M. [Department of Mathematic, Faculty of Education, Ain Shams University, Roxy, Hiliopolis, Cairo (Egypt)
2008-02-25
Based on the closed connections among the homogeneous balance (HB) method, Weiss-Tabor-Carneval (WTC) method and Clarkson-Kruskal (CK) method, we study Baecklund transformation and similarity reductions of the variable coefficients variant Boussinesq system. In the meantime, new exact solutions also are found.
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.
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).
Turbulent Flow over a Flat Plate Using a Three-equation Model
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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.
Periodic solutions of nonlinear equations obtained by linear superposition
International Nuclear Information System (INIS)
We show that a type of linear superposition principle works for several nonlinear differential equations. Using this approach, we find periodic solutions of the Kadomtsev-Petviashvili equation, the nonlinear Schroedinger equation, the λφ4 model, the sine-Gordon equation and the Boussinesq equation by making appropriate linear superpositions of known periodic solutions. This unusual procedure for generating solutions of nonlinear differential equations is successful as a consequence of some powerful, recently discovered, cyclic identities satisfied by the Jacobi elliptic functions
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...
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...
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
-layer hydraulic model will accurately predict a bore's speed of propagation. A two-layer model is required, however, if the densities are more similar. Mass is conserved separately in each layer and momentum is conserved globally, but the model requires for closure an assumption about the loss of energy across a...... bore. In the Boussinesq limit, it is known that there is a decrease of the total energy flux across a bore, but in the expanding layer, turbulent mixing at the interface entrains high speed fluid from the contracting layer, resulting in an increase in the flux of kinetic energy across the expanding...
Candelier, F.; Angilella, J. R.; Souhar, M.
2004-05-01
The trajectory of an isolated solid particle dropped in the core of a vertical vortex is investigated theoretically and experimentally, in order to analyze the effect of the history force on the radial migration of the inclusion. Both the Stokes number (based on the particle radius and the fluid angular velocity) and the particle Reynolds number are small. The particle is heavier than the fluid, and is therefore expelled from the center of the vortex. An experimental device using spherical particles injected in a rotating cylindrical tank filled with silicone oil has been built. Experimental trajectories are compared to analytical solutions of the motion equations, which are obtained by making use of classical Laplace transforms. The analytical expression of the history force and the ejection rate are carried out. This force does not vanish, but increases exponentially and has to be taken into account for efficient predictions. In particular, calculations without history force overestimate particle ejection. The relative difference between the ejection rate with and without history force scales like the square root of the Stokes number, so that differences of the order of 10% are visible as soon as the Stokes number is of the order of 0.01. Also, agreement between experimental and theoretical trajectories is observed only if the acceleration term in the history integral involves the time derivative of the fluid velocity following the particle, rather than the acceleration of fluid points at the particle location, even for small particle Reynolds numbers. Finally, analytical calculations show that the particle ejection rate is more sensitive to the Boussinesq-Basset force than to Saffman's lift.
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.
Travelling Wave Solutions for the Coupled IBq Equations by Using the tanh-coth Method
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Omer Faruk Gozukizil
2014-01-01
Full Text Available Based on the availability of symbolic computation, the tanh-coth method is used to obtain a number of travelling wave solutions for several coupled improved Boussinesq equations. The abundant new solutions can be seen as improvement of the previously known data. The obtained results in this work also demonstrate the efficiency of the method.
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.
Simulation of nonlinear wave run-up with a high-order Boussinesq model
DEFF Research Database (Denmark)
Fuhrman, David R.; Madsen, Per A.
2008-01-01
. As validation, computed results involving the nonlinear run-up of periodic as well as transient waves on a sloping beach are considered in a single horizontal dimension, demonstrating excellent agreement with analytical solutions for both the free surface and horizontal velocity. In two horizontal......This paper considers the numerical simulation of nonlinear wave run-up within a highly accurate Boussinesq-type model. Moving wet–dry boundary algorithms based on so-called extrapolating boundary techniques are utilized, and a new variant of this approach is proposed in two horizontal dimensions...... dimensions cases involving long wave resonance in a parabolic basin, solitary wave evolution in a triangular channel, and solitary wave run-up on a circular conical island are considered. In each case the computed results compare well against available analytical solutions or experimental measurements. The...
High-resolution simulations of non-Boussinesq downslope gravity currents in the acceleration phase
Dai, Albert; Huang, Yu-lin
2016-02-01
Gravity currents generated from an instantaneous buoyancy source of density contrast in the density ratio range of 0.3 ≤ γ ≤ 0.998 propagating downslope in the slope angle range of 0° ≤ θ budgets show that, as the density contrast increases, the heavy fluid retains more fraction of potential energy loss while the ambient fluid receives less fraction of potential energy loss in the process of energy transfer during the propagation of downslope gravity currents. Previously, it was reported that for the Boussinesq case, the downslope gravity currents have a maximum of Uf,max at θ ≈ 40°. It is found, as is also confirmed by the energy budgets in this study, that the slope angle at which the downslope gravity currents have a maximum of Uf,max may increase beyond 40° as the density contrast increases.
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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.
Guevara, Carlos; Graf, Thomas
2013-04-01
Subsurface water systems are endangered due to salt water intrusion in coastal aquifers, leachate infiltration from waste disposal sites and salt transport in agricultural sites. This leads to the situation where more dense fluid overlies a less dense fluid creating a density gradient. Under certain conditions this density gradient produces instabilities in form dense plume fingers that move downwards. This free convection increases solute transport over large distances and shorter times. In cases where a significantly larger density gradient exists, the effect of free convection on transport is non-negligible. The assumption of a constant density distribution in space and time is no longer valid. Therefore variable-density flow must be considered. The flow equation and the transport equation govern the numerical modeling of variable-density flow and solute transport. Computer simulation programs mathematically describe variable-density flow using the Oberbeck-Boussinesq Approximation (OBA). Three levels of simplifications can de considered, which are denoted by OB1, OB2 and OB3. OB1 is the usually applied simplification where variable density is taken into account in the hydraulic potential. In OB2 variable density is considered in the flow equation and in OB3 variable density is additionally considered in the transport equation. Using the results from a laboratory-scale experiment of variable-density flow and solute transport (Simmons et al., Transp. Porous Medium, 2002) it is investigated which level of mathematical accuracy is required to represent the physical experiment the most accurate. Differences between the physical and mathematical model are evaluated using qualitative indicators (e.g. mass fluxes, Nusselt number). Results show that OB1 is required for small density gradients and OB3 is required for large density gradients.
International Nuclear Information System (INIS)
In this Letter, a new method (called generalized sine-Gordon equation expansion method) is proposed by further studying the famous sine-Gordon equation and using a generalized transformation to seek more types of solutions of nonlinear wave equations. With the aid of symbolic computation, we choose (2+1)-dimensional Burgers equation and the variant Boussinesq equations to illustrate the validity and advantages of the algorithm. As a consequence, more types of new solitary wave solutions, singular solitary wave solutions and doubly periodic solutions are obtained. The algorithm can be also extended to many other nonlinear wave equations
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.
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...
International Nuclear Information System (INIS)
The new soliton solutions for the variable-coefficient Boussinesq system, whose applications are seen in fluid dynamics, are studied in this paper with symbolic computation. First, the Painleve analysis is used to investigate its integrability properties. For the identified case we give, the Lax pair of the system is found, and then the Darboux transformation is constructed. At last, some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics. (general)
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.
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.
International Nuclear Information System (INIS)
The ((G')/G )-expansion method is firstly proposed, where G=G(ξ) satisfies a second order linear ordinary differential equation (LODE for short), by which the travelling wave solutions involving parameters of the KdV equation, the mKdV equation, the variant Boussinesq equations and the Hirota-Satsuma equations are obtained. When the parameters are taken as special values the solitary waves are also derived from the travelling waves. The travelling wave solutions are expressed by the hyperbolic functions, the trigonometric functions and the rational functions. The proposed method is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations
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
International Nuclear Information System (INIS)
A new (2+1)-dimensional variant Boussinesq system with its spectral problems is presented in this paper, which has a close connection with the Whitham-Broer-Kaup soliton hierarchy describing long waves in shallow water. Based on the associated spectral problems, the Darboux transformation (DT) with multi-parameters was firstly constructed with the help of symbolic computation. Then, by using the DT, some new one- and two-soliton solutions of the (2+1)-dimensional variant Boussinesq system were obtained and graphically represented. These solutions might be of some value in fluid dynamics.
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.)
Stability of the subseismic wave equation for the Earth's fluid core
Friedlander, Susan
The effects of compressibility on the stability of internal oscillations in the Earth's fluid core are examined in the context of the subseismic approximation for the equations of motion describing a rotating, stratified, self-gravitating, compressible fluid in a thick shell. It is shown that in the case of a bounded fluid the results are closely analogous to those derived under the Boussinesq approximation.
Huang, Aimin
2014-01-01
Global well-posedness of strong solutions and existence of the global attractor to the initial and boundary value problem of 2D Boussinesq system in a periodic channel with non-homogeneous boundary conditions for the temperature and viscosity and thermal diffusivity depending on the temperature are proved.
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...... experimental data in four different cases as well as with the underlying deterministic model. In general, the agreement is found to be acceptable, even far beyond the region where Gaussianity (Gaussian sea state) may be justified. (C) 1998 Elsevier Science B.V....
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...
International Nuclear Information System (INIS)
In this paper, based on the well-known sinh-Gordon equation, a new sinh-Gordon equation expansion method is developed. This method transforms the problem of solving nonlinear partial differential equations into the problem of solving the corresponding systems of algebraic equations. With the aid of symbolic computation, the procedure can be carried out by computer. Many nonlinear wave equations in mathematical physics are chosen to illustrate the method such as the KdV-mKdV equation, (2+1)-dimensional coupled Davey-Stewartson equation, the new integrable Davey-Stewartson-type equation, the modified Boussinesq equation, (2+1)-dimensional mKP equation and (2+1)-dimensional generalized KdV equation. As a consequence, many new doubly-periodic (Jacobian elliptic function) solutions are obtained. When the modulus m → 1 or 0, the corresponding solitary wave solutions and singly-periodic solutions are also found. This approach can also be applied to solve other nonlinear differential equations
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.
The lattice Boltzmann model for the second-order Benjamin–Ono equations
International Nuclear Information System (INIS)
In this paper, in order to extend the lattice Boltzmann method to deal with more complicated nonlinear equations, we propose a 1D lattice Boltzmann scheme with an amending function for the second-order (1 + 1)-dimensional Benjamin–Ono equation. With the Taylor expansion and the Chapman–Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The equilibrium distribution function and the amending function are obtained. Numerical simulations are carried out for the 'good' Boussinesq equation and the 'bad' one to validate the proposed model. It is found that the numerical results agree well with the analytical solutions. The present model can be used to solve more kinds of nonlinear partial differential equations
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.
The staircase method: integrals for periodic reductions of integrable lattice equations
International Nuclear Information System (INIS)
We show, in full generality, that the staircase method (Papageorgiou et al 1990 Phys. Lett. A 147 106-14, Quispel et al 1991 Physica A 173 243-66) provides integrals for mappings, and correspondences, obtained as traveling wave reductions of (systems of) integrable partial difference equations. We apply the staircase method to a variety of equations, including the Korteweg-De Vries equation, the five-point Bruschi-Calogero-Droghei equation, the quotient-difference (QD)-algorithm and the Boussinesq system. We show that, in all these cases, if the staircase method provides r integrals for an n-dimensional mapping, with 2r2 lattice, and we prove linear growth of the multi-valuedness of iterates of high-dimensional correspondences obtained as reductions of the QD-algorithm.
String equation from field equation
Gurovich, V T
1996-01-01
It is shown that the string equation can be obtain from field equations. Such work is performed to scalar field. The equation obtained in nonrelativistic limit describes the nonlinear string. Such string has the effective elasticity connencted with the local string curvature. Some examples of the movement such nonlinear elastic string are considered.
Energy Technology Data Exchange (ETDEWEB)
Chmaissem, W.; Daguenet, M. [Universite de Perpignan, Lab. de Thermodynamique et Energetique, Perpignan, 66 (France)
1999-07-01
The authors present a new calculation code using a two-dimensional finite-element method valid in permanent and laminar flow. They consider elements of symmetry existing in boundary conditions imposed on velocities as well as on temperatures, then elements of symmetry existing only in boundary conditions imposed on velocities and, finally, boundary conditions containing no symmetry. The flow is two-cellular so that the Rayleigh number remained inferior to a value in the order of 10{sup 6}. Beyond this value, secondary cells can appear, following the geometry of the enclosure. (Author)
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
Scalar and vector spherical harmonic spectral equations of rotating magnetohydrodynamics
Ivers, D. J.; Phillips, C. G.
2008-12-01
Vector spherical harmonic analyses have been used effectively to solve laminar and mean-field magnetohydrodynamic dynamo problems with product interactions, such as magnetic induction, anisotropic alpha-effect and anisotropic magnetic diffusion, that are difficult to analyse spectrally in spherical geometries. Spectral forms of the non-linear rotating, Boussinesq and anelastic, momentum, magnetic induction and heat equations are derived for spherical geometries from vector spherical harmonic expansions of the velocity, magnetic induction, vorticity, electrical current and gravitational acceleration and from scalar spherical harmonic expansions of the pressure and temperature. By combining the vector spherical harmonic spectral forms of the momentum equation and the magnetic induction equation with poloidal-toroidal representations of the velocity and the magnetic field, non-linear spherical harmonic spectral equations are also derived for the poloidal-toroidal potentials of the velocity or the momentum density in the anelastic approximation and the magnetic field. Both compact and spectral interaction expansion forms are given. Vector spherical harmonic spectral forms of the linearized rotating magnetic induction, momentum and heat equations for a general basic state can be obtained by linearizing the corresponding non-linear spectral equations. Similarly, the spherical harmonic spectral equations for the poloidal-toroidal potentials of the velocity and the magnetic field may be linearized. However, for computational applications, new alternative hybrid linearized spectral equations are derived. The algorithmically simpler hybrid equations depend on vector spherical harmonic expansions of the velocity, magnetic field, vorticity, electrical current and gravitational acceleration of the basic state and scalar spherical harmonic expansions of the poloidal-toroidal potentials of the perturbation velocity, magnetic field and temperature. The spectral equations derived
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.
Asymptotic reductions and solitons of nonlocal nonlinear Schrödinger equations
Horikis, Theodoros P.; Frantzeskakis, Dimitrios J.
2016-05-01
Asymptotic reductions of a defocusing nonlocal nonlinear Schrödinger model in (3 + 1)-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its far-field, in the form of a variety of Kadomtsev–Petviashvilli (KP) equations for right- and left-going waves, is found. KP models include versions of the KP-I and KP-II equations, in Cartesian and cylindrical geometry. Solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are also predicted to occur. Their nature and stability is determined by a parameter defined by the physical parameters of the original nonlocal system. It is thus found that (dark) anti-dark solitary waves are only supported by a weak (strong) nonlocality, and are unstable (stable) in higher-dimensions. Our analytical predictions are corroborated by direct numerical simulations.
Asymptotic reductions and solitons of nonlocal nonlinear Schr\\"{o}dinger equations
Horikis, Theodoros P
2016-01-01
Asymptotic reductions of a defocusing nonlocal nonlinear Schr\\"{o}dinger model in $(3+1)$-dimensions, in both Cartesian and cylindrical geometry, are presented. First, at an intermediate stage, a Boussinesq equation is derived, and then its far-field, in the form of a variety of Kadomtsev-Petviashvilli (KP) equations for right- and left-going waves, is found. KP models include versions of the KP-I and KP-II equations, in Cartesian and cylindrical geometry. Solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are also predicted to occur. Their nature and stability is determined by a parameter defined by the physical parameters of the original nonlocal system. It is thus found that (dark) anti-dark solitary waves are only supported by a weak (strong) nonlocality, and are unstable (stable) in higher-dimensions. Our analytical predictions are corroborated by direct numerical simulations.
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.
Tricomi, FG
2012-01-01
Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential
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.
Hochstadt, Harry
2012-01-01
Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.
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).
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
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.
Tricomi, Francesco Giacomo
1957-01-01
This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful
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.
Frost, W.; Harper, W. L.
1975-01-01
Flow over surface obstructions can produce significantly large wind shears such that adverse flying conditions can occur for aeronautical systems (helicopters, STOL vehicles, etc.). Atmospheric flow fields resulting from a semi-elliptical surface obstruction in an otherwise horizontally homogeneous statistically stationary flow are modelled with the boundary-layer/Boussinesq-approximation of the governing equation of fluid mechanics. The turbulence kinetic energy equation is used to determine the dissipative effects of turbulent shear on the mean flow. Iso-lines of turbulence kinetic energy and turbulence intensity are plotted in the plane of the flow and highlight regions of high turbulence intensity in the stagnation zone and sharp gradients in intensity along the transition from adverse to favourable pressure gradient. Discussion of the effects of the disturbed wind field in CTOL and STOL aircraft flight path and obstruction clearance standards is given. The results indicate that closer inspection of these presently recommended standards as influenced by wind over irregular terrains is required.
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.
Partial Differential Equations
1988-01-01
The volume contains a selection of papers presented at the 7th Symposium on differential geometry and differential equations (DD7) held at the Nankai Institute of Mathematics, Tianjin, China, in 1986. Most of the contributions are original research papers on topics including elliptic equations, hyperbolic equations, evolution equations, non-linear equations from differential geometry and mechanics, micro-local analysis.
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.
The Generalized Jacobi Equation
Chicone, C.; Mashhoon, B.
2002-01-01
The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighboring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analyzed in this paper. The tidal accelerat...
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
Ray C. Fair
2007-01-01
How inflation and unemployment are related in both the short run and long run is perhaps the key question in macroeconomics. This paper tests various price equations using quarterly U.S. data from 1952 to the present. Issues treated are the following. 1) Estimating price and wage equations in which wages affect prices and vice versa versus estimating "reduced form" price equations with no wage explanatory variables. 2) Estimating price equations in (log) level terms, first difference (i.e., i...
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...
Goncalves, Patricia
2010-01-01
We introduce the notion of energy solutions of the KPZ equation. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, weakly asymmetric, conservative particle systems with respect to the stationary states are given by energy solutions of the KPZ equation. As a consequence, we prove that the Cole-Hofp solutions are also energy solutions of the KPZ equation.
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.
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
Functional equations with causal operators
Corduneanu, C
2003-01-01
Functional equations encompass most of the equations used in applied science and engineering: ordinary differential equations, integral equations of the Volterra type, equations with delayed argument, and integro-differential equations of the Volterra type. The basic theory of functional equations includes functional differential equations with causal operators. Functional Equations with Causal Operators explains the connection between equations with causal operators and the classical types of functional equations encountered by mathematicians and engineers. It details the fundamentals of linear equations and stability theory and provides several applications and examples.
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.
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.
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...
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...
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.
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.
Ordinary differential equations
Pontryagin, Lev Semenovich
1962-01-01
Ordinary Differential Equations presents the study of the system of ordinary differential equations and its applications to engineering. The book is designed to serve as a first course in differential equations. Importance is given to the linear equation with constant coefficients; stability theory; use of matrices and linear algebra; and the introduction to the Lyapunov theory. Engineering problems such as the Watt regulator for a steam engine and the vacuum-tube circuit are also presented. Engineers, mathematicians, and engineering students will find the book invaluable.
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
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…
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.
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.
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.
Applied singular integral equations
Mandal, B N
2011-01-01
The book is devoted to varieties of linear singular integral equations, with special emphasis on their methods of solution. It introduces the singular integral equations and their applications to researchers as well as graduate students of this fascinating and growing branch of applied mathematics.
Alternative equations of gravitation
International Nuclear Information System (INIS)
It is shown, trough a new formalism, that the quantum fluctuation effects of the gravitational field in Einstein's equations are analogs to the effects of a continuum medium in Maxwell's Electrodynamics. Following, a real example of the applications of these equations is studied. Qunatum fluctuations effects as perturbation sources in Minkowski and Friedmann Universes are examined. (L.C.)
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.
The generalized Jacobi equation
International Nuclear Information System (INIS)
The Jacobi equation in pseudo-Riemannian geometry determines the linearized geodesic flow. The linearization ignores the relative velocity of the geodesics. The generalized Jacobi equation takes the relative velocity into account; that is, when the geodesics are neighbouring but their relative velocity is arbitrary the corresponding geodesic deviation equation is the generalized Jacobi equation. The Hamiltonian structure of this nonlinear equation is analysed in this paper. The tidal accelerations for test particles in the field of a plane gravitational wave and the exterior field of a rotating mass are investigated. In the latter case, the existence of an attractor of uniform relative radial motion with speed 2-1/2c ∼ 0.7c is pointed out. The astrophysical implication of this result for the terminal speed of a relativistic jet is briefly explored
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...
International Nuclear Information System (INIS)
The direct use of enlarged subsets of mathematically exact equations of change in moments of the velocity distribution function, each equation corresponding to one of the macroscopic variables to be retained, produces extended MHD models. The first relevant level of closure provides 'ten moment' equations in the density ρ, velocity v, scalar pressure p, and the traceless component of the pressure tensor t. The next 'thirteen moment' level also includes the thermal flux vector q, and further extended MHD models could be developed by including even higher level basic equations of change. Explicit invariant forms for the tensor t and the heat flux vector defining q follow from their respective basic equations of change. Except in the neighbourhood of a magnetic null, in magnetised plasma these forms may be resolved into known sums of their parallel, cross (or transverse) and perpendicular components. Parallel viscosity in an electron-ion plasma is specifically discussed. (author)
Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows
International Nuclear Information System (INIS)
The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial and boundary conditions. Such situations are common in multiscale, intermittent and non-homogeneous fluid and ocean flows. The dynamically orthogonal (DO) field equations provide an adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for the DO methodology applied to unsteady stochastic Navier–Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit projection methods are developed for the mean and for the DO modes, and time-marching schemes of first to fourth order are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with new advection schemes based on total variation diminishing methods. Other results include: (i) the definition of pseudo-stochastic pressures to obtain a number of pressure equations that is linear in the subspace size instead of quadratic; (ii) symmetric advection schemes for the stochastic velocities; (iii) the use of generalized inversion to deal with singular subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal modes at the numerical level. To verify our implementation and study the properties of our schemes and their variations, a set of stochastic flow benchmarks are defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability density functions with the number of stochastic realizations.
Nonlinear gyrokinetic equations
International Nuclear Information System (INIS)
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed
Nonlinear gyrokinetic equations
Energy Technology Data Exchange (ETDEWEB)
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
Standardized Referente Evapotranspiration Equation
M.D. Mundo–Molina
2009-01-01
In this paper is presented a discussion on the necessity to standardize the Penman–Monteith equations in order to estimate ETo. The proposal is to define an accuracy and standarize equation based in Penman–Monteith. The automated weather station named CIANO (27° 22 ' 144 North latitude and 109" 55' west longitude) it was selected tomake comparisons. The compared equations we re: a) CIANO weat her station, b) Penman–Monteith ASCE (PMA), Penman–Monteith FAO 56 (PM FAO 56), Penman–Monteith estan...
Stochastic Schroedinger equations
International Nuclear Information System (INIS)
A derivation of Belavkin's stochastic Schroedinger equations is given using quantum filtering theory. We study an open system in contact with its environment, the electromagnetic field. Continuous observation of the field yields information on the system: it is possible to keep track in real time of the best estimate of the system's quantum state given the observations made. This estimate satisfies a stochastic Schroedinger equation, which can be derived from the quantum stochastic differential equation for the interaction picture evolution of system and field together. Throughout the paper we focus on the basic example of resonance fluorescence
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
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.
Partial differential equations
Friedman, Avner
2008-01-01
This three-part treatment of partial differential equations focuses on elliptic and evolution equations. Largely self-contained, it concludes with a series of independent topics directly related to the methods and results of the preceding sections that helps introduce readers to advanced topics for further study. Geared toward graduate and postgraduate students of mathematics, this volume also constitutes a valuable reference for mathematicians and mathematical theorists.Starting with the theory of elliptic equations and the solution of the Dirichlet problem, the text develops the theory of we
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
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,
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
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…
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.
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.
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.
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...
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...
Resistive ballooning mode equation
Energy Technology Data Exchange (ETDEWEB)
Bateman, G.; Nelson, D. B.
1978-10-01
A second-order ordinary differential equation on each flux surface is derived for the high mode number limit of resistive MHD ballooning modes in tokamaks with arbitrary cross section, aspect ratio, and shear. The equation is structurally similar to that used to study ideal MHD ballooning modes computationally. The model used in this paper indicates that all tokamak plasmas are unstable, with growth rate proportional to resistivity when the pressure gradient is less than the critical value needed for ideal MHD stability.
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.
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.
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.
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)
Transport equation solving methods
International Nuclear Information System (INIS)
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method
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...
The Equations of Magnetoquasigeostrophy
Umurhan, O M
2013-01-01
The dynamics contained in magnetized layers of exoplanet atmospheres are important to understand in order to characterize what observational signatures they may provide for future observations. It is important to develop a framework to begin studying and learning the physical processes possible under those conditions and what, if any, features contained in them may be observed in future observation missions. The aims of this study is to formally derive, from scaling arguments, a manageable reduced set of equations for analysis, i.e. a magnetic formulation of the equations of quasigeostrophy appropriate for a multi-layer atmosphere. The main goal is to provide a simpler theoretical platform to explore the dynamics possible within confined magnetized layers of exoplanet atmospheres. We primarily use scaling arguments to derive the reduced equations of "magnetoquasigeostrophy" which assumes dynamics to take place in an atmospheric layer which is vertically thin compared to its horizontal scales. The derived equa...
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.
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
Mirce Functionability Equation
Directory of Open Access Journals (Sweden)
Dr Jezdimir Knezevic
2014-08-01
Full Text Available Scientific principles and concepts expressed through the laws, equations and formulas are the bedrock for the prediction of the deign-in functionality performance of any engineering creation. However, there is no equivalent when the in-service functionability performance predictions have to be made. Hence, Mirce Mechanics has been created at the MIRCE Akademy to fulfil the roll. The main purpose of this paper is to present the development and application of Mirce Functionability Equation which is the bedrock for the prediction of the functionability performance of maintainable systems.
Obtaining Maxwell's equations heuristically
Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger
2013-02-01
Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.
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
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.
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.
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.
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...
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...
The Statistical Drake Equation
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Comparison of Kernel Equating and Item Response Theory Equating Methods
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
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...
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.
Standardized Referente Evapotranspiration Equation
Directory of Open Access Journals (Sweden)
M.D. Mundo–Molina
2009-04-01
Full Text Available In this paper is presented a discussion on the necessity to standardize the Penman–Monteith equations in order to estimate ETo. The proposal is to define an accuracy and standarize equation based in Penman–Monteith. The automated weather station named CIANO (27° 22 ' 144 North latitude and 109" 55' west longitude it was selected tomake comparisons. The compared equations we re: a CIANO weat her station, b Penman–Monteith ASCE (PMA, Penman–Monteith FAO 56 (PM FAO 56, Penman–Monteith estandarizado ASCE (PM Std. ASCE. The results were: a There are important differences between PMA and CIANO weather station. The differences are attributed to the nonstandardization of the equation CIANO weather station, b The coefficient of correlation between both methods was of 0,92, with a standard deviation of 1,63 mm, an average quadratic error of 0,60 mm and one efficiency in the estimation of ETo with respect to the method pattern of 87%.
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.
DEFF Research Database (Denmark)
Dyre, Jeppe
1995-01-01
energies chosen randomly according to a Gaussian. The random-walk model is here derived from Newton's laws by making a number of simplifying assumptions. In the second part of the paper an approximate low-temperature description of energy fluctuations in the random-walk modelthe energy master equation...
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.
Stochastic nonlinear beam equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Maslowski, Bohdan; Seidler, Jan
2005-01-01
Roč. 132, č. 1 (2005), s. 119-149. ISSN 0178-8051 R&D Projects: GA ČR(CZ) GA201/01/1197 Institutional research plan: CEZ:AV0Z10190503 Keywords : stochastic beam equation * stability Subject RIV: BA - General Mathematics Impact factor: 0.896, year: 2005
On rough differential equations
Lejay, Antoine
2009-01-01
We prove that the Ito map, that is the map that gives the solution of a differential equation controlled by a rough path of finite p-variation with p in [2,3) is locally Lipschitz continuous in all its arguments and could be extended to vector fields that have only a linear growth.
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.
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.
Generalized reduced magnetohydrodynamic equations
International Nuclear Information System (INIS)
A new derivation of reduced magnetohydrodynamic (MHD) equations is presented. A multiple-time-scale expansion is employed. It has the advantage of clearly separating the three time scales of the problem associated with (1) MHD equilibrium, (2) fluctuations whose wave vector is aligned perpendicular to the magnetic field, and (3) those aligned parallel to the magnetic field. The derivation is carried out without relying on a large aspect ratio assumption; therefore this model can be applied to any general configuration. By accounting for the MHD equilibrium and constraints to eliminate the fast perpendicular waves, equations are derived to evolve scalar potential quantities on a time scale associated with the parallel wave vector (shear-Alfven wave time scale), which is the time scale of interest for MHD instability studies. Careful attention is given in the derivation to satisfy energy conservation and to have manifestly divergence-free magnetic fields to all orders in the expansion parameter. Additionally, neoclassical closures and equilibrium shear flow effects are easily accounted for in this model. Equations for the inner resistive layer are derived which reproduce the linear ideal and resistive stability criterion of Glasser, Greene, and Johnson. The equations have been programmed into a spectral initial value code and run with shear flow that is consistent with the equilibrium input into the code. Linear results of tearing modes with shear flow are presented which differentiate the effects of shear flow gradients in the layer with the effects of the shear flow decoupling multiple harmonics
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…
Kinetic equation of sociodynamics
Володимир Олександрович Касьянов
2014-01-01
This article aims to build a theory of social dynamics, similar to the kinetic theory of gases. In general, given model is hybrid because off static mechanics ideas. In particular, Boltsman equation, Jaynes’s principle of entropy optimality have been applied to preference distribution of first and second type.
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
Kinetic equation of sociodynamics
Directory of Open Access Journals (Sweden)
Володимир Олександрович Касьянов
2014-08-01
Full Text Available This article aims to build a theory of social dynamics, similar to the kinetic theory of gases. In general, given model is hybrid because off static mechanics ideas. In particular, Boltsman equation, Jaynes’s principle of entropy optimality have been applied to preference distribution of first and second type.
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.
International Nuclear Information System (INIS)
We present part of our (direct or indirect) knwoledge of the equation of state of nuclear matter in a density-temperature domain for which nucleonic effects are dominant (densities smaller than 2-4 times the saturation density and temperatures smaller than 10-20 MeV). The lectures are divided into three parts corresponding, respectiveley, to direct studies close to the saturation, to the astrophysical case and to the studies involving heavy-ion collisions. In chapter one, after a brief introduction to the concept of equation of state, we discuss the saturation property of nuclear matter. The notion of incompressibility modulus is also introduced and its value is discussed in detail. Nuclear matter calculations trying to reproduce saturation from a nucleon-nucleon interaction are also briefly presented. In chapter two we study the equation of state in the astrophysical context. The role of the nuclear component is discussed in detail for the final phase of the collapse of supernovae cores. A brief presentation of calculations of dense matter constituting neutron stars is also given. Chapter three is devoted to heavy-ion collisions below 500-600 MeV per nucleon. After a brief presentation of both theoretical and experimental frameworks, we focus on three particular aspects which could have a link with the nuclear matter equation of state: the formation of intermediate mass fragments, flow effects and subthreshold particle production
RPA equations and the instantaneous Bethe-Salpeter equation
Resag, J
1993-01-01
We give a derivation of the particle-hole RPA equations for an interacting multi-fermion system by applying the instantaneous approximation to the amputated two-fermion propagator of the system. In relativistic field theory the same approximation leads from the fermion-antifermion Bethe-Salpeter equation to the Salpeter equation. We show that RPA equations and Salpeter equation are indeed equivalent.
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.
Lectures on partial differential equations
Petrovsky, I G
1992-01-01
Graduate-level exposition by noted Russian mathematician offers rigorous, transparent, highly readable coverage of classification of equations, hyperbolic equations, elliptic equations and parabolic equations. Wealth of commentary and insight invaluable for deepening understanding of problems considered in text. Translated from the Russian by A. Shenitzer.
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
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
Chaos in Partial Differential Equations
Li, Y. Charles
2009-01-01
This is a survey on Chaos in Partial Differential Equations. First we classify soliton equations into three categories: 1. (1+1)-dimensional soliton equations, 2. soliton lattices, 3. (1+n)-dimensional soliton equations (n greater than 1). A systematic program has been established by the author and collaborators, for proving the existence of chaos in soliton equations under perturbations. For each category, we pick a representative to present the results. Then we review some initial results o...
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, ...
Multinomial diffusion equation
Balter, Ariel; Tartakovsky, Alexandre M.
2011-06-01
We describe a new, microscopic model for diffusion that captures diffusion induced fluctuations at scales where the concept of concentration gives way to discrete particles. We show that in the limit as the number of particles N→∞, our model is equivalent to the classical stochastic diffusion equation (SDE). We test our new model and the SDE against Langevin dynamics in numerical simulations, and show that our model successfully reproduces the correct ensemble statistics, while the classical model fails.
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
Maxwell Equations as the One Photon Quantum Equation
International Nuclear Information System (INIS)
Maxwell equations (Faraday and Ampere-Maxwell laws) can be presented as a three component equation in a way similar to the two component neutrino equation. However, in this case, the electric and magnetic Gauss's laws can not be derived from first principles. We have shown how all Maxwell equations can be derived simultaneously from first principles, similar to those which have been used to derive the Dirac relativistic electron equation. We have 'also- shown that equations for massless particles, derived by Dirac in 1936, lead to the same result. The complex wave function, being a linear combination of the electric and magnetic fields, is a locally measurable quantity. Therefore Maxwell equations should be used as a guideline for proper interpretations of quantum equations
Bitsadze, A V
1963-01-01
Equations of the Mixed Type compiles a series of lectures on certain fundamental questions in the theory of equations of mixed type. This book investigates the series of problems concerning linear partial differential equations of the second order in two variables, and possessing the property that the type of the equation changes either on the boundary of or inside the considered domain. Topics covered include general remarks on linear partial differential equations of mixed type; study of the solutions of second order hyperbolic equations with initial conditions given along the lines of parab
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.
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
Stochastic Geometric Wave Equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Ondreját, Martin
Cham: Springer, 2015, s. 157-188. (Progress in Probability. 68). ISBN 978-3-0348-0908-5. ISSN 1050-6977. [Stochastic analysis and applications at the Centre Interfacultaire Bernoulli, Ecole Polytechnique Fédérale de Lausanne. Lausanne (CH), 09.01.2012-29.6.2012] R&D Projects: GA ČR GAP201/10/0752 Institutional research plan: CEZ:AV0Z10750506 Institutional support: RVO:67985556 Keywords : Stochastic wave equation * Riemannian manifold * homogeneous space Subject RIV: BA - General Mathematics http://library.utia.cas.cz/separaty/2015/SI/ondrejat-0447803.pdf
The nonlinear fragmentation equation
International Nuclear Information System (INIS)
We study the kinetics of nonlinear irreversible fragmentation. Here, fragmentation is induced by interactions/collisions between pairs of particles and modelled by general classes of interaction kernels, for several types of breakage models. We construct initial value and scaling solutions of the fragmentation equations, and apply the 'non-vanishing mass flux' criterion for the occurrence of shattering transitions. These properties enable us to determine the phase diagram for the occurrence of shattering states and of scaling states in the phase space of model parameters. (fast track communication)
Elliptic differential equations
Hackbusch, Wolfgang; Ion, PDF
2010-01-01
The book offers a simultaneous presentation of the theory and of the numerical treatment of elliptic problems. The author starts with a discussion of the Laplace equation in the classical formulation and its discretisation by finite differences and deals with topics of gradually increasing complexity in the following chapters. He introduces the variational formulation of boundary value problems together with the necessary background from functional analysis and describes the finite element method including the most important error estimates. A more advanced chapter leads the reader into the th
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 ent...
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
Makkonen, Lasse
2016-04-01
Young's construction for a contact angle at a three-phase intersection forms the basis of all fields of science that involve wetting and capillary action. We find compelling evidence from recent experimental results on the deformation of a soft solid at the contact line, and displacement of an elastic wire immersed in a liquid, that Young's equation can only be interpreted by surface energies, and not as a balance of surface tensions. It follows that the a priori variable in finding equilibrium is not the position of the contact line, but the contact angle. This finding provides the explanation for the pinning of a contact line. PMID:26940644
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....
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
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.
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.
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
Functional equations for Feynman integrals
International Nuclear Information System (INIS)
New types of equations for Feynman integrals are found. It is shown that Feynman integrals satisfy functional equations connecting integrals with different kinematics. A regular method is proposed for obtaining such relations. The derivation of functional equations for one-loop two-, three- and four-point functions with arbitrary masses and external momenta is given. It is demonstrated that functional equations can be used for the analytic continuation of Feynman integrals to different kinematic domains
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.
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…
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.
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.
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
Resonantly coupled nonlinear evolution equations
International Nuclear Information System (INIS)
A differential matrix eigenvalue problem is used to generate systems of nonlinear evolution equations. They model triad, multitriad, self-modal, and quartet wave interactions. A nonlinear string equation is also recovered as a special case. A continuum limit of the eigenvalue problem and associated evolution equations are discussed. The initial value solution requires an investigation of the corresponding inverse-scattering problem. (auth)
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.
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...
Quadratic bundle and nonlinear equations
International Nuclear Information System (INIS)
The paper is aimed at giving an exhaustive description of the nonlinear evolution equations (NLEE), connected with the quadratic bundle (the spectral parameter lambda, which enters quadratically into the equations) and at describing Hamiltonian structure of these equations. The equations are solved through the inverse scattering method (ISM). The basic formulae for the scattering problem are given. The spectral expansion of the integrodifferential operator is used so that its eigenfunctions are the squared solutions of the equation. By using the notions of Hamiltonian structure hierarchy and gauge transformations it is shown how to single out physically interesting NLEE
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.
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.
Reduction of infinite dimensional equations
Directory of Open Access Journals (Sweden)
Zhongding Li
2006-02-01
Full Text Available In this paper, we use the general Legendre transformation to show the infinite dimensional integrable equations can be reduced to a finite dimensional integrable Hamiltonian system on an invariant set under the flow of the integrable equations. Then we obtain the periodic or quasi-periodic solution of the equation. This generalizes the results of Lax and Novikov regarding the periodic or quasi-periodic solution of the KdV equation to the general case of isospectral Hamiltonian integrable equation. And finally, we discuss the AKNS hierarchy as a special example.
Integral equations and their applications
Rahman, M
2007-01-01
For many years, the subject of functional equations has held a prominent place in the attention of mathematicians. In more recent years this attention has been directed to a particular kind of functional equation, an integral equation, wherein the unknown function occurs under the integral sign. The study of this kind of equation is sometimes referred to as the inversion of a definite integral. While scientists and engineers can already choose from a number of books on integral equations, this new book encompasses recent developments including some preliminary backgrounds of formulations of in
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
$\\Lambda$ 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.
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.
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...
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. .
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
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.
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.
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.
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...
Differential Equations for Algebraic Functions
Bostan, Alin; Chyzak, Frédéric; Salvy, Bruno; Lecerf, Grégoire; Schost, Éric
2007-01-01
It is classical that univariate algebraic functions satisfy linear differential equations with polynomial coefficients. Linear recurrences follow for the coefficients of their power series expansions. We show that the linear differential equation of minimal order has coefficients whose degree is cubic in the degree of the function. We also show that there exists a linear differential equation of order linear in the degree whose coefficients are only of quadratic degree. Furthermore, we prove ...
Perturbed linear rough differential equations
Coutin, Laure; Lejay, Antoine
2014-01-01
We study linear rough differential equations and we solve perturbed linear rough differential equation using the Duhamel principle. These results provide us with the key technical point to study the regularity of the differential of the Itô map in a subsequent article. Also, the notion of linear rough differential equations leads to consider multiplicative functionals with values in Banach algebra more general than tensor algebra and to consider extensions of classical results such as the Mag...
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), ...
Hyperbolic Methods for Einstein's Equations
Directory of Open Access Journals (Sweden)
Reula Oscar
1998-01-01
Full Text Available 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.
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 ...
The generalized Airy diffusion equation
Directory of Open Access Journals (Sweden)
Frank M. Cholewinski
2003-08-01
Full Text Available Solutions of a generalized Airy diffusion equation and an associated nonlinear partial differential equation are obtained. Trigonometric type functions are derived for a third order generalized radial Euler type operator. An associated complex variable theory and generalized Cauchy-Euler equations are obtained. Further, it is shown that the Airy expansions can be mapped onto the Bessel Calculus of Bochner, Cholewinski and Haimo.
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
Equation with the many fathers
DEFF Research Database (Denmark)
Kragh, Helge
1984-01-01
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 of the us...... electrodynamics. Although this ambitious attempt attracted some interest in 1926, its impact on the mainstream of development in quantum mechanics was virtually nil....
Temporal Fokker-Planck Equations
Boon, Jean Pierre; Lutsko, James F.
2016-01-01
The temporal Fokker-Plank equation [{\\it J. Stat. Phys.}, {\\bf 3/4}, 527 (2003)] or propagation-dispersion equation was derived to describe diffusive processes with temporal dispersion rather than spatial dispersion as in classical diffusion. %\\cite{boon-grosfils-lutsko}. We present two generalizations of the temporal Fokker-Plank equation for the first passage distribution function $f_j(r,t)$ of a particle moving on a substrate with time delays $\\tau_j$. Both generalizations follow from the ...
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
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.
Diffusion equations and turbulent transport
International Nuclear Information System (INIS)
One scrutinized transport equations differing essentially in form from the classical diffusion one. Description of diffusion under strong nonequilibrium and turbulence involved application of equations that took account of transport nonlocality and memory effects. One analyzed ways to derive the mentioned equations starting from quasi-linear approximation and up to equations with fractional derivatives. One points out the generality of the applied theoretical concepts in spite of the essential difference of the exact physical problems. One demonstrated the way of application of the theoretical and probabilistic ideas
Diffusion equations and turbulent transport
International Nuclear Information System (INIS)
Diffusion equations are considered that differ substantially in structure from classical ones. A description of diffusion under strongly nonequilibrium conditions in a highly turbulent plasma requires the use of equations that take into account memory effects and the nonlocal nature of transport. Different methods are developed for constructing such equations, ranging from those in the quasilinear approximation to those with fractional derivatives. It is emphasized that the theoretical concepts underlying the equations proposed are common for a very wide variety of specific physical problems. The ways of applying theoretical probabilistic ideas are demonstrated
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.
ON A CORRELATION BETWEEN DIFFERENTIAL EQUATIONS AND THEIR CHARACTERISTIC EQUATIONS
Boro M. Piperevski
2007-01-01
Abstract: The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.
Tippe Top Equations and Equations for the Related Mechanical Systems
Rutstam, Nils
2012-01-01
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 $\\mathbf{\\hat{3}}$ 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.
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.
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
Solving equations by topological methods
Directory of Open Access Journals (Sweden)
Lech Górniewicz
2005-01-01
Full Text Available In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
Solving equations by topological methods
Lech Górniewicz
2005-01-01
In this paper we survey most important results from topological fixed point theory which can be directly applied to differential equations. Some new formulations are presented. We believe that our article will be useful for analysts applying topological fixed point theory in nonlinear analysis and in differential equations.
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.
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.
Differential equations and moving frames
Abib, Odinette Renée
2006-01-01
The purpose of the paper is to study the relationship between differential equations, Pfaffian systems and geometric structures, via the method of moving frames of E.Cartan. We show a local structure theorem. The Lie algebra aspects differential equations is studied too.
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 of...
Solutions to Arithmetic Convolution Equations
Czech Academy of Sciences Publication Activity Database
Glöckner, H.; Lucht, L.G.; Porubský, Štefan
2007-01-01
Roč. 135, č. 6 (2007), s. 1619-1629. ISSN 0002-9939 R&D Projects: GA ČR GA201/04/0381 Institutional research plan: CEZ:AV0Z10300504 Keywords : arithmetic functions * Dirichlet convolution * polynomial equations * analytic equations * topological algebras * holomorphic functional calculus Subject RIV: BA - General Mathematics Impact factor: 0.520, year: 2007
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
Phenomenological equations for reacting fluids
International Nuclear Information System (INIS)
A nonlocal phenomenological equation is introduced for a multicomponent fluid where chemical or nuclear reactions are taking place. The reciprocity between the nonlocal linear-coefficients is examined closely. An approximation reduces the nonlocal equation to the ordinary phenomenological relation with correction terms which show clearly a coupling of the reaction with the diffusion and the thermal conduction in an isotropic system. (auth.)
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.
Saturation and linear transport equation
Energy Technology Data Exchange (ETDEWEB)
Kutak, K.
2009-03-15
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)
Saturation and linear transport equation
International Nuclear Information System (INIS)
We show that the GBW saturation model provides an exact solution to the one dimensional linear transport equation. We also show that it is motivated by the BK equation considered in the saturated regime when the diffusion and the splitting term in the diffusive approximation are balanced by the nonlinear term. (orig.)
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.
Wigner transforms and Liouville equations
International Nuclear Information System (INIS)
Recent works concerning the semi-classical limit (h barred tending to zero) of the Quantum Mechanics linear and non linear models or equations, are presented. The non linear case is corresponding to mean field (or self consistent) models and gives, at the limit, the Vlasov equations of the Classical Statistical Mechanics. 48 refs
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.
Nonlinear evolution equations and the Painleve test
International Nuclear Information System (INIS)
In this paper a survey is given of new results of the Painleve test and nonlinear evolution equations where ordinary- and partial-differential equations are considered. The authors study the semiclassical Haynes-Cumming model, the energy-eigenvalue-level-motion equation, the Kadomtsev-Petviashvili equation, the nonlinear Klein-Gordon equation and the self-dual Yang-Mills equation
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.
Integrability of equations for soliton's eigenfunctions
International Nuclear Information System (INIS)
Eigenfunctions of the auxiliary linear problems for the soliton equations obey the nonlinear evolution equations. It is shown that these eigenfunction equations are integrable by the inverse spectral transform method. Eigenfunction equations are also the generating equations. Several (1+1) and (2+1) dimensional eigenfunction equations and their properties are considered. 11 refs
Conservation laws of semidiscrete canonical Hamiltonian equations
International Nuclear Information System (INIS)
There are many evolution partial differential equations which can be cast into Hamiltonian form. Conservation laws of these equations are related to one-parameter Hamiltonian symmetries admitted by the PDEs. The same result holds for semidiscrete Hamiltonian equations. In this paper we consider semidiscrete canonical Hamiltonian equations. Using symmetries, we find conservation laws for the semidiscretized nonlinear wave equation and Schroedinger equation. (author)
Conservation Laws of Differential Equations in Finance
International Nuclear Information System (INIS)
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...
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.
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.
Generalizing the cosmic energy equation
International Nuclear Information System (INIS)
We generalize the cosmic energy equation to the case when massive particles interact via a modified gravitational potential of the form φ(a,r), which is allowed to explicitly depend upon the cosmological time through the expansion factor a(t). Using the nonrelativistic approximation for particle dynamics, we derive the equation for the cosmological expansion which has the form of the Friedmann equation with a renormalized gravitational constant. The generalized Layzer-Irvine cosmic energy equation and the associated cosmic virial theorem are applied to some recently proposed modifications of the Newtonian gravitational interaction between dark-matter particles. We also draw attention to the possibility that the cosmic energy equation may be used to probe the expansion history of the universe thereby throwing light on the nature of dark matter and dark energy.
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; 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.
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.
Extended Trial Equation Method for Nonlinear Partial Differential Equations
Gepreel, Khaled A.; Nofal, Taher A.
2015-04-01
The main objective of this paper is to use the extended trial equation method to construct a series of some new solutions for some nonlinear partial differential equations (PDEs) in mathematical physics. We will construct the solutions in many different functions such as hyperbolic function solutions, trigonometric function solutions, Jacobi elliptic function solutions, and rational functional solutions for the nonlinear PDEs when the balance number is a real number via the Zhiber-Shabat nonlinear differential equation. The balance number of this method is not constant as we shown in other methods, but it is changed by changing the trial equation derivative definition. This method allowed us to construct many new types of solutions. It is shown by using the Maple software package that all obtained solutions satisfy the original PDEs.
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
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.
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......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...
Lectures on ordinary differential equations
Hurewicz, Witold
2014-01-01
Hailed by The American Mathematical Monthly as ""a rigorous and lively introduction,"" this text explores a topic of perennial interest in mathematics. The author, a distinguished mathematician and formulator of the Hurewicz theorem, presents a clear and lucid treatment that emphasizes geometric methods. Topics include first-order scalar and vector equations, basic properties of linear vector equations, and two-dimensional nonlinear autonomous systems. Suitable for senior mathematics students, the text begins with an examination of differential equations of the first order in one unknown funct
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.
Loop equations from differential systems
Eynard, Bertrand; Marchal, Olivier
2016-01-01
To any differential system $d\\Psi=\\Phi\\Psi$ where $\\Psi$ belongs to a Lie group (a fiber of a principal bundle) and $\\Phi$ is a Lie algebra $\\mathfrak g$ valued 1-form on a Riemann surface $\\Sigma$, is associated an infinite sequence of "correlators" $W_n$ that are symmetric $n$-forms on $\\Sigma^n$. The goal of this article is to prove that these correlators always satisfy "loop equations", the same equations satisfied by correlation functions in random matrix models, or the same equations as Virasoro or W-algebra constraints in CFT.
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
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.
Symmetry Analysis of Telegraph Equation
Nadjafikhah, Mehdi; Hejazi, Seyed Reza
2010-01-01
Lie symmetry group method is applied to study the Telegraph equation. The symmetry group and its optimal system are given, and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.
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.
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.
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...
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
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...
Improved beam propagation method equations.
Nichelatti, E; Pozzi, G
1998-01-01
Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff-Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens. PMID:18268554
Scattering Equations and KLT Orthogonality
Cachazo, Freddy; He, Song; Yuan, Ellis Ye
2013-01-01
Several recent developments point to the fact that rational maps from n-punctured spheres to the null cone of D dimensional momentum space provide a natural language for describing the scattering of massless particles in D dimensions. In this note we identify and study equations relating the kinematic invariants and the puncture locations, which we call the scattering equations. We provide an inductive algorithm in the number of particles for their solutions and prove a remarkable property wh...
Czech Academy of Sciences Publication Activity Database
Agarwal, R. P.; Bohner, Martin; Řehák, Pavel
Dordrecht : Kluwer Academic Publishers, 2003 - (Agarwal, R.; O´Regan, D.), s. 1-56 Grant ostatní: GA ČR(CZ) GA201/01/0079; GA ČR(CZ) GP201/01/P041 Institutional research plan: CEZ:AV0Z1019905 Keywords : dynamic equation s * time scales * half-linear equation s Subject RIV: BA - General Mathematics
On solving periodic Riccati equations
Varga, Andreas
2008-01-01
Numerically reliable algorithms to compute the periodic non-negative definite stabilizing solutions of the periodic differential Riccati equation (PRDE) and discrete-time periodic Riccati equation (DPRE) are proposed. For the numerical solution of PRDEs, a new multiple shooting-type algorithm is developed to compute the periodic solutions in an arbitrary number of time moments within one period by employing suitable discretizations of the continuous-time problems. In contrast to single shooti...
Stochastic Geometric Partial Differential Equations
Czech Academy of Sciences Publication Activity Database
Brzezniak, Z.; Goldys, B.; Ondreját, Martin
1. Singapore : World Scientific Publishing Company, 2011 - (Zhao, H.; Truman, A.), s. 1-32 ISBN 978-981-4360-91-3. - (Interdisciplinary Mathematical Sciences. 12) R&D Projects: GA ČR GAP201/10/0752 Institutional support: RVO:67985556 Keywords : stochastic geometric * partial differential equations Subject RIV: BA - General Mathematics http://library.utia.cas.cz/separaty/2012/SI/ondrejat-stochastic geometric partial differential equations. pdf
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.
Hydrodynamic equations for granular mixtures
Garzo, V.; Dufty, J. W.
2003-01-01
Many features of granular media can be modeled by a fluid of hard spheres with inelastic collisions. Under rapid flow conditions, the macroscopic behavior of grains can be described through hydrodynamic equations accounting for dissipation among the interacting particles. A basis for the derivation of hydrodynamic equations and explicit expressions appearing in them is provided by the Boltzmann kinetic theory conveniently modified to account for inelastic binary collisions. The goal of this r...
Boltzmann equation and hydrodynamic fluctuations.
Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian
2009-11-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972
Boltzmann equation and hydrodynamic fluctuations
Colangeli, M.; Kroger, M.; Ottinger, H. C.
2009-01-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.
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.
Wave equations for pulse propagation
International Nuclear Information System (INIS)
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. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation
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 solutions of the generalized Bretherton equation
Energy Technology Data Exchange (ETDEWEB)
Kudryashov, Nikolai A., E-mail: nakudr@gmail.co [Department of Applied Mathematics, National Research Nuclear University, MEPHI, 31 Kashirskoe Shosse, 115409 Moscow (Russian Federation); Sinelshchikov, Dmitry I.; Demina, Maria V. [Department of Applied Mathematics, National Research Nuclear University, MEPHI, 31 Kashirskoe Shosse, 115409 Moscow (Russian Federation)
2011-02-14
The generalized Bretherton equation is studied. The Baecklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.
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.)
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.
Linear integral equations and soliton systems
International Nuclear Information System (INIS)
A study is presented of classical integrable dynamical systems in one temporal and one spatial dimension. The direct linearizations are given of several nonlinear partial differential equations, for example the Korteweg-de Vries equation, the modified Korteweg-de Vries equation, the sine-Gordon equation, the nonlinear Schroedinger equation, and the equation of motion for the isotropic Heisenberg spin chain; the author also discusses several relations between these equations. The Baecklund transformations of these partial differential equations are treated on the basis of a singular transformation of the measure (or equivalently of the plane-wave factor) occurring in the corresponding linear integral equations, and the Baecklund transformations are used to derive the direct linearization of a chain of so-called modified partial differential equations. Finally it is shown that the singular linear integral equations lead in a natural way to the direct linearizations of various nonlinear difference-difference equations. (Auth.)
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.
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.
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
Exponential function approach to parabolic equations
Lin, Chin-Yuan
2014-01-01
This volume is on initial-boundary value problems for parabolic partial differential equations of second order. It rewrites the problems as abstract Cauchy problems or evolution equations, and then solves them by the technique of elementary difference equations. Because of this, the volume assumes less background and provides an easy approach for readers to understand. Contents:Existence Theorems for Cauchy ProblemsExistence Theorems for Evolution Equations (I)Linear Autonomous Parabolic EquationsNonlinear Autonomous Parabolic EquationsLinea
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.
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.
Scattering Equations and KLT Orthogonality
Cachazo, Freddy; Yuan, Ellis Ye
2013-01-01
Several recent developments point to the fact that rational maps from n-punctured spheres to the null cone of D dimensional momentum space provide a natural language for describing the scattering of massless particles in D dimensions. In this note we identify and study equations relating the kinematic invariants and the puncture locations, which we call the scattering equations. We provide an inductive algorithm in the number of particles for their solutions and prove a remarkable property which we call KLT Orthogonality. In a nutshell, KLT orthogonality means that "Parke-Taylor" vectors constructed from the solutions to the scattering equations are mutually orthogonal with respect to the Kawai-Lewellen-Tye (KLT) bilinear form. We end with comments on possible connections to gauge theory and gravity amplitudes in any dimension and to the high-energy limit of string theory amplitudes.
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.
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...
Confidence interval in Kirsch equations
Directory of Open Access Journals (Sweden)
Morteza Khodabin
2014-09-01
Full Text Available Rocks at depth are affected by stresses resulting from the weight of the overlying strata and tectonic stresses. When a tunnel is excavated in this rock, the stress field is locally disordered and radial, tangential and shear stresses are induced in the rock around the tunnel. Knowledge of the magnitudes and directions of these induced stresses is essential. Since the measuring of specific gravity and depth are inevitably affected by environmental noise, we consider a random version of P2 in Kirsch equations. By doing this, we define random version of the Kirsch equations. Then we introduce an algorithm to calculate confidence intervals for the Kirsch parameters. Finally we use Alborz tunnel characteristics for creating these confidence intervals as a case study. The results show that the proposed amounts of radial, tangential and shear stresses lie in desired range. Keywords: Kirsch equations; Confidence interval; confidence level; Alborz tunnel; normal distribution.
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.
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.
Dynamics of partial differential equations
Wayne, C Eugene
2015-01-01
This book contains two review articles on the dynamics of partial differential equations that deal with closely related topics but can be read independently. Wayne reviews recent results on the global dynamics of the two-dimensional Navier-Stokes equations. This system exhibits stable vortex solutions: the topic of Wayne's contribution is how solutions that start from arbitrary initial conditions evolve towards stable vortices. Weinstein considers the dynamics of localized states in nonlinear Schrodinger and Gross-Pitaevskii equations that describe many optical and quantum systems. In this contribution, Weinstein reviews recent bifurcations results of solitary waves, their linear and nonlinear stability properties, and results about radiation damping where waves lose energy through radiation. The articles, written independently, are combined into one volume to showcase the tools of dynamical systems theory at work in explaining qualitative phenomena associated with two classes of partial differential equ...
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...
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
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
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
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
Quantum Gross-Pitaevskii Equation
Haegeman, Jutho; Stojevic, Vid; Cirac, J Ignacio; Osborne, Tobias J; Verstraete, Frank
2015-01-01
We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum field theories. This generalization is obtained by applying the Dirac-Frenkel time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of the many body system including entanglement and correlations and thus extends significantly beyond the usual mean-field description of the Gross-Pitaevskii equation, which is known to fail for one-dimensional systems.
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.
Geometric approach to soliton equations
International Nuclear Information System (INIS)
A class of nonlinear equations that can be solved in terms of nxn scattering problem is investigated. A systematic geometric method of exploiting conservation laws and related equations, the so-called prolongation structure, is worked out. The nxn problem is reduced to nsub(n-1)x(n-1) problems and finally to 2x2 problems, which have been comprehensively investigated recently by the author. A general method of deriving the infinite numbers of polynomial conservation laws for an nxn problem is presented. The cases of 3x3 and 2x2 problems are discussed explicitly. (Auth.)
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
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
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.
Modular equations and lattice sums
Rogers, Mathew; Yuttanan, Boonrod
2010-01-01
We highlight modular equations discovered by Somos and Ramanujan, and use them to prove new relations between lattice sums and hypergeometric functions. We also discuss progress towards solving Boyd's Mahler measure conjectures, and we conjecture a new formula for $L(E,2)$ of conductor 17 elliptic curves.
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
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].
Non Monotone Stochastic Evolution Equations
Kenneth L. Kuttler; Li, Ji
2013-01-01
An approach to stochastic evolution equations based on a simple generalization of known embedding theorems is presented. It allows for the inclusion of problems which have nonlinear non monotone operators. This is used to discuss the existence of strong solutions to a stochastic Navier Stokes problem in dimension less than four.
Convexity, Differential Equations, and Games
Flåm, Sjur Didrik
2002-01-01
Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.
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
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…
Conservation laws and kinetic equations
International Nuclear Information System (INIS)
It is shown that the conservation of a magnitude in a kinetic process can be stated at two different levels (microscopic and macroscopic) determining non-equivalent conditions on the associated transition probabilities. These conditions and related examples are explicitly given in the case of the Boltzmann equation. (author). 5 refs
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.
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.
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
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
Thermoviscous Model Equations in Nonlinear Acoustics
DEFF Research Database (Denmark)
Rasmussen, Anders Rønne
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......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 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....
More conservative governing equations in RELAP5: Derivation of equations
International Nuclear Information System (INIS)
Highlights: • The “non-conservative” numerical approximation is used in current versions of RELAP5. • Mass and energy errors increase for some transients due to non-conservativism. • This paper shows the derivation of a numerical approach to eliminate the mass error. • The second article (Fu et al., 2015) shows the (strategic) solution for the approach. - Abstract: The design and analysis of the thermal/hydraulic systems in nuclear power plants necessitate system codes that can be used in the analysis of steady state and transient conditions. RELAP5 is one of the most commonly used system codes in nuclear organizations. RELAP5 is based on a two-fluid, non-equilibrium, non-homogeneous, hydrodynamic model for the transient simulation of the two-phase system behavior. This model includes six governing equations to describe the mass, energy, and momentum of the two fluids. The “non-conservative” numerical approximation form (which is the current version of RELAP5 code) is obtained through the manipulation of selected derivative terms in the equations including the linearization of the product terms in the time derivatives of the equations. In the non-conservative technique, the truncation errors introduced in the linearization process can produce mass and energy errors for some classes of transients during time advancements, either resulting in (a) automatic reduction of time steps used in the advancement of the equations and increased run times or (b) the growth of unacceptably large errors in the transient results. To eliminate these difficulties, an optional numerical approach has been introduced in RELAP/SCDAPSIM/MOD4.0. This approach uses a more consistent set of “conservative” numerical approximations to solve non-linearized mass and energy governing equations. The RELAP/SCDAPSIM/MOD4.0 code, being developed as part of the international Severe Core Damage Analysis Package (SCDAP) Development and Training Program (SDTP), is the first version of
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...
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.
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...
Maxwell's equations of electrodynamics an explanation
Ball, David W
2012-01-01
Maxwell's Equations of Electrodynamics: An Explanation is a concise discussion of Maxwell's four equations of electrodynamics - the fundamental theory of electricity, magnetism, and light. It guides readers step-by-step through the vector calculus and development of each equation. Pictures and diagrams illustrate what the equations mean in basic terms. The book not only provides a fundamental description of our universe but also explains how these equations predict the fact that light is better described as "electromagnetic radiation."
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 ...
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....... Among the special features of this book can be mentioned the presentation of a practical approach to reliable estimates of the global error, including warning signals if the reliability is questionable. The technique is generally applicable for estimating the discretization error in numerical...... expense. Problems in two space dimensions are effectively handled using the Alternating Direction Implicit (ADI) technique. We present a systematic way of incorporating inhomogeneous terms and derivative boundary conditions in ADI methods as well as mixed derivative terms....
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.
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.
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.
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.
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.
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
Lagrange-Poincare field equations
Ellis, David C P; Holm, Darryl D; Ratiu, Tudor S
2009-01-01
The Lagrange-Poincare equations of classical mechanics are cast into a field theoretic context together with their associated constrained variational principle. An integrability/reconstruction condition is established that relates solutions of the original problem with those of the reduced problem. The Kelvin-Noether theorem is formulated in this context. Applications to the isoperimetric problem, the Skyrme model for meson interaction, metamorphosis image dynamics, and molecular strands illustrate various aspects of the theory.
Some Applications of Fractional Equations
Weitzner, H.; Zaslavsky, G. M.
2002-01-01
We present two observations related to theapplication of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power tails in the long time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear media with fractal properties is considere...
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...
Riccati Equations and their Solution
Czech Academy of Sciences Publication Activity Database
Kučera, Vladimír
Control System Advanced Methods. Boca Raton: CRC Press, 2011 - (Lewine, W.), s. 14.1-14.21. (Electrical Engineering Handbook). ISBN 978-1-4200-7366-9 R&D Projects: GA MŠk(CZ) 1M0567 Institutional research plan: CEZ:AV0Z10750506 Institutional support: RVO:67985556 Keywords : Riccati equation * optimal control * solution Subject RIV: BC - Control Systems Theory http://library.utia.cas.cz/separaty/2011/TR/kucera-0436431.pdf
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.
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
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.
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...
The complex chemical Langevin equation
International Nuclear Information System (INIS)
The chemical Langevin equation (CLE) is a popular simulation method to probe the stochastic dynamics of chemical systems. The CLE’s main disadvantage is its break down in finite time due to the problem of evaluating square roots of negative quantities whenever the molecule numbers become sufficiently small. We show that this issue is not a numerical integration problem, rather in many systems it is intrinsic to all representations of the CLE. Various methods of correcting the CLE have been proposed which avoid its break down. We show that these methods introduce undesirable artefacts in the CLE’s predictions. In particular, for unimolecular systems, these correction methods lead to CLE predictions for the mean concentrations and variance of fluctuations which disagree with those of the chemical master equation. We show that, by extending the domain of the CLE to complex space, break down is eliminated, and the CLE’s accuracy for unimolecular systems is restored. Although the molecule numbers are generally complex, we show that the “complex CLE” predicts real-valued quantities for the mean concentrations, the moments of intrinsic noise, power spectra, and first passage times, hence admitting a physical interpretation. It is also shown to provide a more accurate approximation of the chemical master equation of simple biochemical circuits involving bimolecular reactions than the various corrected forms of the real-valued CLE, the linear-noise approximation and a commonly used two moment-closure approximation
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.
Effective Schroedinger equations on submanifolds
International Nuclear Information System (INIS)
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 ε3 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 ε3. 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.
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.
Dynamical equations and approximation methods
International Nuclear Information System (INIS)
The integral equations approach to the three-body problem, decisively stimulated by Faddeev's formulation, provides the most powerful tool for studying the internal structure of this system. An essential step towards a detailed understanding of composite particle dynamics has been done in this way. The search for adequate extensions to the general N-body situation therefore represented, and still represents a natural challenge. For various reasons this transition is non-trivial and non-unique. Emphasizing different aspects of the three-body theory, different generalizations have been found. In particular, it was the concept of connectedness of the (iterated) integral kernel which allows for an arbitrary number of formulations, many of them being presumably only mathematically correct, but physically rather unsatisfactory. Therefore, the present status of the N-body theory is reviewed in a less technical way. Starting from the basic, physically convincing definitions of scattering states, the defining equations are replaced by more appropriate matrix relations. This is done in a reversible way, thus preserving in every step the original structure and information. In order to be as close as possible to the basic definitions, all relations are first derived for scattering states or half-on-shell transition amplitudes. The ambiguity in going over to corresponding operator identities (fully off-shell equations) is demonstrated. (Auth.)
Neutron star equation of state
International Nuclear Information System (INIS)
Experimental information concerning the equation of state in neutron stars is lacking, because of the necessary extrapolations in both density and neutron excess from the nearly symmetric nuclear matter observed in nuclei. However, the combination of new developments in the theory of neutron star structure and in astronomical observations provides important constraints. From a theoretical perspective, it is argued that the extrapolation in neutron excess is more crucial for neutron star structure than is the density extrapolation. For example, the radius of neutron stars is primarily a function of the pressure of matter in the vicinity of nuclear matter density, which is essentially determined by the isospin properties of dense matter. In the absence of extreme softening in the dense matter equation of state, a measurement of the radius of a neutron star more accurate than about 1 km will usefully constrain the equation of state. In addition, the moment of inertial and the binding energy of neutron stars are nearly universal functions of the star's compactness. The potential constraints that can be deduced from observations of thermal emission from young neutron stars, neutrinos from newly born neutron stars, Quasi-Periodic Oscillations from X-ray emitting neutron stars in binaries, and glitches from pulsars are discussed
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
Estimation of saturation and coherence effects in the KGBJS equation - a non-linear CCFM equation
Deak, Michal
2012-01-01
We solve the modified non-linear extension of the CCFM equation - KGBJS equation - numerically for certain initial conditions and compare the resulting gluon Green functions with those obtained from solving the original CCFM equation and the BFKL and BK equations for the same initial conditions. We improve the low transversal momentum behaviour of the KGBJS equation by a small modification.
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)
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
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.
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