A generalized simplest equation method and its application to the Boussinesq-Burgers equation.

Science.gov (United States)

Sudao, Bilige; Wang, Xiaomin

2015-01-01

In this paper, a generalized simplest equation method is proposed to seek exact solutions of nonlinear evolution equations (NLEEs). In the method, we chose a solution expression with a variable coefficient and a variable coefficient ordinary differential auxiliary equation. This method can yield a Bäcklund transformation between NLEEs and a related constraint equation. By dealing with the constraint equation, we can derive infinite number of exact solutions for NLEEs. These solutions include the traveling wave solutions, non-traveling wave solutions, multi-soliton solutions, rational solutions, and other types of solutions. As applications, we obtained wide classes of exact solutions for the Boussinesq-Burgers equation by using the generalized simplest equation method.

Exact periodic solutions of the sixth-order generalized Boussinesq equation

International Nuclear Information System (INIS)

Kamenov, O Y

2009-01-01

This paper examines a class of nonlinear sixth-order generalized Boussinesq-like equations (SGBE): u tt = u xx + 3(u 2 ) xx + u xxxx + αu xxxxxx , 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 Devising Boussinesq-type Equations with Bounded Eigenspectra: Two Horizontal Dimensions

DEFF Research Database (Denmark)

Eskilsson, Claes; Engsig-Karup, Allan Peter

2015-01-01

Boussinesq-type equations are used to describe the propagation and transformation of free-surface waves in the nearshore region. The nonlinear and dispersive performance of the equations are determined by tunable parameters. Recently the authors presented conditions on the free parameters under...... study and provide numerical experimentswhich confirms the theoretical results also is valid in two horizontal dimensions....

problem for the damped Boussinesq equation

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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.

Exact periodic solutions of the sixth-order generalized Boussinesq equation

Energy Technology Data Exchange (ETDEWEB)

Kamenov, O Y [Department of Applied Mathematics and Informatics, Technical University of Sofia, PO Box 384, 1000 Sofia (Bulgaria)], E-mail: okam@abv.bg

2009-09-18

This paper examines a class of nonlinear sixth-order generalized Boussinesq-like equations (SGBE): u{sub tt} = u{sub xx} + 3(u{sup 2}){sub xx} + u{sub xxxx} + {alpha}u{sub xxxxxx}, {alpha} in R, depending on the positive parameter {alpha}. 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.

Symmetry Reductions, Integrability and Solitary Wave Solutions to High-Order Modified Boussinesq Equations with Damping Term

Science.gov (United States)

Yan, Zhen-Ya; Xie, Fu-Ding; Zhang, Hong-Qing

2001-07-01

Both the direct method due to Clarkson and Kruskal and the improved direct method due to Lou are extended to reduce the high-order modified Boussinesq equation with the damping term (HMBEDT) arising in the general Fermi-Pasta-Ulam model. As a result, several types of similarity reductions are obtained. It is easy to show that the nonlinear wave equation is not integrable under the sense of Ablowitz's conjecture from the reduction results obtained. In addition, kink-shaped solitary wave solutions, which are of important physical significance, are found for HMBEDT based on the obtained reduction equation. The project supported by National Natural Science Foundation of China under Grant No. 19572022, the National Key Basic Research Development Project Program of China under Grant No. G1998030600 and Doctoral Foundation of China under Grant No. 98014119

Auto-Baecklund Transformation and Analytic Solutions of (2+1)-Dimensional Boussinesq Equation

International Nuclear Information System (INIS)

Liu Guanting

2008-01-01

Using the truncated Painleve expansion, symbolic computation, and direct integration technique, we study analytic solutions of (2+1)-dimensional Boussinesq equation. An auto-Baecklund transformation and a number of exact solutions of this equation have been found. The set of solutions include solitary wave solutions, solitoff solutions, and periodic solutions in terms of elliptic Jacobi functions and Weierstrass wp function. Some of them are novel.

Some analytical solutions of the linearized Boussinesq equation with recharge for a sloping aquifer

NARCIS (Netherlands)

Verhoest, N.; Troch, P.A.

2000-01-01

Subsurface flow from a hillslope can be described by the hydraulic groundwater theory as formulated by the Boussinesq equation. Several attempts have been made to solve this partial differential equation, and exact solutions have been found for specific situations. In the case of a sloping aquifer,

Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations.

Science.gov (United States)

Vorobev, Anatoliy

2010-11-01

We use the Cahn-Hilliard approach to model the slow dissolution dynamics of binary mixtures. An important peculiarity of the Cahn-Hilliard-Navier-Stokes equations is the necessity to use the full continuity equation even for a binary mixture of two incompressible liquids due to dependence of mixture density on concentration. The quasicompressibility of the governing equations brings a short time-scale (quasiacoustic) process that may not affect the slow dynamics but may significantly complicate the numerical treatment. Using the multiple-scale method we separate the physical processes occurring on different time scales and, ultimately, derive the equations with the filtered-out quasiacoustics. The derived equations represent the Boussinesq approximation of the Cahn-Hilliard-Navier-Stokes equations. This approximation can be further employed as a universal theoretical model for an analysis of slow thermodynamic and hydrodynamic evolution of the multiphase systems with strongly evolving and diffusing interfacial boundaries, i.e., for the processes involving dissolution/nucleation, evaporation/condensation, solidification/melting, polymerization, etc.

On double reductions from symmetries and conservation laws for a damped Boussinesq equation

International Nuclear Information System (INIS)

Gandarias, M.L.; Rosa, M.

2016-01-01

In this work, we study a Boussinesq equation with a strong damping term from the point of view of the Lie theory. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. Some nontrivial conservation laws are derived by using the multipliers method. Taking into account the relationship between symmetries and conservation laws and applying the double reduction method, we obtain a direct reduction of order of the ordinary differential equations and in particular a kink solution.

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

International Nuclear Information System (INIS)

Ma Zhi-Min; Sun Yu-Huai; Liu Fu-Sheng

2013-01-01

In this paper, the generalized Boussinesq wave equation u tt — u xx + a(u m ) xx + bu xxxx = 0 is investigated by using the bifurcation theory and the method of phase portraits analysis. Under the different parameter conditions, the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained. (general)

Blowup with vorticity control for a 2D model of the Boussinesq equations

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Hoang, V.; Orcan-Ekmekci, B.; Radosz, M.; Yang, H.

2018-06-01

We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a vorticity stretching term and a non-local Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.

Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method

Directory of Open Access Journals (Sweden)

Rahmatullah

2018-03-01

Full Text Available We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses. Keywords: Exp-function method, New exact traveling wave solutions, Modified Riemann-Liouville derivative, Fractional complex transformation, Fractional order Boussinesq-like equations, Symbolic computation

DG-FEM solution for nonlinear wave-structure interaction using Boussinesq-type equations

DEFF Research Database (Denmark)

Engsig-Karup, Allan Peter; Hesthaven, Jan; Bingham, Harry B.

2008-01-01

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...... waters within the breaking limit. To demonstrate the current applicability of the model both linear and mildly nonlinear test cases are considered in two horizontal dimensions where the water waves interact with bottom-mounted fully reflecting structures. It is established that, by simple symmetry...... considerations combined with a mirror principle, it is possible to impose weak slip boundary conditions for both structured and general curvilinear wall boundaries while maintaining the accuracy of the scheme. As is standard for current high-order Boussinesq-type models, arbitrary waves can be generated...

On the Boussinesq-Burgers equations driven by dynamic boundary conditions

Science.gov (United States)

Zhu, Neng; Liu, Zhengrong; Zhao, Kun

2018-02-01

We study the qualitative behavior of the Boussinesq-Burgers equations on a finite interval subject to the Dirichlet type dynamic boundary conditions. Assuming H1 ×H2 initial data which are compatible with boundary conditions and utilizing energy methods, we show that under appropriate conditions on the dynamic boundary data, there exist unique global-in-time solutions to the initial-boundary value problem, and the solutions converge to the boundary data as time goes to infinity, regardless of the magnitude of the initial data.

Nodal DG-FEM solution of high-order Boussinesq-type equations

DEFF Research Database (Denmark)

Engsig-Karup, Allan Peter; Hesthaven, Jan S.; Bingham, Harry B.

2006-01-01

We present a discontinuous Galerkin finite element method (DG-FEM) solution to a set of high-order Boussinesq-type equations for modelling highly nonlinear and dispersive water waves in one and two horizontal dimensions. The continuous equations are discretized using nodal polynomial basis...... 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...

Four-level conservative finite-difference schemes for Boussinesq paradigm equation

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Kolkovska, N.

2013-10-01

In this paper a two-parametric family of four level conservative finite difference schemes is constructed for the multidimensional Boussinesq paradigm equation. The schemes are explicit in the sense that no inner iterations are needed for evaluation of the numerical solution. The preservation of the discrete energy with this method is proved. The schemes have been numerically tested on one soliton propagation model and two solitons interaction model. The numerical experiments demonstrate that the proposed family of schemes has second order of convergence in space and time steps in the discrete maximal norm.

Exact solutions of fractional mBBM equation and coupled system of fractional Boussinesq-Burgers

Science.gov (United States)

Javeed, Shumaila; Saif, Summaya; Waheed, Asif; Baleanu, Dumitru

2018-06-01

The new exact solutions of nonlinear fractional partial differential equations (FPDEs) are established by adopting first integral method (FIM). The Riemann-Liouville (R-L) derivative and the local conformable derivative definitions are used to deal with the fractional order derivatives. The proposed method is applied to get exact solutions for space-time fractional modified Benjamin-Bona-Mahony (mBBM) equation and coupled time-fractional Boussinesq-Burgers equation. The suggested technique is easily applicable and effectual which can be implemented successfully to obtain the solutions for different types of nonlinear FPDEs.

Rigorous Numerics for ill-posed PDEs: Periodic Orbits in the Boussinesq Equation

Science.gov (United States)

Castelli, Roberto; Gameiro, Marcio; Lessard, Jean-Philippe

2018-04-01

In this paper, we develop computer-assisted techniques for the analysis of periodic orbits of ill-posed partial differential equations. As a case study, our proposed method is applied to the Boussinesq equation, which has been investigated extensively because of its role in the theory of shallow water waves. The idea is to use the symmetry of the solutions and a Newton-Kantorovich type argument (the radii polynomial approach) to obtain rigorous proofs of existence of the periodic orbits in a weighted ℓ1 Banach space of space-time Fourier coefficients with exponential decay. We present several computer-assisted proofs of the existence of periodic orbits at different parameter values.

Existence and non-uniqueness of global weak solutions to inviscid primitive and Boussinesq equations

Czech Academy of Sciences Publication Activity Database

Chiodaroli, E.; Michálek, Martin

2017-01-01

Roč. 353, č. 3 (2017), s. 1201-1216 ISSN 0010-3616 EU Projects: European Commission(XE) 320078 - MATHEF Institutional support: RVO:67985840 Keywords : Boussinesq equations * global weak solutions Subject RIV: BA - General Mathematics OBOR OECD: Pure mathematics Impact factor: 2.500, year: 2016 https://link.springer.com/article/10.1007%2Fs00220-017-2846-5

Mechanical Balance Laws for Boussinesq Models of Surface Water Waves

Science.gov (United States)

Ali, Alfatih; Kalisch, Henrik

2012-06-01

Depth-integrated long-wave models, such as the shallow-water and Boussinesq equations, are standard fare in the study of small amplitude surface waves in shallow water. While the shallow-water theory features conservation of mass, momentum and energy for smooth solutions, mechanical balance equations are not widely used in Boussinesq scaling, and it appears that the expressions for many of these quantities are not known. This work presents a systematic derivation of mass, momentum and energy densities and fluxes associated with a general family of Boussinesq systems. The derivation is based on a reconstruction of the velocity field and the pressure in the fluid column below the free surface, and the derivation of differential balance equations which are of the same asymptotic validity as the evolution equations. It is shown that all these mechanical quantities can be expressed in terms of the principal dependent variables of the Boussinesq system: the surface excursion η and the horizontal velocity w at a given level in the fluid.

Analytical approach to (2+1)-dimensional Boussinesq equation and (3+1)-dimensional Kadomtsev-Petviashvili equation

Energy Technology Data Exchange (ETDEWEB)

Sariaydin, Selin; Yildirim, Ahmet [Ege Univ., Dept. of Mathematics, Bornova-Izmir (Turkey)

2010-05-15

In this paper, we studied the solitary wave solutions of the (2+1)-dimensional Boussinesq equation u{sub tt} - u{sub xx} - u{sub yy} - (u{sup 2}){sub xx} - u{sub xxxx} = 0 and the (3+1)-dimensional Kadomtsev-Petviashvili (KP) equation u{sub xt} - 6u{sub x}{sup 2} + 6uu{sub xx} - u{sub xxxx} - u{sub yy} - u{sub zz} = 0. By using this method, an explicit numerical solution is calculated in the form of a convergent power series with easily computable components. To illustrate the application of this method numerical results are derived by using the calculated components of the homotopy perturbation series. The numerical solutions are compared with the known analytical solutions. Results derived from our method are shown graphically. (orig.)

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Science.gov (United States)

Liu, Ping; Wang, Ya-Xiong; Ren, Bo; Li, Jin-Hua

2016-12-01

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space. Supported by the National Natural Science Foundation of China under Grant Nos. 11305031 and 11305106, and Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province under Grant No. Yq2013205

Smooth and non-smooth travelling waves in a nonlinearly dispersive Boussinesq equation

International Nuclear Information System (INIS)

Shen Jianwei; Xu Wei; Lei Youming

2005-01-01

The dynamical behavior and special exact solutions of nonlinear dispersive Boussinesq equation (B(m,n) equation), u tt -u xx -a(u n ) xx +b(u m ) xxxx =0, is studied by using bifurcation theory of dynamical system. As a result, all possible phase portraits in the parametric space for the travelling wave system, solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth and non-smooth periodic wave solutions are obtained. It can be shown that the existence of singular straight line in the travelling wave system is the reason why smooth waves converge to cusp waves, finally. When parameter are varied, under different parametric conditions, various sufficient conditions guarantee the existence of the above solutions are given

Analytic self-similar solutions of the Oberbeck–Boussinesq equations

International Nuclear Information System (INIS)

Barna, I.F.; Mátyás, L.

2015-01-01

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Bénard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes

New solitary solutions with compact support for Boussinesq-like B(2n, 2n) equations with fully nonlinear dispersion

International Nuclear Information System (INIS)

Zhu Yonggui; Lu Chao

2007-01-01

In this paper, the Boussinesq-like equations with fully nonlinear dispersion, B(2n, 2n) equations: u tt + (u 2n ) xx + (u 2n ) xxxx 0 which exhibit compactons: solitons with compact support, are studied. New exact solitary solutions with compact support are found. The special case B(2, 2) is chosen to illustrate the concrete scheme of the decomposition method in B(2n, 2n) equations. General formulas for the solutions of B(2n, 2n) equations are established

Exact Solutions of Generalized Modified Boussinesq, Kuramoto-Sivashinsky, and Camassa-Holm Equations via Double Reduction Theory

Directory of Open Access Journals (Sweden)

Zulfiqar Ali

2013-01-01

Full Text Available We find exact solutions of the Generalized Modified Boussinesq (GMB equation, the Kuromoto-Sivashinsky (KS equation the and, Camassa-Holm (CH equation by utilizing the double reduction theory related to conserved vectors. The fourth order GMB equation involves the arbitrary function and mixed derivative terms in highest derivative. The partial Noether’s approach yields seven conserved vectors for GMB equation and one conserved for vector KS equation. Due to presence of mixed derivative term the conserved vectors for GMB equation derived by the Noether like theorem do not satisfy the divergence relationship. The extra terms that constitute the trivial part of conserved vectors are adjusted and the resulting conserved vectors satisfy the divergence property. The double reduction theory yields two independent solutions and one reduction for GMB equation and one solution for KS equation. For CH equation two independent solutions are obtained elsewhere by double reduction theory with the help of conserved Vectors.

Solitary wave solutions of the fourth order Boussinesq equation through the exp(-Ф(η))-expansion method.

Science.gov (United States)

Akbar, M Ali; Hj Mohd Ali, Norhashidah

2014-01-01

The exp(-Ф(η))-expansion method is an ascending method for obtaining exact and solitary wave solutions for nonlinear evolution equations. In this article, we implement the exp(-Ф(η))-expansion method to build solitary wave solutions to the fourth order Boussinesq equation. The procedure is simple, direct and useful with the help of computer algebra. By using this method, we obtain solitary wave solutions in terms of the hyperbolic functions, the trigonometric functions and elementary functions. The results show that the exp(-Ф(η))-expansion method is straightforward and effective mathematical tool for the treatment of nonlinear evolution equations in mathematical physics and engineering. 35C07; 35C08; 35P99.

Singular solitons and other solutions to a couple of nonlinear wave equations

International Nuclear Information System (INIS)

Inc Mustafa; Ulutaş Esma; Biswas Anjan

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

Traveling wave solutions of the Boussinesq equation via the new approach of generalized (G'/G)-expansion method.

Science.gov (United States)

Alam, Md Nur; Akbar, M Ali; Roshid, Harun-Or-

2014-01-01

Exact solutions of nonlinear evolution equations (NLEEs) play a vital role to reveal the internal mechanism of complex physical phenomena. In this work, the exact traveling wave solutions of the Boussinesq equation is studied by using the new generalized (G'/G)-expansion method. Abundant traveling wave solutions with arbitrary parameters are successfully obtained by this method and the wave solutions are expressed in terms of the hyperbolic, trigonometric, and rational functions. It is shown that the new approach of generalized (G'/G)-expansion method is a powerful and concise mathematical tool for solving nonlinear partial differential equations in mathematical physics and engineering. 05.45.Yv, 02.30.Jr, 02.30.Ik.

The Boussinesq Debate: Reversibility, Instability, and Free Will.

Science.gov (United States)

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.

Exact traveling wave solutions of fractional order Boussinesq-like equations by applying Exp-function method

Science.gov (United States)

Rahmatullah; Ellahi, Rahmat; Mohyud-Din, Syed Tauseef; Khan, Umar

2018-03-01

We have computed new exact traveling wave solutions, including complex solutions of fractional order Boussinesq-Like equations, occurring in physical sciences and engineering, by applying Exp-function method. The method is blended with fractional complex transformation and modified Riemann-Liouville fractional order operator. Our obtained solutions are verified by substituting back into their corresponding equations. To the best of our knowledge, no other technique has been reported to cope with the said fractional order nonlinear problems combined with variety of exact solutions. Graphically, fractional order solution curves are shown to be strongly related to each other and most importantly, tend to fixate on their integer order solution curve. Our solutions comprise high frequencies and very small amplitude of the wave responses.

Behaviour of the extended Toda lattice

Science.gov (United States)

Wattis, Jonathan A. D.; Gordoa, Pilar R.; Pickering, Andrew

2015-11-01

We consider the first member of an extended Toda lattice hierarchy. This system of equations is differential with respect to one independent variable and differential-delay with respect to a second independent variable. We use asymptotic analysis to consider the long wavelength limits of the system. By considering various magnitudes for the parameters involved, we derive reduced equations related to the Korteweg-de Vries and potential Boussinesq equations.

Parallelization of elliptic solver for solving 1D Boussinesq model

Science.gov (United States)

Tarwidi, D.; Adytia, D.

2018-03-01

In this paper, a parallel implementation of an elliptic solver in solving 1D Boussinesq model is presented. Numerical solution of Boussinesq model is obtained by implementing a staggered grid scheme to continuity, momentum, and elliptic equation of Boussinesq model. Tridiagonal system emerging from numerical scheme of elliptic equation is solved by cyclic reduction algorithm. The parallel implementation of cyclic reduction is executed on multicore processors with shared memory architectures using OpenMP. To measure the performance of parallel program, large number of grids is varied from 28 to 214. Two test cases of numerical experiment, i.e. propagation of solitary and standing wave, are proposed to evaluate the parallel program. The numerical results are verified with analytical solution of solitary and standing wave. The best speedup of solitary and standing wave test cases is about 2.07 with 214 of grids and 1.86 with 213 of grids, respectively, which are executed by using 8 threads. Moreover, the best efficiency of parallel program is 76.2% and 73.5% for solitary and standing wave test cases, respectively.

The formation of shocks and fundamental solution of a fourth-order quasilinear Boussinesq-type equation

Science.gov (United States)

Galaktionov, Victor A.

2009-02-01

As a basic higher-order model, the fourth-order Boussinesq-type quasilinear wave equation (the QWE-4) \\[ \\begin{equation*}\\fl u_{tt} = -(|u|^n u)_{xxxx} \\tqs in\\ \\mathbb{R} \\times \\mathbb{R}_+, \\quad with\\ exponent\\ n > 0,\\end{equation*} \\] is considered. Self-similar blow-up solutions \\[ \\begin{eqnarray*}\\tqs\\tqs u_-(x,t)=g(z), \\quad\\, z=\\frac x{\\sqrt{T-t}},\\\\ where\\ g\\ solved\\ the\\ ODE\\ \\frac 14 g'' z^2 + \\frac 34 g'z = -(|g|^n g)^{(4)},\\end{eqnarray*} \\] are shown to exist that generate as t → T- discontinuous shock waves. The QWE-4 is also shown to admit a smooth (for t > 0) global 'fundamental solution' \\[ \\begin{eqnarray*}\\fl b_n(x,t)= t^{\\frac{2}{n+4}} F_n(y),\\ y = x/t^{\\frac{n+2}{n+4}},\\ such\\ that\\ b_{n}(x,0)= 0,\\ b_{nt}(x,0)= {\\delta}(x),\\end{eqnarray*} \\] i.e. having a measure as initial data. A 'homotopic' limit n → 0 is used to get b_0(x,t)= \\sqrt t \\, F_0(x/\\sqrt t) being the classic fundamental solution of the 1D linear beam equation \\[ \\begin{equation*}u_{tt} = -u_{xxxx} \\tqs in\\ \\mathbb{R} \\times \\mathbb{R}_+.\\end{equation*} \\

Approximate analytical solution to the Boussinesq equation with a sloping water-land boundary

Science.gov (United States)

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.

Numerical doubly-periodic solution of the (2+1)-dimensional Boussinesq equation with initial conditions by the variational iteration method

International Nuclear Information System (INIS)

Inc, Mustafa

2007-01-01

In this Letter, a scheme is developed to study numerical doubly-periodic solutions of the (2+1)-dimensional Boussinesq equation with initial condition by the variational iteration method. As a result, the approximate and exact doubly-periodic solutions are obtained. For different modulus m, comparison between the approximate solution and the exact solution is made graphically, revealing that the variational iteration method is a powerful and effective tool to non-linear problems

Tsunami generation, propagation, and run-up with a high-order Boussinesq model

DEFF Research Database (Denmark)

Fuhrman, David R.; Madsen, Per A.

2009-01-01

In this work we extend a high-order Boussinesq-type (finite difference) model, capable of simulating waves out to wavenumber times depth kh landslide-induced tsunamis. The extension is straight forward, requiring only....... The Boussinesq-type model is then used to simulate numerous tsunami-type events generated from submerged landslides, in both one and two horizontal dimensions. The results again compare well against previous experiments and/or numerical simulations. The new extension compliments recently developed run...

Time-splitting combined with exponential wave integrator fourier pseudospectral method for Schrödinger-Boussinesq system

Science.gov (United States)

Liao, Feng; Zhang, Luming; Wang, Shanshan

2018-02-01

In this article, we formulate an efficient and accurate numerical method for approximations of the coupled Schrödinger-Boussinesq (SBq) system. The main features of our method are based on: (i) the applications of a time-splitting Fourier spectral method for Schrödinger-like equation in SBq system, (ii) the utilizations of exponential wave integrator Fourier pseudospectral for spatial derivatives in the Boussinesq-like equation. The scheme is fully explicit and efficient due to fast Fourier transform. The numerical examples are presented to show the efficiency and accuracy of our method.

Application of the bifurcation method to the modified Boussinesq equation

Directory of Open Access Journals (Sweden)

Shaoyong Li

2014-08-01

Firstly, we give a property of the solutions of the equation, that is, if $1+u(x, t$ is a solution, so is $1-u(x, t$. Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended.

Explorations of the extended ncKP hierarchy

International Nuclear Information System (INIS)

Dimakis, Aristophanes; Mueller-Hoissen, Folkert

2004-01-01

A recently obtained extension (xncKP) of the Moyal-deformed KP hierarchy (ncKP hierarchy) by a set of evolution equations in the Moyal-deformation parameters is further explored. Formulae are derived to compute these equations efficiently. Reductions of the xncKP hierarchy are treated, in particular to the extended ncKdV and ncBoussinesq hierarchies. Furthermore, a good part of the Sato formalism for the KP hierarchy is carried over to the generalized framework. In particular, the well-known bilinear identity theorem for the KP hierarchy, expressed in terms of the (formal) Baker-Akhiezer function, extends to the xncKP hierarchy. Moreover, it is demonstrated that N-soliton solutions of the ncKP equation are also solutions of the first few deformation equations. This is shown to be related to the existence of certain families of algebraic identities

Some Aspects of Extended Kinetic Equation

Directory of Open Access Journals (Sweden)

Dilip Kumar

2015-09-01

Full Text Available Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.

The lattice Boltzmann model for the second-order Benjamin–Ono equations

International Nuclear Information System (INIS)

Lai, Huilin; Ma, Changfeng

2010-01-01

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

On devising Boussinesq-type models with bounded eigenspectra: One horizontal dimension

DEFF Research Database (Denmark)

Eskilsson, Claes; Engsig-Karup, Allan Peter

2014-01-01

) are introduced. Using spectral element simulations of stream function waves it is illustrated that (i) the bounded equations capture the physics of the wave motion as well as the standard unbounded equations, and (ii) the bounded equations are computationally more efficient when explicit time-stepping schemes...... using a spectral element method of arbitrary spatial order p. It is shown that existing sets of parameters, found by optimising the linear dispersion relation, give rise to unbounded eigenspectra which govern stability. For explicit time-stepping schemes the global CFL time-step restriction typically...... 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...

Rogue waves in the multicomponent Mel'nikov system and multicomponent Schrödinger-Boussinesq system

Science.gov (United States)

Sun, Baonan; Lian, Zhan

2018-02-01

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel'nikov equation and the multicomponent Schrödinger-Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the multicomponent Mel'nikov equation, the fundamental rational solutions possess two different behaviours: lump and rogue wave. It is shown that the fundamental (simplest) rogue waves are line localised waves which arise from the constant background with a line profile and then disappear into the constant background again. The fundamental line rogue waves can be classified into three: bright, intermediate and dark line rogue waves. Two subclasses of non-fundamental rogue waves, i.e., multirogue waves and higher-order rogue waves are discussed. The multirogue waves describe interaction of several fundamental line rogue waves, in which interesting wave patterns appear in the intermediate time. Higher-order rogue waves exhibit dynamic behaviours that the wave structures start from lump and then retreat back to it. Moreover, by taking the parameter constraints further, general higher-order rogue wave solutions for the multicomponent Schrödinger-Boussinesq system are generated.

A Note of Extended Proca Equations and Superconductivity

Directory of Open Access Journals (Sweden)

Christianto V.

2009-01-01

Full Text Available It has been known for quite long time that the electrodynamics of Maxwell equations can be extended and generalized further into Proca equations. The implications of in- troducing Proca equations include an alternative description of superconductivity, via extending London equations. In the light of another paper suggesting that Maxwell equations can be written using quaternion numbers, then we discuss a plausible exten- sion of Proca equation using biquaternion number. Further implications and experi- ments are recommended.

Celeris: A GPU-accelerated open source software with a Boussinesq-type wave solver for real-time interactive simulation and visualization

Science.gov (United States)

Tavakkol, Sasan; Lynett, Patrick

2017-08-01

In this paper, we introduce an interactive coastal wave simulation and visualization software, called Celeris. Celeris is an open source software which needs minimum preparation to run on a Windows machine. The software solves the extended Boussinesq equations using a hybrid finite volume-finite difference method and supports moving shoreline boundaries. The simulation and visualization are performed on the GPU using Direct3D libraries, which enables the software to run faster than real-time. Celeris provides a first-of-its-kind interactive modeling platform for coastal wave applications and it supports simultaneous visualization with both photorealistic and colormapped rendering capabilities. We validate our software through comparison with three standard benchmarks for non-breaking and breaking waves.

Reduced Braginskii equations

International Nuclear Information System (INIS)

Yagi, M.; Horton, W.

1994-01-01

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 the perpendicular component of Ohm's law be solved to ensure ∇·j=0 for energy conservation

A Short Proof of the Large Time Energy Growth for the Boussinesq System

Science.gov (United States)

Brandolese, Lorenzo; Mouzouni, Charafeddine

2017-10-01

We give a direct proof of the fact that the L^p-norms of global solutions of the Boussinesq system in R^3 grow large as t→ ∞ for 1R+× R3. In particular, the kinetic energy blows up as \\Vert u(t)\\Vert _2^2˜ ct^{1/2} for large time. This contrasts with the case of the Navier-Stokes equations.

Investigation of behavior of the dynamic contact angle on the basis of the Oberbeck-Boussinesq approximation of the Navier-Stokes equations

Directory of Open Access Journals (Sweden)

Goncharova Olga

2016-01-01

Full Text Available Flows of a viscous incompressible liquid with a thermocapillary boundary are investigated numerically on the basis of the mathematical model that consists of the Oberbeck-Boussinesq approximation of the Navier-Stokes equations, kinematic and dynamic conditions at the free boundary and of the slip boundary conditions at solid walls. We assume that the constant temperature is kept on the solid walls. On the thermocapillary gas-liquid interface the condition of the third order for temperature is imposed. The numerical algorithm based on a finite-difference scheme of the second order approximation on space and time has been constructed. The numerical experiments are performed for water under conditions of normal and low gravity for different friction coefficients and different values of the interphase heat transfer coefficient.

Renormalization Group Theory of Bolgiano Scaling in Boussinesq Turbulence

Science.gov (United States)

Rubinstein, Robert

1994-01-01

Bolgiano scaling in Boussinesq turbulence is analyzed using the Yakhot-Orszag renormalization group. For this purpose, an isotropic model is introduced. Scaling exponents are calculated by forcing the temperature equation so that the temperature variance flux is constant in the inertial range. Universal amplitudes associated with the scaling laws are computed by expanding about a logarithmic theory. Connections between this formalism and the direct interaction approximation are discussed. It is suggested that the Yakhot-Orszag theory yields a lowest order approximate solution of a regularized direct interaction approximation which can be corrected by a simple iterative procedure.

Reduced Braginskii equations

International Nuclear Information System (INIS)

Yagi, M.; Horton, W.

1993-11-01

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

Reduced Braginskii equations

Energy Technology Data Exchange (ETDEWEB)

Yagi, M. [Japan Atomic Energy Research Inst., Naka, Ibaraki (Japan). Naka Fusion Research Establishment; Horton, W. [Texas Univ., Austin, TX (United States). Inst. for Fusion Studies

1993-11-01

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 {beta} that we solve the perpendicular component of Ohm`s law to conserve the physical energy while ensuring the relation {del} {center_dot} j = 0.

Extended Thermodynamics: a Theory of Symmetric Hyperbolic Field Equations

Science.gov (United States)

Müller, Ingo

2008-12-01

Extended thermodynamics is based on a set of equations of balance which are supplemented by local and instantaneous constitutive equations so that the field equations are quasi-linear first order differential equations. If the constitutive functions are subject to the requirements of the entropy principle, one may write them in symmetric hyperbolic form by a suitable choice of fields. The kinetic theory of gases, or the moment theories based on the Boltzmann equation provide an explicit example for extended thermodynamics. The theory proves its usefulness and practicality in the successful treatment of light scattering in rarefied gases. This presentation is based upon the book [1] of which the author of this paper is a co-author. For more details about the motivation and exploitation of the basic principles the interested reader is referred to that reference. It would seem that extended thermodynamics is worthy of the attention of mathematicians. It may offer them a non-trivial field of study concerning hyperbolic equations, if ever they get tired of the Burgers equation. Physicists may prefer to appreciate the success of extended thermodynamics in light scattering and to work on the open problems concerning the modification of the Navier-Stokes-Fourier theory in rarefied gases as predicted by extended thermodynamics of 13, 14, and more moments.

The super-classical-Boussinesq hierarchy and its super-Hamiltonian structure

International Nuclear Information System (INIS)

Si-Xing, Tao; Tie-Cheng, Xia

2010-01-01

Based on the constructed Lie superalgebra, the super-classical-Boussinesq hierarchy is obtained. Then, its super-Hamiltonian structure is obtained by making use of super-trace identity. Furthermore, the super-classical-Boussinesq hierarchy is also integrable in the sense of Liouville. (general)

Non-Oberbeck-Boussinesq Effects in Gaseous Rayleigh-Bénard Convection

NARCIS (Netherlands)

Ahlers, Günter; Fontenele Araujo Junior, F.; Funfschilling, Denis; Grossmann, Siegfried; Lohse, Detlef

2007-01-01

Non-Oberbeck-Boussinesq (NOB) effects are measured experimentally and calculated theoretically for strongly turbulent Rayleigh-Be´nard convection of ethane gas under pressure where the material properties strongly depend on the temperature. Relative to the Oberbeck-Boussinesq case we find a decrease

Sun's pole-equator flux differences

Energy Technology Data Exchange (ETDEWEB)

Belvedere, G [Istituto di Astronomia dell' Universita di Catania, Italy; Paterno, L [Osservatorio Astrofisico di Catania, Italy

1977-04-01

The possibility that large flux differences between the poles and the equator at the bottom of the solar convective zone are compatible with the small differences observed at the surface is studied. The consequences of increasing the depth of the convective zone due to overshooting are explored. A Boussinesq model is used for the convective zone and it is assumed that the interaction of the global convection with rotation is modelled through a convective flux coefficient whose perturbed part is proportional to the local Taylor number. The numerical integration of the equations of motion and energy shows that coexistence between large pole-equator flux differences at the bottom and small ones at the surface is possible if the solar convective zone extends to a depth of 0.4 R(Sun). The angular velocity distribution inside the convective zone is in agreement with the ..cap alpha omega..-dynamo theories of the solar cycle.

A new sub-equation method applied to obtain exact travelling wave solutions of some complex nonlinear equations

International Nuclear Information System (INIS)

Zhang Huiqun

2009-01-01

By using a new coupled Riccati equations, a direct algebraic method, which was applied to obtain exact travelling wave solutions of some complex nonlinear equations, is improved. And the exact travelling wave solutions of the complex KdV equation, Boussinesq equation and Klein-Gordon equation are investigated using the improved method. The method presented in this paper can also be applied to construct exact travelling wave solutions for other nonlinear complex equations.

Integrable coupling system of fractional soliton equation hierarchy

Energy Technology Data Exchange (ETDEWEB)

Yu Fajun, E-mail: yfajun@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)

2009-10-05

In this Letter, we consider the derivatives and integrals of fractional order and present a class of the integrable coupling system of the fractional order soliton equations. The fractional order coupled Boussinesq and KdV equations are the special cases of this class. Furthermore, the fractional AKNS soliton equation hierarchy is obtained.

Self-consistence equations for extended Feynman rules in quantum chromodynamics

International Nuclear Information System (INIS)

Wielenberg, A.

2005-01-01

In this thesis improved solutions for Green's functions are obtained. First the for this thesis essential techniques and concepts of QCD as euclidean field theory are presented. After a discussion of the foundations of the extended approach for the Feynman rules of QCD with a systematic approach for the 4-gluon vertex a modified renormalization scheme for the extended approach is developed. Thereafter the resummation of the Dyson-Schwinger equations (DSE) by the appropriately modified Bethe-Salpeter equation is discussed. Then the leading divergences for the 1-loop graphs of the resummed DSE are determined. Thereafter the equation-of-motion condensate is defined as result of an operator-product expansion. Then the self-consistency equations for the extended approaches are defined and numerically solved. (HSI)

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

International Nuclear Information System (INIS)

El-Sabbagh, Mostafa F.; Ahmad, Ali T.

2011-01-01

The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The (2+1)-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples illustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries. (general)

Influence of Boussinesq coefficient on depth-averaged modelling of rapid flows

Science.gov (United States)

Yang, Fan; Liang, Dongfang; Xiao, Yang

2018-04-01

The traditional Alternating Direction Implicit (ADI) scheme has been proven to be incapable of modelling trans-critical flows. Its inherent lack of shock-capturing capability often results in spurious oscillations and computational instabilities. However, the ADI scheme is still widely adopted in flood modelling software, and various special treatments have been designed to stabilise the computation. Modification of the Boussinesq coefficient to adjust the amount of fluid inertia is a numerical treatment that allows the ADI scheme to be applicable to rapid flows. This study comprehensively examines the impact of this numerical treatment over a range of flow conditions. A shock-capturing TVD-MacCormack model is used to provide reference results. For unsteady flows over a frictionless bed, such as idealised dam-break floods, the results suggest that an increase in the value of the Boussinesq coefficient reduces the amplitude of the spurious oscillations. The opposite is observed for steady rapid flows over a frictional bed. Finally, a two-dimensional urban flooding phenomenon is presented, involving unsteady flow over a frictional bed. The results show that increasing the value of the Boussinesq coefficient can significantly reduce the numerical oscillations and reduce the predicted area of inundation. In order to stabilise the ADI computations, the Boussinesq coefficient could be judiciously raised or lowered depending on whether the rapid flow is steady or unsteady and whether the bed is frictional or frictionless. An increase in the Boussinesq coefficient generally leads to overprediction of the propagating speed of the flood wave over a frictionless bed, but the opposite is true when bed friction is significant.

Extended irreversible thermodynamics and the Jeffreys type constitutive equations

International Nuclear Information System (INIS)

Serdyukov, S.I.

2003-01-01

A postulate of extended irreversible thermodynamics is considered, according to which the entropy density is a function of the internal energy, the specific volume, and their material time derivatives. On the basis of this postulate, entropy balance equations and phenomenological equations are obtained, which directly lead to the Jeffreys type constitutive equations

Optimal control problem for the extended Fisher–Kolmogorov equation

Indian Academy of Sciences (India)

In this paper, the optimal control problem for the extended Fisher–Kolmogorov equation is studied. The optimal control under boundary condition is given, the existence of optimal solution to the equation is proved and the optimality system is established.

Numerical resolution of the Navier-Stokes equations for a low Mach number by a spectral method

International Nuclear Information System (INIS)

Frohlich, Jochen

1990-01-01

The low Mach number approximation of the Navier-Stokes equations, also called isobar, is an approximation which is less restrictive than the one due to Boussinesq. It permits strong density variations while neglecting acoustic phenomena. We present a numerical method to solve these equations in the unsteady, two dimensional case with one direction of periodicity. The discretization uses a semi-implicit finite difference scheme in time and a Fourier-Chebycheff pseudo-spectral method in space. The solution of the equations of motion is based on an iterative algorithm of Uzawa type. In the Boussinesq limit we obtain a direct method. A first application is concerned with natural convection in the Rayleigh-Benard setting. We compare the results of the low Mach number equations with the ones in the Boussinesq case and consider the influence of variable fluid properties. A linear stability analysis based on a Chebychev-Tau method completes the study. The second application that we treat is a case of isobaric combustion in an open domain. We communicate results for the hydrodynamic Darrieus-Landau instability of a plane laminar flame front. [fr

Evolution equations for extended dihadron fragmentation functions

International Nuclear Information System (INIS)

Ceccopieri, F.A.; Bacchetta, A.

2007-03-01

We consider dihadron fragmentation functions, describing the fragmentation of a parton in two unpolarized hadrons, and in particular extended dihadron fragmentation functions, explicitly dependent on the invariant mass, M h , of the hadron pair. We first rederive the known results on M h -integrated functions using Jet Calculus techniques, and then we present the evolution equations for extended dihadron fragmentation functions. Our results are relevant for the analysis of experimental measurements of two-particle-inclusive processes at different energies. (orig.)

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

International Nuclear Information System (INIS)

Dai Chaoqing; Zhang Jiefang

2006-01-01

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.

Propagation and quenching in a reactive Burgers–Boussinesq system

International Nuclear Information System (INIS)

Constantin, Peter; Ryzhik, Lenya; Roquejoffre, Jean-Michel; Vladimirova, Natalia

2008-01-01

We investigate the qualitative behaviour of solutions of a Burgers–Boussinesq system—a reaction–diffusion equation coupled via gravity to a Burgers equation—by a combination of numerical, asymptotic and mathematical techniques. Numerical simulations suggest that when the gravity ρ is small the solutions decompose into a travelling wave and an accelerated shock wave moving in opposite directions. There exists ρ cr1 so that, when ρ > ρ cr1 , this structure changes drastically, and the solutions become more complicated. The solutions are composed of three elementary pieces: a wave fan, a combustion travelling wave and an accelerating shock, the whole structure travelling in the same direction. There exists ρ cr2 so that when ρ > ρ cr2 , the wave fan catches up with the accelerating shock wave and the solution is quenched, no matter how large the support of the initial temperature. We prove that the three building blocks (wave fans, combustion travelling waves and shocks) exist and we construct asymptotic solutions made up of these three elementary pieces. We finally prove, in a mathematically rigorous way, a quenching result irrespective of the size of the region where the temperature was above ignition—a major difference from what happens in advection–reaction–diffusion equations where an incompressible flow is imposed

The Extended Fractional Subequation Method for Nonlinear Fractional Differential Equations

OpenAIRE

Zhao, Jianping; Tang, Bo; Kumar, Sunil; Hou, Yanren

2012-01-01

An extended fractional subequation method is proposed for solving fractional differential equations by introducing a new general ansätz and Bäcklund transformation of the fractional Riccati equation with known solutions. Being concise and straightforward, this method is applied to the space-time fractional coupled Burgers’ equations and coupled MKdV equations. As a result, many exact solutions are obtained. It is shown that the considered method provides a very effective, convenient, and powe...

Variational Boussinesq model for strongly nonlinear dispersive waves

NARCIS (Netherlands)

Lawrence, C.; Adytia, D.; van Groesen, E.

2018-01-01

For wave tank, coastal and oceanic applications, a fully nonlinear Variational Boussinesq model with optimized dispersion is derived and a simple Finite Element implementation is described. Improving a previous weakly nonlinear version, high waves over flat and varying bottom are shown to be

New exact solutions to MKDV-Burgers equation and (2 + 1)-dimensional dispersive long wave equation via extended Riccati equation method

International Nuclear Information System (INIS)

Kong Cuicui; Wang Dan; Song Lina; Zhang Hongqing

2009-01-01

In this paper, with the aid of symbolic computation and a general ansaetz, we presented a new extended rational expansion method to construct new rational formal exact solutions to nonlinear partial differential equations. In order to illustrate the effectiveness of this method, we apply it to the MKDV-Burgers equation and the (2 + 1)-dimensional dispersive long wave equation, then several new kinds of exact solutions are successfully obtained by using the new ansaetz. The method can also be applied to other nonlinear partial differential equations.

Extended rate equations

International Nuclear Information System (INIS)

Shore, B.W.

1981-01-01

The equations of motion are discussed which describe time dependent population flows in an N-level system, reviewing the relationship between incoherent (rate) equations, coherent (Schrodinger) equations, and more general partially coherent (Bloch) equations. Approximations are discussed which replace the elaborate Bloch equations by simpler rate equations whose coefficients incorporate long-time consequences of coherence

A numerical study on the non-Boussinesq effect in the natural convection in horizontal annulus

Science.gov (United States)

Zhang, Yu; Cao, Yuhui

2018-04-01

In the present study, the non-Boussinesq effect in the thermal convection in an air-filled horizontal concentric annulus is studied numerically by using the variable property-based lattice Boltzmann flux solver (VPLBFS), with the radial temperature difference ratio of 1.0, the radius ratio of 2.0, and the Rayleigh number in the range 104 ≤ Ra ≤ 106. Several solutions are obtained by using the standard form or simplified versions of the VPLBFS, including the real solution with the total variation in fluid properties considered, named as the variable property solution (VPS), the constant property solution (CPS) based on the Boussinesq approximation, the solution with variable dynamic viscosity (VVS), the solution based on the partial Boussinesq approximation (PBAS), the solution with variable thermal conductivity (VCS) and the solution with variable fluid density (VDS). The discrepancy between these solutions is analyzed to illuminate the influence of the non-Boussinesq effects induced by partial or total variation in fluid properties on flow instability behaviors and heat transfer characteristics. The present study reveals the complicated flow instability behavior under non-Boussinesq conditions and its tight association with heat transfer characteristics. Also, it demonstrates the necessity of considering the integral effect of the total variation in fluid properties and highlights the essential role of the fluid density variation.

The effects of the Boussinesq model to the rising of the explosion clouds

International Nuclear Information System (INIS)

Li Xiaoli; Zheng Yi

2010-01-01

It is to study the rising of the explosion clouds in the normal atmosphere using Boussinesq model and the Incompressible model, the numerical model is based on the assumption that effects the clouds are gravity and buoyancy. By comparing the evolvement of different density cloud, and gives the conclusion-the Boussinesq model and the Incompressible model is accord when the cloud's density is larger compared to the density of the environment. (authors)

Extended sine-Gordon Equation Method and Its Application to Maccari's System

International Nuclear Information System (INIS)

Song Lina; Zhang Hongqing

2005-01-01

An extended sine-Gordon equation method is proposed to construct exact travelling wave solutions to Maccari's equation based upon a generalized sine-Gordon equation. It is shown that more new travelling wave solutions can be found by this new method, which include bell-shaped soliton solutions, kink-shaped soliton solutions, periodic wave solution, and new travelling waves.

A new modification in the exponential rational function method for nonlinear fractional differential equations

Science.gov (United States)

Ahmed, Naveed; Bibi, Sadaf; Khan, Umar; Mohyud-Din, Syed Tauseef

2018-02-01

We have modified the traditional exponential rational function method (ERFM) and have used it to find the exact solutions of two different fractional partial differential equations, one is the time fractional Boussinesq equation and the other is the (2+1)-dimensional time fractional Zoomeron equation. In both the cases it is observed that the modified scheme provides more types of solutions than the traditional one. Moreover, a comparison of the recent solutions is made with some already existing solutions. We can confidently conclude that the modified scheme works better and provides more types of solutions with almost similar computational cost. Our generalized solutions include periodic, soliton-like, singular soliton and kink solutions. A graphical simulation of all types of solutions is provided and the correctness of the solution is verified by direct substitution. The extended version of the solutions is expected to provide more flexibility to scientists working in the relevant field to test their simulation data.

Integro-differential equation approach extended to larger nuclei

International Nuclear Information System (INIS)

Adam, R.M.; Sofianos, S.A.; Fiedeldey, H.; Fabre de la Ripelle, M.

1992-01-01

We extend the integro-differential equation approach (IDEA) from few-nucleon to closed-shell and closed-subshell nuclei and outline the analytical methods required for the calculation of the density functions, which enter into the integro-differential equations. These contain all the physics for a system of fermions associated with the Pauli principle. In order to test the accuracy of the IDEA comparisons are made of the binding energies of 4 He, 12 C and 16 O obtained with effective potentials using the hypercentral approximation (HCA) providing a variational solution without correlations, the IDEA which fully includes the two-body correlations, the S-states integro-differential equation (SIDE) valid for potentials operating only on pairs in the S-state and those calculated by several variational or perturbative methods in the literature. (author)

High-order Boussinesq-type modelling of nonlinear wave phenomena in deep and shallow water

DEFF Research Database (Denmark)

Madsen, Per A.; Fuhrman, David R.

2010-01-01

In this work, we start with a review of the development of Boussinesq theory for water waves covering the period from 1872 to date. Previous reviews have been given by Dingemans,1 Kirby,2,3 and Madsen & Schäffer.4 Next, we present our most recent high-order Boussinesq-type formulation valid for f...... from an undular sea bed; (8) Run-up of non-breaking solitary waves on a beach; and (9) Tsunami generation from submerged landslides....

Earth's core convection: Boussinesq approximation or incompressible approach?

Czech Academy of Sciences Publication Activity Database

Anufriev, A. P.; Hejda, Pavel

2010-01-01

Roč. 104, č. 1 (2010), s. 65-83 ISSN 0309-1929 R&D Projects: GA AV ČR IAA300120704 Grant - others:INTAS(XE) 03-51-5807 Institutional research plan: CEZ:AV0Z30120515 Keywords : geodynamic models * core convection * Boussinesq approximation Subject RIV: DE - Earth Magnetism, Geodesy, Geography Impact factor: 0.831, year: 2010

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.

A depth semi-averaged model for coastal dynamics

Science.gov (United States)

Antuono, M.; Colicchio, G.; Lugni, C.; Greco, M.; Brocchini, M.

2017-05-01

The present work extends the semi-integrated method proposed by Antuono and Brocchini ["Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics," Phys. Fluids 25(1), 016603 (2013)], which comprises a subset of depth-averaged equations (similar to Boussinesq-like models) and a Poisson equation that accounts for vertical dynamics. Here, the subset of depth-averaged equations has been reshaped in a conservative-like form and both the Poisson equation formulations proposed by Antuono and Brocchini ["Beyond Boussinesq-type equations: Semi-integrated models for coastal dynamics," Phys. Fluids 25(1), 016603 (2013)] are investigated: the former uses the vertical velocity component (formulation A) and the latter a specific depth semi-averaged variable, ϒ (formulation B). Our analyses reveal that formulation A is prone to instabilities as wave nonlinearity increases. On the contrary, formulation B allows an accurate, robust numerical implementation. Test cases derived from the scientific literature on Boussinesq-type models—i.e., solitary and Stokes wave analytical solutions for linear dispersion and nonlinear evolution and experimental data for shoaling properties—are used to assess the proposed solution strategy. It is found that the present method gives reliable predictions of wave propagation in shallow to intermediate waters, in terms of both semi-averaged variables and conservation properties.

Unstructured nodal DG-FEM solution of high-order Boussinesq-type equations

DEFF Research Database (Denmark)

Engsig-Karup, Allan Peter

2007-01-01

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 interactions and interaction with structures. The main goal can be reached by using multi-domain methods...... 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...... 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 as the stable time increment...

The modified extended Fan's sub-equation method and its application to (2 + 1)-dimensional dispersive long wave equation

International Nuclear Information System (INIS)

Yomba, Emmanuel

2005-01-01

By using a modified extended Fan's sub-equation method, we have obtained new and more general solutions including a series of non-travelling wave and coefficient function solutions namely: soliton-like solutions, triangular-like solutions, single and combined non-degenerative Jacobi elliptic wave function-like solutions for the (2 + 1)-dimensional dispersive long wave equation. The most important achievement of this method lies on the fact that, we have succeeded in one move to give all the solutions which can be previously obtained by application of at least four methods (method using Riccati equation, or first kind elliptic equation, or auxiliary ordinary equation, or generalized Riccati equation as mapping equation)

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

Double diffusive unsteady convective micropolar flow past a vertical porous plate moving through binary mixture using modified Boussinesq approximation

Directory of Open Access Journals (Sweden)

Isaac Lare Animasaun

2016-06-01

Full Text Available The problem of unsteady convective with thermophoresis, chemical reaction and radiative heat transfer in a micropolar fluid flow past a vertical porous surface moving through binary mixture considering temperature dependent dynamic viscosity and constant vortex viscosity has been investigated theoretically. For proper and correct analysis of fluid flow along vertical surface with a temperature lesser than that of the free stream, Boussinesq approximation and temperature dependent viscosity model were modified and incorporated into the governing equations. The governing equations are converted to systems of ordinary differential equations by applying suitable similarity transformations and solved numerically using fourth-order Runge–Kutta method along with shooting technique. The results of the numerical solution are presented graphically and in tabular forms for different values of parameters. Velocity profile increases with temperature dependent variable fluid viscosity parameter. Increase of suction parameter corresponds to an increase in both temperature and concentration within the thin boundary layer.

Extended trigonometric Cherednik algebras and nonstationary Schrödinger equations with delta-potentials

International Nuclear Information System (INIS)

Hartwig, J. T.; Stokman, J. V.

2013-01-01

We realize an extended version of the trigonometric Cherednik algebra as affine Dunkl operators involving Heaviside functions. We use the quadratic Casimir element of the extended trigonometric Cherednik algebra to define an explicit nonstationary Schrödinger equation with delta-potential. We use coordinate Bethe ansatz methods to construct solutions of the nonstationary Schrödinger equation in terms of generalized Bethe wave functions. It is shown that the generalized Bethe wave functions satisfy affine difference Knizhnik-Zamolodchikov equations as functions of the momenta. The relation to the vector valued root system analogs of the quantum Bose gas on the circle with delta-function interactions is indicated.

Nonlocal Symmetries, Consistent Riccati Expansion, and Analytical Solutions of the Variant Boussinesq System

Science.gov (United States)

Feng, Lian-Li; Tian, Shou-Fu; Zhang, Tian-Tian; Zhou, Jun

2017-07-01

Under investigation in this paper is the variant Boussinesq system, which describes the propagation of surface long wave towards two directions in a certain deep trough. With the help of the truncated Painlevé expansion, we construct its nonlocal symmetry, Bäcklund transformation, and Schwarzian form, respectively. The nonlocal symmetries can be localised to provide the corresponding nonlocal group, and finite symmetry transformations and similarity reductions are computed. Furthermore, we verify that the variant Boussinesq system is solvable via the consistent Riccati expansion (CRE). By considering the consistent tan-function expansion (CTE), which is a special form of CRE, the interaction solutions between soliton and cnoidal periodic wave are explicitly studied.

Soliton-like solutions to the GKdV equation by extended mapping method

International Nuclear Information System (INIS)

Wu Ranchao; Sun Jianhua

2007-01-01

In this note, many new exact solutions of the generalized KdV equation, such as rational solutions, periodic solutions like Jacobian elliptic and triangular functions, soliton-like solutions, are constructed by symbolic computation and the extended mapping method, with the auxiliary ordinary equation replaced by a more general one

Inverse problems for the Boussinesq system

International Nuclear Information System (INIS)

Fan, Jishan; Jiang, Yu; Nakamura, Gen

2009-01-01

We obtain two results on inverse problems for a 2D Boussinesq system. One is that we prove the Lipschitz stability for the inverse source problem of identifying a time-independent external force in the system with observation data in an arbitrary sub-domain over a time interval of the velocity and the data of velocity and temperature at a fixed positive time t 0 > 0 over the whole spatial domain. The other one is that we prove a conditional stability estimate for an inverse problem of identifying the two initial conditions with a single observation on a sub-domain

Singular limits of the equations of magnetohydrodynamics

Czech Academy of Sciences Publication Activity Database

Kukučka, Peter

2011-01-01

Roč. 13, č. 2 (2011), s. 173-189 ISSN 1422-6928 R&D Projects: GA MŠk LC06052 Institutional research plan: CEZ:AV0Z10190503 Keywords : Navier-Stokes-Fourier system * Oberbeck -Boussinesq approximation * Maxwell equations Subject RIV: BA - General Mathematics Impact factor: 0.768, year: 2011 http://www.springerlink.com/content/e14w1h5x188142n6/

Equations of motion and conservation laws in a theory of stable stratified turbulence

NARCIS (Netherlands)

L'vov, V.S.; Rudenko, O.

2008-01-01

This paper is part of an invited talk given at the international conference 'Turbulent Mixing and Beyond'. We consider non-isothermal fluid flows and revise simplifications of basic hydrodynamic equations for such flows, arriving eventually at a generalization of the Oberbeck–Boussinesq

Variational Boussinesq model for simulation of coastal waves and tsunamis

NARCIS (Netherlands)

Adytia, D.; Adytia, Didit; van Groesen, Embrecht W.C.; 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

Non-Oberbeck-Boussinesq effects in strongly turbulent Rayleigh-Bénard convection

NARCIS (Netherlands)

Ahlers, Günter; Brown, Eric; Fontenele Araujo Junior, F.; Funfschilling, Denis; Grossmann, Siegfried; Lohse, Detlef

2006-01-01

Non-Oberbeck–Boussinesq (NOB) effects on the Nusselt number $Nu$ and Reynolds number $\\hbox{\\it Re}$ in strongly turbulent Rayleigh–Bénard (RB) convection in liquids were investigated both experimentally and theoretically. In the experiments the heat current, the temperature difference, and the

Dispersive optical soliton solutions for the hyperbolic and cubic-quintic nonlinear Schrödinger equations via the extended sinh-Gordon equation expansion method

Science.gov (United States)

Seadawy, Aly R.; Kumar, Dipankar; Chakrabarty, Anuz Kumar

2018-05-01

The (2+1)-dimensional hyperbolic and cubic-quintic nonlinear Schrödinger equations describe the propagation of ultra-short pulses in optical fibers of nonlinear media. By using an extended sinh-Gordon equation expansion method, some new complex hyperbolic and trigonometric functions prototype solutions for two nonlinear Schrödinger equations were derived. The acquired new complex hyperbolic and trigonometric solutions are expressed by dark, bright, combined dark-bright, singular and combined singular solitons. The obtained results are more compatible than those of other applied methods. The extended sinh-Gordon equation expansion method is a more powerful and robust mathematical tool for generating new optical solitary wave solutions for many other nonlinear evolution equations arising in the propagation of optical pulses.

The Investigation of Solutions to the Coupled Schrödinger-Boussinesq Equations

Directory of Open Access Journals (Sweden)

Xin Huang

2013-01-01

equations. The hyperbolic function solutions, trigonometric function solutions, and rational function solutions to the equations are obtained. The decaying properties of several solutions are analyzed.

Turbulent thermal convection at high Rayleigh numbers for a Boussinesq fluid of constant Prandtl number

International Nuclear Information System (INIS)

Amati, G.; Koal, K.; Massaioli, F.; Sreenivasan, K.R.; Verzicco, R.

2006-12-01

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 2x10 6 to 2x10 14 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)

New extended (G'/G)-expansion method to solve nonlinear evolution equation: the (3 + 1)-dimensional potential-YTSF equation.

Science.gov (United States)

Roshid, Harun-Or-; Akbar, M Ali; Alam, Md Nur; Hoque, Md Fazlul; Rahman, Nizhum

2014-01-01

In this article, a new extended (G'/G) -expansion method has been proposed for constructing more general exact traveling wave solutions of nonlinear evolution equations with the aid of symbolic computation. In order to illustrate the validity and effectiveness of the method, we pick the (3 + 1)-dimensional potential-YTSF equation. As a result, abundant new and more general exact solutions have been achieved of this equation. It has been shown that the proposed method provides a powerful mathematical tool for solving nonlinear wave equations in applied mathematics, engineering and mathematical physics.

Soliton-like structures and the connection between the Bq and KP equations

International Nuclear Information System (INIS)

Bernal, J.; Agueero, Maximo A.; Flores-Romero, Erick

2003-01-01

We study the (2+1)-dimensional model proposed by Kadomtsev and Petviashvili (KP) to describe slowly varying nonlinear waves in a dispersive medium. Applying an appropriate Lie transformation and following the method introduced by Tajiri et al., the KP equation is reduced to a one-dimensional equation, that is, to a certain version of the Boussinesq equation (BqE). Then, we solve the BqE by the Hirota method, and finally we use the inverse transformation in order to obtain de KP solutions. We analyze some remarkable properties of the solutions found in this work

Soliton-like structures and the connection between the Bq and KP equations

CERN Document Server

Bernal, J; Flores-Romero, E

2003-01-01

We study the (2+1)-dimensional model proposed by Kadomtsev and Petviashvili (KP) to describe slowly varying nonlinear waves in a dispersive medium. Applying an appropriate Lie transformation and following the method introduced by Tajiri et al., the KP equation is reduced to a one-dimensional equation, that is, to a certain version of the Boussinesq equation (BqE). Then, we solve the BqE by the Hirota method, and finally we use the inverse transformation in order to obtain de KP solutions. We analyze some remarkable properties of the solutions found in this work.

Integrability of an extended (2+1)-dimensional shallow water wave equation with Bell polynomials

International Nuclear Information System (INIS)

Wang Yun-Hu; Chen Yong

2013-01-01

We investigate the extended (2+1)-dimensional shallow water wave equation. The binary Bell polynomials are used to construct bilinear equation, bilinear Bäcklund transformation, Lax pair, and Darboux covariant Lax pair for this equation. Moreover, the infinite conservation laws of this equation are found by using its Lax pair. All conserved densities and fluxes are given with explicit recursion formulas. The N-soliton solutions are also presented by means of the Hirota bilinear method. (general)

Toward Analytic Solution of Nonlinear Differential Difference Equations via Extended Sensitivity Approach

International Nuclear Information System (INIS)

Darmani, G.; Setayeshi, S.; Ramezanpour, H.

2012-01-01

In this paper an efficient computational method based on extending the sensitivity approach (SA) is proposed to find an analytic exact solution of nonlinear differential difference equations. In this manner we avoid solving the nonlinear problem directly. By extension of sensitivity approach for differential difference equations (DDEs), the nonlinear original problem is transformed into infinite linear differential difference equations, which should be solved in a recursive manner. Then the exact solution is determined in the form of infinite terms series and by intercepting series an approximate solution is obtained. Numerical examples are employed to show the effectiveness of the proposed approach. (general)

Numerical study on turbulent flow inside a channel with an extended chamber

International Nuclear Information System (INIS)

Lee, Young Tae; Lim, Hee Chang

2009-01-01

The paper presents a LES numerical simulation of turbulent flow around an extended chamber. The simulations are carried out on a series of 3-dimensional cavities placed in a turbulent boundary layer at a Reynolds number of 1.0x10 5 based on U and h, which are the velocity at the upper top of the cavity and the depth height, respectively. In order to get an appropriate solution in the Filtered Navier-Stokes equation for the incompressible flow, the computational mesh is densely attracted to the cavity surface and coarsely far-field, as this aids saving the computation cost and rapid convergence. The Boussinesq hypothesis is employed in the subgrid-scale turbulence model. In order to obtain the subgrid-scale turbulent viscosity, the Smagorinsky-Lilly SGS model is applied and the CFL number for time marching is 1.0. The results include the flow variations inside a cavity with the different sizes and shapes.

Exact Solutions of Fractional Burgers and Cahn-Hilliard Equations Using Extended Fractional Riccati Expansion Method

Directory of Open Access Journals (Sweden)

Wei Li

2014-01-01

Full Text Available Based on a general fractional Riccati equation and with Jumarie’s modified Riemann-Liouville derivative to an extended fractional Riccati expansion method for solving the time fractional Burgers equation and the space-time fractional Cahn-Hilliard equation, the exact solutions expressed by the hyperbolic functions and trigonometric functions are obtained. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

Boussinesq Modeling of Wave Propagation and Runup over Fringing Coral Reefs, Model Evaluation Report

National Research Council Canada - National Science Library

Demirbilek, Zeki; Nwogu, Okey G

2007-01-01

This report describes evaluation of a two-dimensional Boussinesq-type wave model, BOUSS-2D, with data obtained from two laboratory experiments and two field studies at the islands of Guam and Hawaii...

The variational 2D Boussinesq model for wave propagation over a shoal

NARCIS (Netherlands)

Adytia, D.; van Groesen, Embrecht W.C.

2011-01-01

The Variational Boussinesq Model (VBM) for waves (Klopman et al. 2010) is based on the Hamiltonian structure of gravity surface waves. In its approximation, the fluid potential in the kinetic energy is approximated by the sum of its value at the free surface and a linear combination of vertical

Differential equations extended to superspace

International Nuclear Information System (INIS)

Torres, J.; Rosu, H.C.

2003-01-01

We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)

Differential equations extended to superspace

Energy Technology Data Exchange (ETDEWEB)

Torres, J. [Instituto de Fisica, Universidad de Guanajuato, A.P. E-143, Leon, Guanajuato (Mexico); Rosu, H.C. [Instituto Potosino de Investigacion Cientifica y Tecnologica, A.P. 3-74, Tangamanga, San Luis Potosi (Mexico)

2003-07-01

We present a simple SUSY Ns = 2 superspace extension of the differential equations in which the sought solutions are considered to be real superfields but maintaining the common derivative operators and the coefficients of the differential equations unaltered. In this way, we get self consistent systems of coupled differential equations for the components of the superfield. This procedure is applied to the Riccati equation, for which we obtain in addition the system of coupled equations corresponding to the components of the general superfield solution. (Author)

Exact Solutions for Fractional Differential-Difference Equations by an Extended Riccati Sub-ODE Method

International Nuclear Information System (INIS)

Feng Qinghua

2013-01-01

In this paper, an extended Riccati sub-ODE method is proposed to establish new exact solutions for fractional differential-difference equations in the sense of modified Riemann—Liouville derivative. By a fractional complex transformation, a given fractional differential-difference equation can be turned into another differential-difference equation of integer order. The validity of the method is illustrated by applying it to solve the fractional Hybrid lattice equation and the fractional relativistic Toda lattice system. As a result, some new exact solutions including hyperbolic function solutions, trigonometric function solutions and rational solutions are established. (general)

Optimized Variational 1D Boussinesq Modelling for broad-band waves over flat bottom

NARCIS (Netherlands)

Lakhturov, I.; Adytia, D.; van Groesen, Embrecht W.C.

The Variational Boussinesq Model (VBM) for waves above a layer of ideal fluid conserves mass, momentum, energy, and has decreased dimensionality compared to the full problem. It is derived from the Hamiltonian formulation via an approximation of the kinetic energy, and can provide approximate

Auto-Baecklund transformation and similarity reductions to the variable coefficients variant Boussinesq system

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.

Non-Oberbeck-Boussinesq effects in two-dimensional Rayleigh-Bénard convection in glycerol

NARCIS (Netherlands)

Sugiyama, K.; Calzavarini, E.; Grossmann, S.; Lohse, Detlef

2007-01-01

We numerically analyze Non-Oberbeck-Boussinesq (NOB) effects in two-dimensional Rayleigh-Benard flow in glycerol, which shows a dramatic change in the viscosity with temperature. The results are presented both as functions of the Rayleigh number Ra up to 108 (for fixed temperature difference �

Filtering of sound from the Navier-Stokes equations. [An approximation for describing thermal convection in a compressible fluid

Energy Technology Data Exchange (ETDEWEB)

Paolucci, S.

1982-12-01

An approximation leading to anelastic equations capable of describing thermal convection in a compressible fluid is given. These equations are more general than the Oberbeck-Boussinesq equations and different than the standard anelastic equations in that they can be used for the computation of convection in a fluid with large density gradients present. We show that the equations do not contain acoustic waves, while at the same time they can still describe the propagation of internal waves. Throughout we show that the filtering of acoustic waves, within the limits of the approximation, does not appreciably alter the description of the physics.

On wave breaking for Boussinesq-type models

Science.gov (United States)

Kazolea, M.; Ricchiuto, M.

2018-03-01

We consider the issue of wave breaking closure for Boussinesq type models, and attempt at providing some more understanding of the sensitivity of some closure approaches to the numerical set-up, and in particular to mesh size. For relatively classical choices of weakly dispersive propagation models, we compare two closure strategies. The first is the hybrid method consisting in suppressing the dispersive terms in breaking regions, as initially suggested by Tonelli and Petti in 2009. The second is an eddy viscosity approach based on the solution of a a turbulent kinetic energy. The formulation follows early work by O. Nwogu in the 90's, and some more recent developments by Zhang and co-workers (Ocean Mod. 2014), adapting it to be consistent with the wave breaking detection used here. We perform a study of the behaviour of the two closures for different mesh sizes, with attention to the possibility of obtaining grid independent results. Based on a classical shallow water theory, we also suggest some monitors to quantify the different contributions to the dissipation mechanism, differentiating those associated to the scheme from those of the partial differential equation. These quantities are used to analyze the dynamics of dissipation in some classical benchmarks, and its dependence on the mesh size. Our main results show that numerical dissipation contributes very little to the the results obtained when using eddy viscosity method. This closure shows little sensitivity to the grid, and may lend itself to the development and use of non-dissipative/energy conserving numerical methods. The opposite is observed for the hybrid approach, for which numerical dissipation plays a key role, and unfortunately is sensitive to the size of the mesh. In particular, when working, the two approaches investigated provide results which are in the same ball range and which agree with what is usually reported in literature. With the hybrid method, however, the inception of instabilities

Equations of motion and conservation laws in a theory of stably stratified turbulence

Energy Technology Data Exchange (ETDEWEB)

L' vov, Victor S; Rudenko, Oleksii [Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100 (Israel)], E-mail: oleksii.rudenko@weizmann.ac.il

2008-12-15

This paper is part of an invited talk given at the international conference 'Turbulent Mixing and Beyond'. We consider non-isothermal fluid flows and revise simplifications of basic hydrodynamic equations for such flows, arriving eventually at a generalization of the Oberbeck-Boussinesq approximation valid for arbitrary equation of state including both non-ideal gases as well as liquids. The proposed approach is based on a suggested general definition of potential temperature. Special attention is paid to the energy conservation principle: the proposed approximation exactly preserves the total mechanical energy by approximate equations of motion. It is emphasized explicitly the importance for any turbulent boundary layer model to respect the conservation laws.

Classification of kink type solutions to the extended derivative nonlinear Schrödinger equation

DEFF Research Database (Denmark)

Wyller, J.; Fla, T.; Juul Rasmussen, J.

1998-01-01

The Raman Extended Derivative Non Linear Schrodinger (R-EDNLS) equation which models single mode propagation in optical fibers, is shown to possess travelling and stationary kink envelope solutions of monotonic and oscillatory type. These structures have been called optical shocks in analogy...

Spin and energy evolution equations for a wide class of extended bodies

International Nuclear Information System (INIS)

Racine, Etienne

2006-01-01

We give a surface integral derivation of the leading-order evolution equations for the spin and energy of a relativistic body interacting with other bodies in the post-Newtonian expansion scheme. The bodies can be arbitrarily shaped and can be strongly self-gravitating. The effects of all mass and current multipoles are taken into account. As part of the computation one of the 2PN potentials parametrizing the metric is obtained. The formulae obtained here for spin and energy evolution coincide with those obtained by Damour, Soffel and Xu for the case of weakly self-gravitating bodies. By combining an Einstein-Infeld-Hoffman-type surface integral approach with multipolar expansions we extend the domain of validity of these evolution equations to a wide class of strongly self-gravitating bodies. This paper completes in a self-contained way a previous work by Racine and Flanagan on translational equations of motion for compact objects

Non-Oberbeck-Boussinesq effects in turbulent thermal convection in ethane close to the critical point

NARCIS (Netherlands)

Ahlers, Günter; Calzavarini, E.; Fontenele Araujo Junior, F.; Funfschilling, Denis; Grossmann, Siegfried; Lohse, Detlef; Sugiyama, K.

2008-01-01

As shown in earlier work [Ahlers et al., J. Fluid Mech. 569, 409 (2006)], non-Oberbeck-Boussinesq (NOB) corrections to the center temperature in turbulent Rayleigh-Bénard convection in water and also in glycerol are governed by the temperature dependences of the kinematic viscosity and the thermal

Computation of nonlinear water waves with a high-order Boussinesq model

DEFF Research Database (Denmark)

Fuhrman, David R.; Madsen, Per A.; Bingham, Harry

2005-01-01

Computational highlights from a recently developed high-order Boussinesq model are shown. The model is capable of treating fully nonlinear waves (up to the breaking point) out to dimensionless depths of (wavenumber times depth) kh \\approx 25. Cases considered include the study of short......-crested waves in shallow/deep water, resulting in hexagonal/rectangular surface patterns; crescent waves, resulting from unstable perturbations of plane progressive waves; and highly-nonlinear wave-structure interactions. The emphasis is on physically demanding problems, and in eachcase qualitative and (when...

The Oberbeck-Boussinesq approximation as a singular limit of the full Navier-Stokes-Fourier system

Czech Academy of Sciences Publication Activity Database

Feireisl, Eduard; Novotný, A.

2009-01-01

Roč. 11, č. 2 (2009), s. 274-302 ISSN 1422-6928 R&D Projects: GA ČR GA201/05/0164 Institutional research plan: CEZ:AV0Z10190503 Keywords : singular limit * Navier-Stokes-Fourier system * Oberbeck -Boussinesq approximation Subject RIV: BA - General Mathematics Impact factor: 1.214, year: 2009

Solution of Fractional Partial Differential Equations in Fluid Mechanics by Extension of Some Iterative Method

Directory of Open Access Journals (Sweden)

A. A. Hemeda

2013-01-01

Full Text Available An extension of the so-called new iterative method (NIM has been used to handle linear and nonlinear fractional partial differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. Therefore, a general framework of the NIM is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation, and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the NIM with those obtained by both Adomian decomposition method (ADM and the variational iteration method (VIM reveals that the NIM is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

Application of the Extended Grunwald-Winstein Equation to Solvolyses of n-Propyl Chloroformate

OpenAIRE

Kyong, Jin Burm; Won, Hoshik; Kevill, Dennis N.

2005-01-01

Abstract: Application of the extended Grunwald-Winstein equation to solvolyses of n-propyl chloroformate in a variety of pure and binary solvents indicates an addition-elimination pathway in the majority of the solvents but an ionization pathway in the solvents of highest ionizing power and lowest nucleophilicity. For methanolysis, a solvent deuterium isotope effect of 2.17 is compatible with the incorporation of general-base catalysis into the substitution process. Activation parameters are ...

The extended hyperbolic function method and exact solutions of the long-short wave resonance equations

International Nuclear Information System (INIS)

Shang Yadong

2008-01-01

The extended hyperbolic functions method for nonlinear wave equations is presented. Based on this method, we obtain a multiple exact explicit solutions for the nonlinear evolution equations which describe the resonance interaction between the long wave and the short wave. The solutions obtained in this paper include (a) the solitary wave solutions of bell-type for S and L, (b) the solitary wave solutions of kink-type for S and bell-type for L, (c) the solitary wave solutions of a compound of the bell-type and the kink-type for S and L, (d) the singular travelling wave solutions, (e) periodic travelling wave solutions of triangle function types, and solitary wave solutions of rational function types. The variety of structure to the exact solutions of the long-short wave equation is illustrated. The methods presented here can also be used to obtain exact solutions of nonlinear wave equations in n dimensions

A relativistic extended Fermi-Thomas-like equation for a self-gravitating system of fermions

International Nuclear Information System (INIS)

Merloni, A.; Ruffini, R.; Torroni, V.

1998-01-01

The authors extend previous results of a Fermi-Thomas model, describing self-gravitating fermions in their ground state, to a relativistic gravitational theory in Minkowski space. In such a theory the source term of the gravitational potential depends both on the pressure and the density of the fluid. It is shown that, in correspondence of this relativistic treatment, still a Fermi-Thomas-like equation can be derived for the self-gravitating system, though the non-linearities are much more complex. No Fermi-Thomas-like equation can be obtained in the General Relativistic treatment. The canonical results for neutron stars and white dwarfs are recovered and also some erroneous statements in the scientific literature are corrected

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

Science.gov (United States)

Şengül, Taylan; Wang, Shouhong

2018-02-01

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criterion for local nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next, we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength of the basic flow crosses a critical threshold. Also, we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition number, fully characterizing the nonlinear interactions of different modes. Both the critical shear strength and the transition number are functions of the system parameters. A systematic numerical method is carried out to explore transition in different flow parameter regimes. In particular, our numerical investigations show the existence of a hypersurface which separates the parameter space into regions where the basic shear flow is stable and unstable. Numerical investigations also yield that the selection of horizontal wave indices is determined only by the aspect ratio of the box. We find that the system admits only critical eigenmodes with roll patterns aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple

Design of TIR collimating lens for ordinary differential equation of extended light source

Science.gov (United States)

Zhan, Qianjing; Liu, Xiaoqin; Hou, Zaihong; Wu, Yi

2017-10-01

The source of LED has been widely used in our daily life. The intensity angle distribution of single LED is lambert distribution, which does not satisfy the requirement of people. Therefore, we need to distribute light and change the LED's intensity angle distribution. The most commonly method to change its intensity angle distribution is the free surface. Generally, using ordinary differential equations to calculate free surface can only be applied in a point source, but it will lead to a big error for the expand light. This paper proposes a LED collimating lens based on the ordinary differential equation, combined with the LED's light distribution curve, and adopt the method of calculating the center gravity of the extended light to get the normal vector. According to the law of Snell, the ordinary differential equations are constructed. Using the runge-kutta method for solution of ordinary differential equation solution, the curve point coordinates are gotten. Meanwhile, the edge point data of lens are imported into the optical simulation software TracePro. Based on 1mm×1mm single lambert body for light conditions, The degrees of collimating light can be close to +/-3. Furthermore, the energy utilization rate is higher than 85%. In this paper, the point light source is used to calculate partial differential equation method and compared with the simulation of the lens, which improve the effect of 1 degree of collimation.

Modification of 2-D Time-Domain Shallow Water Wave Equation using Asymptotic Expansion Method

Science.gov (United States)

Khairuman, Teuku; Nasruddin, MN; Tulus; Ramli, Marwan

2018-01-01

Generally, research on the tsunami wave propagation model can be conducted by using a linear model of shallow water theory, where a non-linear side on high order is ignored. In line with research on the investigation of the tsunami waves, the Boussinesq equation model underwent a change aimed to obtain an improved quality of the dispersion relation and non-linearity by increasing the order to be higher. To solve non-linear sides at high order is used a asymptotic expansion method. This method can be used to solve non linear partial differential equations. In the present work, we found that this method needs much computational time and memory with the increase of the number of elements.

NASA Trapezoidal Wing Simulation Using Stress-w and One- and Two-Equation Turbulence Models

Science.gov (United States)

Rodio, J. J.; Xiao, X; Hassan, H. A.; Rumsey, C. L.

2014-01-01

The Wilcox 2006 stress-omega model (also referred to as WilcoxRSM-w2006) has been implemented in the NASA Langley code CFL3D and used to study a variety of 2-D and 3-D configurations. It predicted a variety of basic cases reasonably well, including secondary flow in a supersonic rectangular duct. One- and two-equation turbulence models that employ the Boussinesq constitutive relation were unable to predict this secondary flow accurately because it is driven by normal turbulent stress differences. For the NASA trapezoidal wing at high angles of attack, the WilcoxRSM-w2006 model predicted lower maximum lift than experiment, similar to results of a two-equation model.

Unsteady free surface flow in porous media: One-dimensional model equations including vertical effects and seepage face

Science.gov (United States)

Di Nucci, Carmine

2018-05-01

This note examines the two-dimensional unsteady isothermal free surface flow of an incompressible fluid in a non-deformable, homogeneous, isotropic, and saturated porous medium (with zero recharge and neglecting capillary effects). Coupling a Boussinesq-type model for nonlinear water waves with Darcy's law, the two-dimensional flow problem is solved using one-dimensional model equations including vertical effects and seepage face. In order to take into account the seepage face development, the system equations (given by the continuity and momentum equations) are completed by an integral relation (deduced from the Cauchy theorem). After testing the model against data sets available in the literature, some numerical simulations, concerning the unsteady flow through a rectangular dam (with an impermeable horizontal bottom), are presented and discussed.

Modelling of natural convection flows with large temperature differences: a benchmark problem for low Mach number solvers. Part. 1 reference solutions

International Nuclear Information System (INIS)

Le Quere, P.; Weisman, C.; Paillere, H.; Vierendeels, J.; Dick, E.; Becker, R.; Braack, M.; Locke, J.

2005-01-01

Heat transfer by natural convection and conduction in enclosures occurs in numerous practical situations including the cooling of nuclear reactors. For large temperature difference, the flow becomes compressible with a strong coupling between the continuity, the momentum and the energy equations through the equation of state, and its properties (viscosity, heat conductivity) also vary with the temperature, making the Boussinesq flow approximation inappropriate and inaccurate. There are very few reference solutions in the literature on non-Boussinesq natural convection flows. We propose here a test case problem which extends the well-known De Vahl Davis differentially heated square cavity problem to the case of large temperature differences for which the Boussinesq approximation is no longer valid. The paper is split in two parts: in this first part, we propose as yet unpublished reference solutions for cases characterized by a non-dimensional temperature difference of 0.6, Ra 10 6 (constant property and variable property cases) and Ra = 10 7 (variable property case). These reference solutions were produced after a first international workshop organized by Cea and LIMSI in January 2000, in which the above authors volunteered to produce accurate numerical solutions from which the present reference solutions could be established. (authors)

Stability analysis solutions and optical solitons in extended nonlinear Schrödinger equation with higher-order odd and even terms

Science.gov (United States)

Peng, Wei-Qi; Tian, Shou-Fu; Zou, Li; Zhang, Tian-Tian

2018-01-01

In this paper, the extended nonlinear Schrödinger equation with higher-order odd (third order) and even (fourth order) terms is investigated, whose particular cases are the Hirota equation, the Sasa-Satsuma equation and Lakshmanan-Porsezian-Daniel equation by selecting some specific values on the parameters of higher-order terms. We first study the stability analysis of the equation. Then, using the ansatz method, we derive its bright, dark solitons and some constraint conditions which can guarantee the existence of solitons. Moreover, the Ricatti equation extension method is employed to derive some exact singular solutions. The outstanding characteristics of these solitons are analyzed via several diverting graphics.

Rational extended thermodynamics

CERN Document Server

Müller, Ingo

1998-01-01

Ordinary thermodynamics provides reliable results when the thermodynamic fields are smooth, in the sense that there are no steep gradients and no rapid changes. In fluids and gases this is the domain of the equations of Navier-Stokes and Fourier. Extended thermodynamics becomes relevant for rapidly varying and strongly inhomogeneous processes. Thus the propagation of high­ frequency waves, and the shape of shock waves, and the regression of small-scale fluctuation are governed by extended thermodynamics. The field equations of ordinary thermodynamics are parabolic while extended thermodynamics is governed by hyperbolic systems. The main ingredients of extended thermodynamics are • field equations of balance type, • constitutive quantities depending on the present local state and • entropy as a concave function of the state variables. This set of assumptions leads to first order quasi-linear symmetric hyperbolic systems of field equations; it guarantees the well-posedness of initial value problems and f...

Velocity potential formulations of highly accurate Boussinesq-type models

DEFF Research Database (Denmark)

Bingham, Harry B.; Madsen, Per A.; Fuhrman, David R.

2009-01-01

, B., 2006. A Boussinesq-type method for fully nonlinear waves interacting with a rapidly varying bathymetry. Coast. Eng. 53, 487-504); Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave processes on the weather side of reflective...... with the kinematic bottom boundary condition. The true behaviour of the velocity potential formulation with respect to linear shoaling is given for the first time, correcting errors made by Jamois et al. (Jamois, E., Fuhrman, D.R., Bingham, H.B., Molin, B., 2006. Wave-structure interactions and nonlinear wave...... processes on the weather side of reflective structures. Coast. Eng. 53, 929-945). An exact infinite series solution for the potential is obtained via a Taylor expansion about an arbitrary vertical position z=(z) over cap. For practical implementation however, the solution is expanded based on a slow...

New solitary wave solutions of (3 + 1)-dimensional nonlinear extended Zakharov-Kuznetsov and modified KdV-Zakharov-Kuznetsov equations and their applications

Science.gov (United States)

Lu, Dianchen; Seadawy, A. R.; Arshad, M.; Wang, Jun

In this paper, new exact solitary wave, soliton and elliptic function solutions are constructed in various forms of three dimensional nonlinear partial differential equations (PDEs) in mathematical physics by utilizing modified extended direct algebraic method. Soliton solutions in different forms such as bell and anti-bell periodic, dark soliton, bright soliton, bright and dark solitary wave in periodic form etc are obtained, which have large applications in different branches of physics and other areas of applied sciences. The obtained solutions are also presented graphically. Furthermore, many other nonlinear evolution equations arising in mathematical physics and engineering can also be solved by this powerful, reliable and capable method. The nonlinear three dimensional extended Zakharov-Kuznetsov dynamica equation and (3 + 1)-dimensional modified KdV-Zakharov-Kuznetsov equation are selected to show the reliability and effectiveness of the current method.

Solitary waves on nonlinear elastic rods. I

DEFF Research Database (Denmark)

Sørensen, Mads Peter; Christiansen, Peter Leth; Lomdahl, P. S.

1984-01-01

Acoustic waves on elastic rods with circular cross section are governed by improved Boussinesq equations when transverse motion and nonlinearity in the elastic medium are taken into account. Solitary wave solutions to these equations have been found. The present paper treats the interaction betwe...... nonlinearity. The balance between dispersion and nonlinearity in the equation is investigated.......Acoustic waves on elastic rods with circular cross section are governed by improved Boussinesq equations when transverse motion and nonlinearity in the elastic medium are taken into account. Solitary wave solutions to these equations have been found. The present paper treats the interaction between...... the solitary waves numerically. It is demonstrated that the waves behave almost like solitons in agreement with the fact that the improved Boussinesq equations are nearly integrable. Thus three conservation theorems can be derived from the equations. A new subsonic quasibreather is found in the case of a cubic...

Equi-frequency contour of photonic crystals with the extended Dirichlet-to-Neumann wave vector eigenvalue equation method

International Nuclear Information System (INIS)

Jiang Bin; Zhang Yejing; Wang Yufei; Liu Anjin; Zheng Wanhua

2012-01-01

We present the extended Dirichlet-to-Neumann wave vector eigenvalue equation (DtN-WVEE) method to calculate the equi-frequency contour (EFC) of square lattice photonic crystals (PhCs). With the extended DtN-WVEE method and Snell's law, the effective refractive index of the mode with a circular EFC can be obtained, which is further validated with the refractive index weighted by the electric field or magnetic field. To further verify the EFC calculated by the DtN-WVEE method, the finite-difference time-domain method is also used. Compared with other wave vector eigenvalue equation methods that calculate EFC directly, the size of the eigenmatrix used in the DtN-WVEE method is much smaller, and the computation time is significantly reduced. Since the DtN-WVEE method solves wave vectors for given arbitrary frequencies, it can also find applications in studying the optical properties of a PhC with dispersive, lossy and magnetic materials. (paper)

The application of Fast Fourier transforms to the primitive equations of Boussinesq convection

International Nuclear Information System (INIS)

Parrott, A.K.

1976-01-01

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.) [de

Application of the Extended Grunwald-Winstein Equation to Solvolyses of n-Propyl Chloroformate

Directory of Open Access Journals (Sweden)

Dennis N. Kevill

2005-01-01

Full Text Available Abstract: Application of the extended Grunwald-Winstein equation to solvolyses of n-propyl chloroformate in a variety of pure and binary solvents indicates an addition-elimination pathway in the majority of the solvents but an ionization pathway in the solvents of highest ionizing power and lowest nucleophilicity. For methanolysis, a solvent deuterium isotope effect of 2.17 is compatible with the incorporation of general-base catalysis into the substitution process. Activation parameters are consistent with the duality of mechanism. Very modest positive salt effects are observed on adding chloride or bromide salts to the ethanolysis.

Coupling of atom-by-atom calculations of extended defects with B kick-out equations: application to the simulation of boron ted

International Nuclear Information System (INIS)

Lampin, E.; Cristiano, F.; Lamrani, Y.; Colombeau, B.

2004-01-01

We present simulations of B TED based on a complete calculation of the extended defect growth/shrinkage during annealing. The Si self-interstitial supersaturation calculated at the extended defect depth is coupled to the set of equations for the B kick-out diffusion through a generation/recombination term in the diffusion equation of the Si self-interstitials. The simulations are compared to the measurements performed on a Si wafer containing several B marker layers, where the amount of TED varies from one peak to the other. The good agreement obtained on this experiment is very promising for the application of these calculations to the case of ultra-shallow B + implants

Variable separation solutions for the Nizhnik-Novikov-Veselov equation via the extended tanh-function method

International Nuclear Information System (INIS)

Zhang Jiefang; Dai Chaoqing; Zong Fengde

2007-01-01

In this paper, with the variable separation approach and based on the general reduction theory, we successfully generalize this extended tanh-function method to obtain new types of variable separation solutions for the following Nizhnik-Novikov-Veselov (NNV) equation. Among the solutions, two solutions are new types of variable separation solutions, while the last solution is similar to the solution given by Darboux transformation in Hu et al 2003 Chin. Phys. Lett. 20 1413

Simulating run-up on steep slopes with operational Boussinesq models; capabilities, spurious effects and instabilities

Directory of Open Access Journals (Sweden)

F. Løvholt

2013-06-01

Full Text Available Tsunamis induced by rock slides plunging into fjords constitute a severe threat to local coastal communities. The rock slide impact may give rise to highly non-linear waves in the near field, and because the wave lengths are relatively short, frequency dispersion comes into play. Fjord systems are rugged with steep slopes, and modeling non-linear dispersive waves in this environment with simultaneous run-up is demanding. We have run an operational Boussinesq-type TVD (total variation diminishing model using different run-up formulations. Two different tests are considered, inundation on steep slopes and propagation in a trapezoidal channel. In addition, a set of Lagrangian models serves as reference models. Demanding test cases with solitary waves with amplitudes ranging from 0.1 to 0.5 were applied, and slopes were ranging from 10 to 50°. Different run-up formulations yielded clearly different accuracy and stability, and only some provided similar accuracy as the reference models. The test cases revealed that the model was prone to instabilities for large non-linearity and fine resolution. Some of the instabilities were linked with false breaking during the first positive inundation, which was not observed for the reference models. None of the models were able to handle the bore forming during drawdown, however. The instabilities are linked to short-crested undulations on the grid scale, and appear on fine resolution during inundation. As a consequence, convergence was not always obtained. It is reason to believe that the instability may be a general problem for Boussinesq models in fjords.

Buckling analysis of laminated plates using the extended Kantorovich method and a system of first-order differential equations

Energy Technology Data Exchange (ETDEWEB)

Singhatanadgid, Pairod; Jommalai, Panupan [Chulalongkorn University, Bangkok (Thailand)

2016-05-15

The extended Kantorovich method using multi-term displacement functions is applied to the buckling problem of laminated plates with various boundary conditions. The out-of-plane displacement of the buckled plate is written as a series of products of functions of parameter x and functions of parameter y. With known functions in parameter x or parameter y, a set of governing equations and a set of boundary conditions are obtained after applying the variational principle to the total potential energy of the system. The higher order differential equations are then transformed into a set of first-order differential equations and solved for the buckling load and mode. Since the governing equations are first-order differential equations, solutions can be obtained analytically with the out-of-plane displacement written in the form of an exponential function. The solutions from the proposed technique are verified with solutions from the literature and FEM solutions. The bucking loads correspond very well to other available solutions in most of the comparisons. The buckling modes also compare very well with the finite element solutions. The proposed solution technique transforms higher-order differential equations to first-order differential equations, and they are analytically solved for out-of-plane displacement in the form of an exponential function. Therefore, the proposed solution technique yields a solution which can be considered as an analytical solution.

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

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....... 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 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 ability...

Equations of motion derived from a generalization of Einstein's equation for the gravitational field

International Nuclear Information System (INIS)

Mociutchi, C.

1980-01-01

The extended Einstein's equation, combined with a vectorial theory of maxwellian type of the gravitational field, leads to: a) the equation of motion; b) the equation of the trajectory for the static case of spherical symmetry, the test particle having a rest mass other than zero, and c) the propagation of light on null geodesics. All the basic tests of the theory given by Einstein's extended equation. Thus, the new theory of gravitation suggested by us is competitive. (author)

Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

International Nuclear Information System (INIS)

Li Hongzhe; Tian Bo; Li Lili; Zhang Haiqiang

2010-01-01

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)

Extended thermodynamics

CERN Document Server

Müller, Ingo

1993-01-01

Physicists firmly believe that the differential equations of nature should be hyperbolic so as to exclude action at a distance; yet the equations of irreversible thermodynamics - those of Navier-Stokes and Fourier - are parabolic. This incompatibility between the expectation of physicists and the classical laws of thermodynamics has prompted the formulation of extended thermodynamics. After describing the motifs and early evolution of this new branch of irreversible thermodynamics, the authors apply the theory to mon-atomic gases, mixtures of gases, relativistic gases, and "gases" of phonons and photons. The discussion brings into perspective the various phenomena called second sound, such as heat propagation, propagation of shear stress and concentration, and the second sound in liquid helium. The formal mathematical structure of extended thermodynamics is exposed and the theory is shown to be fully compatible with the kinetic theory of gases. The study closes with the testing of extended thermodynamics thro...

Optimal control problem for the extended Fisher–Kolmogorov equation

Indian Academy of Sciences (India)

by methods of optimal control, such as chemical engineering and vehicle ... ern optimal control theories and applied models are not only represented by .... Obviously, equation (2.5) is an ordinary differential equation and according to ODE.

On turbulent motion caused by temperature fluctuations - a critical review on the Boussinesq approximation

International Nuclear Information System (INIS)

Ruediger, R.

1977-01-01

Fluctuating motions which are caused by a given stochastical temperature field acting in a gas with gravitation and T = constant are dealt with. It results that the often used Boussinesq approximation much underestimates the horizontal motions in case wide-spread temperature fluctuations occur. For sufficiently large scales the horizontal motion exceeds the vertical ones even in the case of the temperature field fluctuating completely isotropically. Scales of 1,000 km and 1 day in the Earth atmosphere lead to the observed value u'(horizontal)/u'(vertical) approximately 10. Finally besides the relation between density correlation and pressure correlation the expression for the turbulent mass transport vanishing with the molecular viscosity is determined. (author)

Adiabatic invariants of the extended KdV equation

Energy Technology Data Exchange (ETDEWEB)

Karczewska, Anna [Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra (Poland); Rozmej, Piotr, E-mail: p.rozmej@if.uz.zgora.pl [Institute of Physics, Faculty of Physics and Astronomy, University of Zielona Góra, Szafrana 4a, 65-246 Zielona Góra (Poland); Infeld, Eryk [National Centre for Nuclear Research, Hoża 69, 00-681 Warszawa (Poland); Rowlands, George [Department of Physics, University of Warwick, Coventry, CV4 7A (United Kingdom)

2017-01-30

When the Euler equations for shallow water are taken to the next order, beyond KdV, momentum and energy are no longer exact invariants. (The only one is mass.) However, adiabatic invariants (AI) can be found. When the KdV expansion parameters are zero, exact invariants are recovered. Existence of adiabatic invariants results from general theory of near-identity transformations (NIT) which allow us to transform higher order nonintegrable equations to asymptotically equivalent (when small parameters tend to zero) integrable form. Here we present a direct method of calculations of adiabatic invariants. It does not need a transformation to a moving reference frame nor performing a near-identity transformation. Numerical tests show that deviations of AI from constant values are indeed small. - Highlights: • We suggest a new and simple method for calculating adiabatic invariants of second order wave equations. • It is easy to use and we hope that it will be useful if published. • Interesting numerics included.

Rhodium SPND's Error Reduction using Extended Kalman Filter combined with Time Dependent Neutron Diffusion Equation

International Nuclear Information System (INIS)

Lee, Jeong Hun; Park, Tong Kyu; Jeon, Seong Su

2014-01-01

The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core

Rhodium SPND's Error Reduction using Extended Kalman Filter combined with Time Dependent Neutron Diffusion Equation

Energy Technology Data Exchange (ETDEWEB)

Lee, Jeong Hun; Park, Tong Kyu; Jeon, Seong Su [FNC Technology Co., Ltd., Yongin (Korea, Republic of)

2014-05-15

The Rhodium SPND is accurate in steady-state conditions but responds slowly to changes in neutron flux. The slow response time of Rhodium SPND precludes its direct use for control and protection purposes specially when nuclear power plant is used for load following. To shorten the response time of Rhodium SPND, there were some acceleration methods but they could not reflect neutron flux distribution in reactor core. On the other hands, some methods for core power distribution monitoring could not consider the slow response time of Rhodium SPND and noise effect. In this paper, time dependent neutron diffusion equation is directly used to estimate reactor power distribution and extended Kalman filter method is used to correct neutron flux with Rhodium SPND's and to shorten the response time of them. Extended Kalman filter is effective tool to reduce measurement error of Rhodium SPND's and even simple FDM to solve time dependent neutron diffusion equation can be an effective measure. This method reduces random errors of detectors and can follow reactor power level without cross-section change. It means monitoring system may not calculate cross-section at every time steps and computing time will be shorten. To minimize delay of Rhodium SPND's conversion function h should be evaluated in next study. Neutron and Rh-103 reaction has several decay chains and half-lives over 40 seconds causing delay of detection. Time dependent neutron diffusion equation will be combined with decay chains. Power level and distribution change corresponding movement of control rod will be tested with more complicated reference code as well as xenon effect. With these efforts, final result is expected to be used as a powerful monitoring tool of nuclear reactor core.

Modeling of large amplitude plasma blobs in three-dimensions

Energy Technology Data Exchange (ETDEWEB)

Angus, Justin R. [Naval Research Laboratory, 4555 Overlook Avenue, Washington, DC 20375 (United States); Umansky, Maxim V. [Lawrence Livermore National Laboratory, 7000 East Avenue, Livermore, California 94550 (United States)

2014-01-15

Fluctuations in fusion boundary and similar plasmas often have the form of filamentary structures, or blobs, that convectively propagate radially. This may lead to the degradation of plasma facing components as well as plasma confinement. Theoretical analysis of plasma blobs usually takes advantage of the so-called Boussinesq approximation of the potential vorticity equation, which greatly simplifies the treatment analytically and numerically. This approximation is only strictly justified when the blob density amplitude is small with respect to that of the background plasma. However, this is not the case for typical plasma blobs in the far scrape-off layer region, where the background density is small compared to that of the blob, and results obtained based on the Boussinesq approximation are questionable. In this report, the solution of the full vorticity equation, without the usual Boussinesq approximation, is proposed via a novel numerical approach. The method is used to solve for the evolution of 2D and 3D plasma blobs in a regime where the Boussinesq approximation is not valid. The Boussinesq solution under predicts the cross field transport in 2D. However, in 3D, for parameters typical of current tokamaks, the disparity between the radial cross field transport from the Boussinesq approximation and full solution is virtually non-existent due to the effects of the drift wave instability.

Modeling of large amplitude plasma blobs in three-dimensions

International Nuclear Information System (INIS)

Angus, Justin R.; Umansky, Maxim V.

2014-01-01

Fluctuations in fusion boundary and similar plasmas often have the form of filamentary structures, or blobs, that convectively propagate radially. This may lead to the degradation of plasma facing components as well as plasma confinement. Theoretical analysis of plasma blobs usually takes advantage of the so-called Boussinesq approximation of the potential vorticity equation, which greatly simplifies the treatment analytically and numerically. This approximation is only strictly justified when the blob density amplitude is small with respect to that of the background plasma. However, this is not the case for typical plasma blobs in the far scrape-off layer region, where the background density is small compared to that of the blob, and results obtained based on the Boussinesq approximation are questionable. In this report, the solution of the full vorticity equation, without the usual Boussinesq approximation, is proposed via a novel numerical approach. The method is used to solve for the evolution of 2D and 3D plasma blobs in a regime where the Boussinesq approximation is not valid. The Boussinesq solution under predicts the cross field transport in 2D. However, in 3D, for parameters typical of current tokamaks, the disparity between the radial cross field transport from the Boussinesq approximation and full solution is virtually non-existent due to the effects of the drift wave instability

Theory of Brownian motion with the Alder-Wainwright effect

International Nuclear Information System (INIS)

Okabe, Y.

1986-01-01

The Stokes-Boussinesq-Langevin equation, which describes the time evolution of Brownian motion with the Alder-Wainwright effect, can be treated in the framework of the theory of KMO-Langevin equations which describe the time evolution of a real, stationary Gaussian process with T-positivity (reflection positivity) originating in axiomatic quantum field theory. After proving the fluctuation-dissipation theorems for KMO-Langevin equations, the authors obtain an explicit formula for the deviation from the classical Einstein relation that occurs in the Stokes-Boussinesq-Langevin equation with a white noise as its random force. The authors interested in whether or not it can be measured experimentally

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Science.gov (United States)

Vitanov, Nikolay K.; Dimitrova, Zlatinka I.

2018-03-01

We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

Superfield realizations of N=2 super-W3

International Nuclear Information System (INIS)

Ivanov, E.; Krivonos, S.

1992-04-01

We present a manifestly N=2 supersymmetric formulation of N=2 super-W 3 algebra (its classical version) in terms of the spin 1 and spin 2 supercurrents. Two closely related types of the Feigin-Fuchs representation for these supercurrents are found: via two chiral spin 1/2 superfields generating N=2 extended U(1) Kac-Moody algebras and via two free chiral spin 0 superfields. We also construct a one-parameter family of N=2 super Boussinesq equations for which N=2 super-W 3 provides the second Hamiltonian structure. (author). 17 refs

Modified Method of Simplest Equation Applied to the Nonlinear Schrödinger Equation

Directory of Open Access Journals (Sweden)

Vitanov Nikolay K.

2018-03-01

Full Text Available We consider an extension of the methodology of the modified method of simplest equation to the case of use of two simplest equations. The extended methodology is applied for obtaining exact solutions of model nonlinear partial differential equations for deep water waves: the nonlinear Schrödinger equation. It is shown that the methodology works also for other equations of the nonlinear Schrödinger kind.

On the mechanical prototypes of fundamental hydrodynamic invariants and slow manifolds

Energy Technology Data Exchange (ETDEWEB)

Dolzhansky, Feliks V

2005-12-31

Arnol'd's group-theoretical concept of generalized rigid body includes the Euler equations of motion of the classical gyroscope and ideal homogeneous fluid as particular representatives. Here, this concept is extended to motion in force fields with a scalar or vector potential and in a Coriolis force field. The concepts of generalized heavy top and generalized MHD system are introduced. As particular cases, they include, on the one hand, the Euler-Poisson equations of the classical heavy top and the Kirchhoff equations of motion of a solid body in a potential flow of an ideal incompressible fluid and, on the other hand, the Oberbeck-Boussinesq equations of motion of a heavy fluid and MHD equations. On this basis, mechanical prototypes are constructed for all known fundamental hydrodynamic invariants and global geophysical flows, including a prototype of the general atmospheric circulation. (reviews of topical problems)

On the mechanical prototypes of fundamental hydrodynamic invariants and slow manifolds

International Nuclear Information System (INIS)

Dolzhansky, Feliks V

2005-01-01

Arnol'd's group-theoretical concept of generalized rigid body includes the Euler equations of motion of the classical gyroscope and ideal homogeneous fluid as particular representatives. Here, this concept is extended to motion in force fields with a scalar or vector potential and in a Coriolis force field. The concepts of generalized heavy top and generalized MHD system are introduced. As particular cases, they include, on the one hand, the Euler-Poisson equations of the classical heavy top and the Kirchhoff equations of motion of a solid body in a potential flow of an ideal incompressible fluid and, on the other hand, the Oberbeck-Boussinesq equations of motion of a heavy fluid and MHD equations. On this basis, mechanical prototypes are constructed for all known fundamental hydrodynamic invariants and global geophysical flows, including a prototype of the general atmospheric circulation. (reviews of topical problems)

A new extended elliptic equation rational expansion method and its application to (2 + 1)-dimensional Burgers equation

International Nuclear Information System (INIS)

Wang Baodong; Song Lina; Zhang Hongqing

2007-01-01

In this paper, we present a new elliptic equation rational expansion method to uniformly construct a series of exact solutions for nonlinear partial differential equations. As an application of the method, we choose the (2 + 1)-dimensional Burgers equation to illustrate the method and successfully obtain some new and more general solutions

On a functional equation related to the intermediate long wave equation

International Nuclear Information System (INIS)

Hone, A N W; Novikov, V S

2004-01-01

We resolve an open problem stated by Ablowitz et al (1982 J. Phys. A: Math. Gen. 15 781) concerning the integral operator appearing in the intermediate long wave equation. We explain how this is resolved using the perturbative symmetry approach introduced by one of us with Mikhailov. By solving a certain functional equation, we prove that the intermediate long wave equation and the Benjamin-Ono equation are the unique integrable cases within a particular class of integro-differential equations. Furthermore, we explain how the perturbative symmetry approach is naturally extended to treat equations on a periodic domain. (letter to the editor)

An Operator Method for Field Moments from the Extended Parabolic Wave Equation and Analytical Solutions of the First and Second Moments for Atmospheric Electromagnetic Wave Propagation

Science.gov (United States)

Manning, Robert M.

2004-01-01

The extended wide-angle parabolic wave equation applied to electromagnetic wave propagation in random media is considered. A general operator equation is derived which gives the statistical moments of an electric field of a propagating wave. This expression is used to obtain the first and second order moments of the wave field and solutions are found that transcend those which incorporate the full paraxial approximation at the outset. Although these equations can be applied to any propagation scenario that satisfies the conditions of application of the extended parabolic wave equation, the example of propagation through atmospheric turbulence is used. It is shown that in the case of atmospheric wave propagation and under the Markov approximation (i.e., the delta-correlation of the fluctuations in the direction of propagation), the usual parabolic equation in the paraxial approximation is accurate even at millimeter wavelengths. The comprehensive operator solution also allows one to obtain expressions for the longitudinal (generalized) second order moment. This is also considered and the solution for the atmospheric case is obtained and discussed. The methodology developed here can be applied to any qualifying situation involving random propagation through turbid or plasma environments that can be represented by a spectral density of permittivity fluctuations.

Development of Robust Boundary Layer Controllers

National Research Council Canada - National Science Library

Speyer, Jason

2002-01-01

.... The three-dimensional Navier-Stokes equations of channel flow, linearized about a Poisueille profile, and Oberbeck-Boussinesq equations of a layer of fluid, linearized about the no motion state...

On the relation between elementary partial difference equations and partial differential equations

NARCIS (Netherlands)

van den Berg, I.P.

1998-01-01

The nonstandard stroboscopy method links discrete-time ordinary difference equations of first-order and continuous-time, ordinary differential equations of first order. We extend this method to the second order, and also to an elementary, yet general class of partial difference/differential

Reconstruction of extended sources for the Helmholtz equation

KAUST Repository

Kress, Rainer; Rundell, William

2013-01-01

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our

Quantum Potential and Symmetries in Extended Phase Space

Directory of Open Access Journals (Sweden)

Sadollah Nasiri

2006-06-01

Full Text Available The behavior of the quantum potential is studied for a particle in a linear and a harmonic potential by means of an extended phase space technique. This is done by obtaining an expression for the quantum potential in momentum space representation followed by the generalization of this concept to extended phase space. It is shown that there exists an extended canonical transformation that removes the expression for the quantum potential in the dynamical equation. The situation, mathematically, is similar to disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates that changes the physical potential to an effective one. The representation where the quantum potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form, is one in which the dynamical equation turns out to be the Wigner equation.

Spectral theories for linear differential equations

International Nuclear Information System (INIS)

Sell, G.R.

1976-01-01

The use of spectral analysis in the study of linear differential equations with constant coefficients is not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour of the solutions. The spectral analysis, via the Jordan canonical form, will not only lead to a representation theorem for a basis of solutions, but will also give a rather precise statement of the (exponential) growth rates of various solutions. Various attempts have been made to extend this analysis to linear differential equations with time-varying coefficients. The most complete such extensions is the Floquet theory for equations with periodic coefficients. For time-varying linear differential equations with aperiodic coefficients several authors have attempted to ''extend'' the Foquet theory. The precise meaning of such an extension is itself a problem, and we present here several attempts in this direction that are related to the general problem of extending the spectral analysis of equations with constant coefficients. The main purpose of this paper is to introduce some problems of current research. The primary problem we shall examine occurs in the context of linear differential equations with almost periodic coefficients. We call it ''the Floquet problem''. (author)

Stochastic Differential Equations and Kondratiev Spaces

Energy Technology Data Exchange (ETDEWEB)

Vaage, G.

1995-05-01

The purpose of this mathematical thesis was to improve the understanding of physical processes such as fluid flow in porous media. An example is oil flowing in a reservoir. In the first of five included papers, Hilbert space methods for elliptic boundary value problems are used to prove the existence and uniqueness of a large family of elliptic differential equations with additive noise without using the Hermite transform. The ideas are then extended to the multidimensional case and used to prove existence and uniqueness of solution of the Stokes equations with additive noise. The second paper uses functional analytic methods for partial differential equations and presents a general framework for proving existence and uniqueness of solutions to stochastic partial differential equations with multiplicative noise, for a large family of noises. The methods are applied to equations of elliptic, parabolic as well as hyperbolic type. The framework presented can be extended to the multidimensional case. The third paper shows how the ideas from the second paper can be extended to study the moving boundary value problem associated with the stochastic pressure equation. The fourth paper discusses a set of stochastic differential equations. The fifth paper studies the relationship between the two families of Kondratiev spaces used in the thesis. 102 refs.

On the Inclusion of Difference Equation Problems and Z Transform Methods in Sophomore Differential Equation Classes

Science.gov (United States)

Savoye, Philippe

2009-01-01

In recent years, I started covering difference equations and z transform methods in my introductory differential equations course. This allowed my students to extend the "classical" methods for (ordinary differential equation) ODE's to discrete time problems arising in many applications.

Algebraic entropy for differential-delay equations

OpenAIRE

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.

Cellular solutions for the Poisson equation in extended systems

International Nuclear Information System (INIS)

Zhang, X.; Butler, W.H.; MacLaren, J.M.; van Ek, J.

1994-01-01

The Poisson equation for the electrostatic potential in a solid is solved using three different cellular techniques. The relative merits of these different approaches are discussed for two test charge densities for which an analytic solution to the Poisson equation is known. The first approach uses full-cell multiple-scattering theory and results in the famililar structure constant and multipole moment expansion. This solution is shown to be valid everywhere inside the cell, although for points outside the muffin-tin sphere but inside the cell the sums must be performed in the correct order to yield meaningful results. A modification of the multiple-scattering-theory approach yields a second method, a Green-function cellular method, which only requires the solution of a nearest-neighbor linear system of equations. A third approach, a related variational cellular method, is also derived. The variational cellular approach is shown to be the most accurate and reliable, and to have the best convergence in angular momentum of the three methods. Coulomb energies accurate to within 10 -6 hartree are easily achieved with the variational cellular approach, demonstrating the practicality of the approach in electronic structure calculations

The next step in coastal numerical models: spectral/hp element methods?

DEFF Research Database (Denmark)

Eskilsson, Claes; Engsig-Karup, Allan Peter; Sherwin, Spencer J.

2005-01-01

In this paper we outline the application of spectral/hp element methods for modelling nonlinear and dispersive waves. We present one- and two-dimensional test cases for the shallow water equations and Boussinesqtype equations – including highly dispersive Boussinesq-type equations....

A Novel Biped Pattern Generator Based on Extended ZMP and Extended Cart-Table Model

Directory of Open Access Journals (Sweden)

Guangbin Sun

2015-07-01

Full Text Available This paper focuses on planning patterns for biped walking on complex terrains. Two problems are solved: ZMP (zero moment point cannot be used on uneven terrain, and the conventional cart-table model does not allow vertical CM (centre of mass motion. For the ZMP definition problem, we propose the extended ZMP (EZMP concept as an extension of ZMP to uneven terrains. It can be used to judge dynamic balance on universal terrains. We achieve a deeper insight into the connection and difference between ZMP and EZMP by adding different constraints. For the model problem, we extend the cart-table model by using a dynamic constraint instead of constant height constraint, which results in a mathematically symmetric set of three equations. In this way, the vertical motion is enabled and the resultant equations are still linear. Based on the extended ZMP concept and extended cart-table model, a biped pattern generator using triple preview controllers is constructed and implemented simultaneously to three dimensions. Using the proposed pattern generator, the Atlas robot is simulated. The simulation results show the robot can walk stably on rather complex terrains by accurately tracking extended ZMP.

Numerical Study on Turbulent Flow Inside a Channel with an Extended Chamber

International Nuclear Information System (INIS)

Lee, Young Tae; Lim, Hee Chang

2010-01-01

The paper describes a Large Eddy Simulation (LES) study of turbulent flow around a cavity. A series of three-dimensional cavities placed in a turbulent boundary layer are simulated at a Reynolds number of 1.0 x 10 5 by considering U and h, which represent the velocity at the top and the depth of the cavity, respectively. In order to obtain the appropriate solution for the filtered Navier-Stokes equation for incompressible flow, the computational mesh forms dense close to the wall of the cavity but relatively coarse away from the wall; this helps reduce computation cost and ensure rapid convergence. The Boussinesq hypothesis is employed in the subgrid-scale turbulence model. In order to determine the subgrid-scale turbulent viscosity, the Smagorinsky-Lilly SGS model is applied and the CFL number for time marching is set as 1.0. The results show the flow variations inside cavities of different sizes and shapes

Extended N=2 supersymmetric matrix (1, s)-KdV hierarchies

International Nuclear Information System (INIS)

Krivonos, S.O.; Sorin, A.S.

1997-01-01

We propose the Lax operators for N=2 supersymmetric matrix generalization of the bosonic (1, s)-KdV hierarchies. The simplest examples - the N=2 supersymmetric a=4 KdV and a=5/2 Boussinesq hierarchies - are discussed in detail

Integrability and Poisson Structures of Three Dimensional Dynamical Systems and Equations of Hydrodynamic Type

Science.gov (United States)

Gumral, Hasan

Poisson structure of completely integrable 3 dimensional dynamical systems can be defined in terms of an integrable 1-form. We take advantage of this fact and use the theory of foliations in discussing the geometrical structure underlying complete and partial integrability. We show that the Halphen system can be formulated in terms of a flat SL(2,R)-valued connection and belongs to a non-trivial Godbillon-Vey class. On the other hand, for the Euler top and a special case of 3-species Lotka-Volterra equations which are contained in the Halphen system as limiting cases, this structure degenerates into the form of globally integrable bi-Hamiltonian structures. The globally integrable bi-Hamiltonian case is a linear and the sl_2 structure is a quadratic unfolding of an integrable 1-form in 3 + 1 dimensions. We complete the discussion of the Hamiltonian structure of 2-component equations of hydrodynamic type by presenting the Hamiltonian operators for Euler's equation and a continuum limit of Toda lattice. We present further infinite sequences of conserved quantities for shallow water equations and show that their generalizations by Kodama admit bi-Hamiltonian structure. We present a simple way of constructing the second Hamiltonian operators for N-component equations admitting some scaling properties. The Kodama reduction of the dispersionless-Boussinesq equations and the Lax reduction of the Benney moment equations are shown to be equivalent by a symmetry transformation. They can be cast into the form of a triplet of conservation laws which enable us to recognize a non-trivial scaling symmetry. The resulting bi-Hamiltonian structure generates three infinite sequences of conserved densities.

On reduction and exact solutions of nonlinear many-dimensional Schroedinger equations

International Nuclear Information System (INIS)

Barannik, A.F.; Marchenko, V.A.; Fushchich, V.I.

1991-01-01

With the help of the canonical decomposition of an arbitrary subalgebra of the orthogonal algebra AO(n) the rank n and n-1 maximal subalgebras of the extended isochronous Galileo algebra, the rank n maximal subalgebras of the generalized extended classical Galileo algebra AG(a,n) the extended special Galileo algebra AG(2,n) and the extended whole Galileo algebra AG(3,n) are described. By using the rank n subalgebras, ansatze reducing the many dimensional Schroedinger equations to ordinary differential equations is found. With the help of the reduced equation solutions exact solutions of the Schroedinger equation are considered

Calculation of propellant gas pressure by simple extended corresponding state principle

OpenAIRE

Bin Xu; San-jiu Ying; Xin Liao

2016-01-01

The virial equation can well describe gas state at high temperature and pressure, but the difficulties in virial coefficient calculation limit the use of virial equation. Simple extended corresponding state principle (SE-CSP) is introduced in virial equation. Based on a corresponding state equation, including three characteristic parameters, an extended parameter is introduced to describe the second virial coefficient expressions of main products of propellant gas. The modified SE-CSP second ...

Experimental and numerical study of the wave run-up along a vertical plate

DEFF Research Database (Denmark)

Molin, Bernard; Kimmoun, O.; Liu, Y.

2010-01-01

Results from experiments on wave interaction with a rigid vertical plate are reported. The 5m long plate is set against the wall of a 30m wide basin, at 100m from the wavemaker. This set-up is equivalent to a 10m plate in the middle of a 60m wide basin. Regular waves are produced, with wavelength...... on extended Boussinesq equations. In most of the experimental tests, despite the large distance from the wavemaker to the plate and the small amplitude of the incident wave, no steady state is attained by the end of the exploitable part of the records....

Nonlinear integrodifferential equations as discrete systems

Science.gov (United States)

Tamizhmani, K. M.; Satsuma, J.; Grammaticos, B.; Ramani, A.

1999-06-01

We analyse a class of integrodifferential equations of the `intermediate long wave' (ILW) type. We show that these equations can be formally interpreted as discrete, differential-difference systems. This allows us to link equations of this type with previous results of ours involving differential-delay equations and, on the basis of this, propose new integrable equations of ILW type. Finally, we extend this approach to pure difference equations and propose ILW forms for the discrete lattice KdV equation.

Analysis of atmospheric flow over a surface protrusion using the turbulence kinetic energy equation with reference to aeronautical operating systems

Science.gov (United States)

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.

Non-Boussinesq Dissolution-Driven Convection in Porous Media

Science.gov (United States)

Amooie, M. A.; Soltanian, M. R.; Moortgat, J.

2017-12-01

Geological carbon dioxide (CO2) sequestration in deep saline aquifers has been increasingly recognized as a feasible technology to stabilize the atmospheric carbon concentrations and subsequently mitigate the global warming. Solubility trapping is one of the most effective storage mechanisms, which is associated initially with diffusion-driven slow dissolution of gaseous CO2 into the aqueous phase, followed by density-driven convective mixing of CO2 throughout the aquifer. The convection includes both diffusion and fast advective transport of the dissolved CO2. We study the fluid dynamics of CO2 convection in the underlying single aqueous-phase region. Two modeling approaches are employed to define the system: (i) a constant-concentration condition for CO2 in aqueous phase at the top boundary, and (ii) a sufficiently low, constant injection-rate for CO2 from top boundary. The latter allows for thermodynamically consistent evolution of the CO2 composition and the aqueous phase density against the rate at which the dissolved CO2 convects. Here we accurately model the full nonlinear phase behavior of brine-CO2 mixture in a confined domain altered by dissolution and compressibility, while relaxing the common Boussinesq approximation. We discover new flow regimes and present quantitative scaling relations for global characters of spreading, mixing, and dissolution flux in two- and three-dimensional media for the both model types. We then revisit the universal Sherwood-Rayleigh scaling that is under debate for porous media convective flows. Our findings confirm the sublinear scaling for the constant-concentration case, while reconciling the classical linear scaling for the constant-injection model problem. The results provide a detailed perspective into how the available modeling strategies affect the prediction ability for the total amount of CO2 dissolved in the long term within saline aquifers of different permeabilities.

Exact Solutions of a Fractional-Type Differential-Difference Equation Related to Discrete MKdV Equation

International Nuclear Information System (INIS)

Aslan İsmail

2014-01-01

The extended simplest equation method is used to solve exactly a new differential-difference equation of fractional-type, proposed by Narita [J. Math. Anal. Appl. 381 (2011) 963] quite recently, related to the discrete MKdV equation. It is shown that the model supports three types of exact solutions with arbitrary parameters: hyperbolic, trigonometric and rational, which have not been reported before. (general)

A new sine-Gordon equation expansion algorithm to investigate some special nonlinear differential equations

International Nuclear Information System (INIS)

Yan Zhenya

2005-01-01

A new transformation method is developed using the general sine-Gordon travelling wave reduction equation and a generalized transformation. With the aid of symbolic computation, this method can be used to seek more types of solutions of nonlinear differential equations, which include not only the known solutions derived by some known methods but new solutions. Here we choose the double sine-Gordon equation, the Magma equation and the generalized Pochhammer-Chree (PC) equation to illustrate the method. As a result, many types of new doubly periodic solutions are obtained. Moreover when using the method to these special nonlinear differential equations, some transformations are firstly needed. The method can be also extended to other nonlinear differential equations

Transformation of nonlinear discrete-time system into the extended observer form

Science.gov (United States)

Kaparin, V.; Kotta, Ü.

2018-04-01

The paper addresses the problem of transforming discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. Necessary and sufficient conditions for the existence of both the extended coordinate and output transformations, solving the problem, are formulated in terms of differential one-forms, associated with the input-output equation, corresponding to the state equations. An algorithm for transformation of state equations into the extended observer form is proposed and illustrated by an example. Moreover, the considered approach is compared with the method of dynamic observer error linearisation, which likewise is intended to enlarge the class of systems transformable into an observer form.

Energy exchange analysis in droplet dynamics via the Navier–Stokes–Cahn–Hilliard model

KAUST Repository

Espath, L. F. R.; Sarmiento, Adel; Vignal, Philippe; Varga, B. O. N.; Cortes, Adriano Mauricio; Dalcin, Lisandro; Calo, Victor M.

2016-01-01

We develop the energy budget equation of the coupled Navier-Stokes-Cahn-Hilliard (NSCH) system. We use the NSCH equations to model the dynamics of liquid droplets in a liquid continuum. Buoyancy effects are accounted for through the Boussinesq

Clarkson-Kruskal Direct Similarity Approach for Differential-Difference Equations

Institute of Scientific and Technical Information of China (English)

SHEN Shou-Feng

2005-01-01

In this letter, the Clarkson-Kruskal direct method is extended to similarity reduce some differentialdifference equations. As examples, the differential-difference KZ equation and KP equation are considered.

Nonlinear von Neumann equations for quantum dissipative systems

International Nuclear Information System (INIS)

Messer, J.; Baumgartner, B.

1978-01-01

For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Auth.)

Nonlinear von Neumann equations for quantum dissipative systems

International Nuclear Information System (INIS)

Messer, J.; Baumgartner, B.

For pure states nonlinear Schroedinger equations, the so-called Schroedinger-Langevin equations are well-known to model quantum dissipative systems of the Langevin type. For mixtures it is shown that these wave equations do not extend to master equations, but to corresponding nonlinear von Neumann equations. Solutions for the damped harmonic oscillator are discussed. (Author)

A Modified Homogeneous Balance Method and Its Applications

International Nuclear Information System (INIS)

Liu Chunping

2011-01-01

A modified homogeneous balance method is proposed by improving some key steps in the homogeneous balance method. Bilinear equations of some nonlinear evolution equations are derived by using the modified homogeneous balance method. Generalized Boussinesq equation, KP equation, and mKdV equation are chosen as examples to illustrate our method. This approach is also applicable to a large variety of nonlinear evolution equations. (general)

A note on the extended dispersionless Toda hierarchy

Science.gov (United States)

Lee, Niann-Chern; Tu, Ming-Hsien

2013-04-01

We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological ℂP1 model. Using the dispersionless Hirota equations, we compute the twopoint functions and express the result in terms of Catalan numbers

Geometrical-integrability constraints and equations of motion in four plus extended super spaces

International Nuclear Information System (INIS)

Chau, L.L.

1987-01-01

It is pointed out that many equations of motion in physics, including gravitational and Yang-Mills equations, have a common origin: i.e. they are the results of certain geometrical integrability conditions. These integrability conditions lead to linear systems and conservation laws that are important in integrating these equations of motion

On polynomial solutions of the Heun equation

International Nuclear Information System (INIS)

Gurappa, N; Panigrahi, Prasanta K

2004-01-01

By making use of a recently developed method to solve linear differential equations of arbitrary order, we find a wide class of polynomial solutions to the Heun equation. We construct the series solution to the Heun equation before identifying the polynomial solutions. The Heun equation extended by the addition of a term, -σ/x, is also amenable for polynomial solutions. (letter to the editor)

Tailoring discrete quantum walk dynamics via extended initial conditions

Energy Technology Data Exchange (ETDEWEB)

Valcarcel, German J de; Roldan, Eugenio [Departament d' Optica, Universitat de Valencia, Dr Moliner 50, 46100-Burjassot, Spain, EU (Spain); Romanelli, Alejandro, E-mail: german.valcarcel@uv.es, E-mail: eugenio.roldan@uv.es, E-mail: alejo@fing.edu.uy [Instituto de Fisica, Facultad de IngenierIa, Universidad de la Republica, CC 30, CP 11000, Montevideo (Uruguay)

2010-12-15

We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schroedinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.

Tailoring discrete quantum walk dynamics via extended initial conditions

International Nuclear Information System (INIS)

Valcarcel, German J de; Roldan, Eugenio; Romanelli, Alejandro

2010-01-01

We study the evolution of initially extended distributions in the coined quantum walk (QW) on the line. By analysing the dispersion relation of the process, continuous wave equations are derived whose form depends on the initial distribution shape. In particular, for a class of initial conditions, the evolution is dictated by the Schroedinger equation of a free particle. As that equation also governs paraxial optical diffraction, all of the phenomenology of the latter can be implemented in the QW. This allows us, in particular, to devise an initially extended condition leading to a uniform probability distribution whose width increases linearly with time, with increasing homogeneity.

Rogue waves in the multicomponent Mel'nikov system and ...

Indian Academy of Sciences (India)

By virtue of the bilinear method and the KP hierarchy reduction technique, exact explicit rational solutions of the multicomponent Mel'nikov equation and the multicomponent Schrödinger–Boussinesq equation are constructed, which contain multicomponent short waves and single-component long wave. For the ...

Correlation of the Rates of Solvolyses of 4-Methylthiophene-2-carbonyl Chloride Using the Extended Grunwald-Winstein Equation

International Nuclear Information System (INIS)

Choi, Ho June; Koo, In Sun

2012-01-01

The specific rates of sovolysis of 4-methylthiophene-2-carbonyl chloride (1) have been determined in 26 pure and binary solvents at 25.0 .deg. C. Product selectivities are reported for solvolyses of 1 in aqueous ethanol and methanol binary mixtures. Comparison of the specific rates of solvolyses of 1 with those for p-methoxybenzoyl chloride (2) in terms of linear free energy relationships (LFER) are helpful in mechanistic considerations, as is also treatment in terms of the extended Grunwald-Winstein equation. It is proposed that the solvolyses of 1 in binary aqueous solvent mixtures proceed through an S N 1 and/or ionization (I) pathway rather than through an associative S N 2 and/or addition-elimination (A-E) pathway

Differential equation analysis in biomedical science and engineering partial differential equation applications with R

CERN Document Server

Schiesser, William E

2014-01-01

Features a solid foundation of mathematical and computational tools to formulate and solve real-world PDE problems across various fields With a step-by-step approach to solving partial differential equations (PDEs), Differential Equation Analysis in Biomedical Science and Engineering: Partial Differential Equation Applications with R successfully applies computational techniques for solving real-world PDE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the com

Nonlinear electrostatic wave equations for magnetized plasmas - II

DEFF Research Database (Denmark)

Dysthe, K. B.; Mjølhus, E.; Pécseli, H. L.

1985-01-01

For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent (electrosta......For pt.I see ibid., vol.26, p.443-7 (1984). The problem of extending the high frequency part of the Zakharov equations for nonlinear electrostatic waves to magnetized plasmas, is considered. Weak electromagnetic and thermal effects are retained on an equal footing. Direction dependent...... (electrostatic) cut-off implies that various cases must be considered separately, leading to equations with rather different properties. Various equations encountered previously in the literature are recovered as limiting cases....

Generalized estimating equations

CERN Document Server

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

Extended common-image-point gathers for anisotropic wave-equation migration

KAUST Repository

Sava, Paul C.; Alkhalifah, Tariq Ali

2010-01-01

In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes

Soliton solutions and chaotic motion of the extended Zakharov-Kuznetsov equations in a magnetized two-ion-temperature dusty plasma

Energy Technology Data Exchange (ETDEWEB)

Zhen, Hui-Ling; Tian, Bo, E-mail: tian-bupt@163.com; Wang, Yu-Feng; Sun, Wen-Rong; Liu, Li-Cai [State Key Laboratory of Information Photonics and Optical Communications, and School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 (China)

2014-07-15

The extended Zakharov-Kuznetsov (eZK) equation for the magnetized two-ion-temperature dusty plasma is studied in this paper. With the help of Hirota method, bilinear forms and N-soliton solutions are given, and soliton propagation is graphically analyzed. We find that the soliton amplitude is positively related to the nonlinear coefficient A, while inversely related to the dispersion coefficients B and C. We obtain that the soliton amplitude will increase with the mass of the jth dust grain and the average charge number residing on the dust grain decreased, but the soliton amplitude will increase with the equilibrium number density of the jth dust grain increased. Upon the introduction of the periodic external forcing term, both the weak and developed chaotic motions can occur. Difference between the two chaotic motions roots in the inequality between the nonlinear coefficient l{sub 2} and perturbed term h{sub 1}. The developed chaos can be weakened with B or C decreased and A increased. Periodic motion of the perturbed eZK equation can be observed when there is a balance between l{sub 2} and h{sub 1}.

A second order splitting algorithm for thermally-driven flow problems

NARCIS (Netherlands)

Minev, P.D.; Vosse, van de F.N.; Timmermans, L.J.P.; Steenhoven, van A.A.

1995-01-01

A splitting technique for solutions of the Navier—Stokes and the energy equations, in Boussinesq approximately, is presented. The equations are first integrated in time using a splitting procedure and then discretized spatially by means of a high-order spectral element method. The whole technique is

Solution of differential equations by application of transformation groups

Science.gov (United States)

Driskell, C. N., Jr.; Gallaher, L. J.; Martin, R. H., Jr.

1968-01-01

Report applies transformation groups to the solution of systems of ordinary differential equations and partial differential equations. Lies theorem finds an integrating factor for appropriate invariance group or groups can be found and can be extended to partial differential equations.

Reconstruction of extended sources for the Helmholtz equation

International Nuclear Information System (INIS)

Kress, Rainer; Rundell, William

2013-01-01

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our inclusions are interior forces with compact support and our data consist of a single measurement of near-field Cauchy data on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler ‘equivalent point source’ problem, and which uses a Newton scheme to improve the corresponding initial approximation. (paper)

Reconstruction of extended sources for the Helmholtz equation

KAUST Repository

Kress, Rainer

2013-02-26

The basis of most imaging methods is to detect hidden obstacles or inclusions within a body when one can only make measurements on an exterior surface. Our underlying model is that of inverse acoustic scattering based on the Helmholtz equation. Our inclusions are interior forces with compact support and our data consist of a single measurement of near-field Cauchy data on the external boundary. We propose an algorithm that under certain assumptions allows for the determination of the support set of these forces by solving a simpler \\'equivalent point source\\' problem, and which uses a Newton scheme to improve the corresponding initial approximation. © 2013 IOP Publishing Ltd.

On reductions of the discrete Kadomtsev-Petviashvili-type equations

Science.gov (United States)

Fu, Wei; Nijhoff, Frank W.

2017-12-01

The reduction by restricting the spectral parameters k and k\\prime on a generic algebraic curve of degree N is performed for the discrete AKP, BKP and CKP equations, respectively. A variety of two-dimensional discrete integrable systems possessing a more general solution structure arise from the reduction, and in each case a unified formula for the generic positive integer N≥slant 2 is given to express the corresponding reduced integrable lattice equations. The obtained extended two-dimensional lattice models give rise to many important integrable partial difference equations as special degenerations. Some new integrable lattice models such as the discrete Sawada-Kotera, Kaup-Kupershmidt and Hirota-Satsuma equations in extended form are given as examples within the framework.

Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows

International Nuclear Information System (INIS)

Ueckermann, M.P.; Lermusiaux, P.F.J.; Sapsis, T.P.

2013-01-01

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.

Backlund transformations as canonical transformations

International Nuclear Information System (INIS)

Villani, A.; Zimerman, A.H.

1977-01-01

Toda and Wadati as well as Kodama and Wadati have shown that the Backlund transformations, for the exponential lattice equation, sine-Gordon equation, K-dV (Korteweg de Vries) equation and modifies K-dV equation, are canonical transformation. It is shown that the Backlund transformation for the Boussinesq equation, for a generalized K-dV equation, for a model equation for shallow water waves and for the nonlinear Schroedinger equation are also canonical transformations [pt

The classification of the single travelling wave solutions to the ...

Indian Academy of Sciences (India)

2016-09-21

Sep 21, 2016 ... For example,. Fan used Liu's method [11,12] to invest the generalized equal width equation and Pochhammer–Chree equa- tion, and she obtained all the possible travelling wave solutions including elliptic functions and hyperelliptic functions. In this paper, we consider the variant Boussinesq equations [13].

An integral transform of the Salpeter equation

International Nuclear Information System (INIS)

Krolikowski, W.

1980-03-01

We find a new form of relativistic wave equation for two spin-1/2 particles, which arises by an integral transformation (in the position space) of the wave function in the Salpeter equation. The non-locality involved in this transformation is extended practically over the Compton wavelength of the lighter of two particles. In the case of equal masses the new equation assumes the form of the Breit equation with an effective integral interaction. In the one-body limit it reduces to the Dirac equation also with an effective integral interaction. (author)

Extended common-image-point gathers for anisotropic wave-equation migration

KAUST Repository

Sava, Paul C.

2010-01-01

In regions characterized by complex subsurface structure, wave-equation depth migration is a powerful tool for accurately imaging the earth’s interior. The quality of the final image greatly depends on the quality of the model which includes anisotropy parameters (Gray et al., 2001). In particular, it is important to construct subsurface velocity models using techniques that are consistent with the methods used for imaging. Generally speaking, there are two possible strategies for velocity estimation from surface seismic data in the context of wavefield-based imaging (Sava et al., 2010). One possibility is to formulate an objective function in the data space, prior to migration, by matching the recorded data with simulated data. Techniques in this category are known by the name of waveform inversion. Another possibility is to formulate an objective function in the image space, after migration, by measuring and correcting image features that indicate model inaccuracies. Techniques in this category are known as wave-equation migration velocity analysis (MVA).

Tangent Lines without Derivatives for Quadratic and Cubic Equations

Science.gov (United States)

Carroll, William J.

2009-01-01

In the quadratic equation, y = ax[superscript 2] + bx + c, the equation y = bx + c is identified as the equation of the line tangent to the parabola at its y-intercept. This is extended to give a convenient method of graphing tangent lines at any point on the graph of a quadratic or a cubic equation. (Contains 5 figures.)

Study on the Variation of Groundwater Level under Time-varying Recharge

Science.gov (United States)

Wu, Ming-Chang; Hsieh, Ping-Cheng

2017-04-01

The slopes of the suburbs come to important areas by focusing on the work of soil and water conservation in recent years. The water table inside the aquifer is affected by rainfall, geology and topography, which will result in the change of groundwater discharge and water level. Currently, the way to obtain water table information is to set up the observation wells; however, owing to that the cost of equipment and the wells excavated is too expensive, we develop a mathematical model instead, which might help us to simulate the groundwater level variation. In this study, we will discuss the groundwater level change in a sloping unconfined aquifer with impermeable bottom under time-varying rainfall events. Referring to Child (1971), we employ the Boussinesq equation as the governing equation, and apply the General Integral Transforms Method (GITM) to analyzing the groundwater level after linearizing the Boussinesq equation. After comparing the solution with Verhoest & Troch (2000) and Bansal & Das (2010), we get satisfactory results. To sum up, we have presented an alternative approach to solve the linearized Boussinesq equation for the response of groundwater level in a sloping unconfined aquifer. The present analytical results combine the effect of bottom slope and the time-varying recharge pattern on the water table fluctuations. Owing to the limitation and difficulty of measuring the groundwater level directly, we develop such a mathematical model that we can predict or simulate the variation of groundwater level affected by any rainfall events in advance.

Isomorphism of Intransitive Linear Lie Equations

Directory of Open Access Journals (Sweden)

Jose Miguel Martins Veloso

2009-11-01

Full Text Available We show that formal isomorphism of intransitive linear Lie equations along transversal to the orbits can be extended to neighborhoods of these transversal. In analytic cases, the word formal is dropped from theorems. Also, we associate an intransitive Lie algebra with each intransitive linear Lie equation, and from the intransitive Lie algebra we recover the linear Lie equation, unless of formal isomorphism. The intransitive Lie algebra gives the structure functions introduced by É. Cartan.

On implicit abstract neutral nonlinear differential equations

Energy Technology Data Exchange (ETDEWEB)

Hernández, Eduardo, E-mail: lalohm@ffclrp.usp.br [Universidade de São Paulo, Departamento de Computação e Matemática, Faculdade de Filosofia Ciências e Letras de Ribeirão Preto (Brazil); O’Regan, Donal, E-mail: donal.oregan@nuigalway.ie [National University of Ireland, School of Mathematics, Statistics and Applied Mathematics (Ireland)

2016-04-15

In this paper we continue our developments in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) on the existence of solutions for abstract neutral differential equations. In particular we extend the results in Hernández and O’Regan (J Funct Anal 261:3457–3481, 2011) for the case of implicit nonlinear neutral equations and we focus on applications to partial “nonlinear” neutral differential equations. Some applications involving partial neutral differential equations are presented.

Multi-component WKI equations and their conservation laws

Energy Technology Data Exchange (ETDEWEB)

Qu Changzheng [Department of Mathematics, Northwest University, Xi' an 710069 (China) and Center for Nonlinear Studies, Northwest University, Xi' an 710069 (China)]. E-mail: qu_changzheng@hotmail.com; Yao Ruoxia [Department of Computer Sciences, East China Normal University, Shanghai 200062 (China); Department of Computer Sciences, Weinan Teacher' s College, Weinan 715500 (China); Liu Ruochen [Department of Mathematics, Northwest University, Xi' an 710069 (China)

2004-10-25

In this Letter, a two-component WKI equation is obtained by using the fact that when curvature and torsion of a space curve satisfy the vector modified KdV equation, a graph of the curve satisfies the two-component WKI equation, which is a natural generalization to the WKI equation. It is shown that the two-component WKI equation can be solved in terms of the extended WKI scheme, and it admits an infinite number of conservation laws. In the same vein, a n-component generalization to the WKI equation is proposed.

OpenMx: An Open Source Extended Structural Equation Modeling Framework

Science.gov (United States)

Boker, Steven; Neale, Michael; Maes, Hermine; Wilde, Michael; Spiegel, Michael; Brick, Timothy; Spies, Jeffrey; Estabrook, Ryne; Kenny, Sarah; Bates, Timothy; Mehta, Paras; Fox, John

2011-01-01

OpenMx is free, full-featured, open source, structural equation modeling (SEM) software. OpenMx runs within the "R" statistical programming environment on Windows, Mac OS-X, and Linux computers. The rationale for developing OpenMx is discussed along with the philosophy behind the user interface. The OpenMx data structures are…

Integral equation for Coulomb problem

International Nuclear Information System (INIS)

Sasakawa, T.

1986-01-01

For short range potentials an inhomogeneous (homogeneous) Lippmann-Schwinger integral equation of the Fredholm type yields the wave function of scattering (bound) state. For the Coulomb potential, this statement is no more valid. It has been felt difficult to express the Coulomb wave function in a form of an integral equation with the Coulomb potential as the perturbation. In the present paper, the author shows that an inhomogeneous integral equation of a Volterra type with the Coulomb potential as the perturbation can be constructed both for the scattering and the bound states. The equation yielding the binding energy is given in an integral form. The present treatment is easily extended to the coupled Coulomb problems

Extension of Gibbs-Duhem equation including influences of external fields

Science.gov (United States)

Guangze, Han; Jianjia, Meng

2018-03-01

Gibbs-Duhem equation is one of the fundamental equations in thermodynamics, which describes the relation among changes in temperature, pressure and chemical potential. Thermodynamic system can be affected by external field, and this effect should be revealed by thermodynamic equations. Based on energy postulate and the first law of thermodynamics, the differential equation of internal energy is extended to include the properties of external fields. Then, with homogeneous function theorem and a redefinition of Gibbs energy, a generalized Gibbs-Duhem equation with influences of external fields is derived. As a demonstration of the application of this generalized equation, the influences of temperature and external electric field on surface tension, surface adsorption controlled by external electric field, and the derivation of a generalized chemical potential expression are discussed, which show that the extended Gibbs-Duhem equation developed in this paper is capable to capture the influences of external fields on a thermodynamic system.

Properties of meromorphic solutions to certain differential-difference equations

Directory of Open Access Journals (Sweden)

Xiaoguang Qi

2013-06-01

Full Text Available We consider the properties of meromorphic solutions to certain type of non-linear difference equations. Also we show the existence of meromorphic solutions with finite order for differential-difference equations related to the Fermat type functional equation. This article extends earlier results by Liu et al [12].

Quantum Gross-Pitaevskii Equation

Directory of Open Access Journals (Sweden)

Jutho Haegeman, Damian Draxler, Vid Stojevic, J. Ignacio Cirac, Tobias J. Osborne, Frank Verstraete

2017-07-01

Full Text Available We introduce a non-commutative generalization of the Gross-Pitaevskii equation for one-dimensional quantum gasses and quantum liquids. This generalization is obtained by applying the time-dependent variational principle to the variational manifold of continuous matrix product states. This allows for a full quantum description of 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 (quasi one-dimensional systems. By linearizing around a stationary solution, we furthermore derive an associated generalization of the Bogoliubov -- de Gennes equations. This framework is applied to compute the steady state response amplitude to a periodic perturbation of the potential.

Travelling wave solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations

Directory of Open Access Journals (Sweden)

M. Arshad

Full Text Available In this manuscript, we constructed different form of new exact solutions of generalized coupled Zakharov–Kuznetsov and dispersive long wave equations by utilizing the modified extended direct algebraic method. New exact traveling wave solutions for both equations are obtained in the form of soliton, periodic, bright, and dark solitary wave solutions. There are many applications of the present traveling wave solutions in physics and furthermore, a wide class of coupled nonlinear evolution equations can be solved by this method. Keywords: Traveling wave solutions, Elliptic solutions, Generalized coupled Zakharov–Kuznetsov equation, Dispersive long wave equation, Modified extended direct algebraic method

On the exact solutions of high order wave equations of KdV type (I)

Science.gov (United States)

Bulut, Hasan; Pandir, Yusuf; Baskonus, Haci Mehmet

2014-12-01

In this paper, by means of a proper transformation and symbolic computation, we study high order wave equations of KdV type (I). We obtained classification of exact solutions that contain soliton, rational, trigonometric and elliptic function solutions by using the extended trial equation method. As a result, the motivation of this paper is to utilize the extended trial equation method to explore new solutions of high order wave equation of KdV type (I). This method is confirmed by applying it to this kind of selected nonlinear equations.

Analytical solution of groundwater waves in unconfined aquifers with ...

Indian Academy of Sciences (India)

Selva Balaji Munusamy

2017-07-29

Jul 29, 2017 ... higher-order Boussinesq equation. The homotopy perturbation solution is derived using a virtual perturbation .... reality, seepage face formation is common for tide–aquifer interaction problems. To simplify the complexity of the.

A Finite Difference Scheme for Double-Diffusive Unsteady Free Convection from a Curved Surface to a Saturated Porous Medium with a Non-Newtonian Fluid

KAUST Repository

El-Amin, Mohamed; Sun, Shuyu

2011-01-01

and the stability conditions are determined for each equation. Boundary layer and Boussinesq approximations have been incorporated. Numerical calculations are carried out for the various parameters entering into the problem. Velocity, temperature and concentration

Extending Teach and Repeat to Pivoting Wheelchairs

Directory of Open Access Journals (Sweden)

Guillermo Del Castillo

2003-02-01

Full Text Available The paper extends the teach-and-repeat paradigm that has been successful for the control of holonomic robots to nonholonomic wheelchairs which may undergo pivoting action over the course of their taught movement. Due to the nonholonomic nature of the vehicle kinematics, estimation is required -- in the example given herein, based upon video detection of wall-mounted cues -- both in the teaching and the tracking events. In order to accommodate motion that approaches pivoting action as well as motion that approaches straight-line action, the estimation equations of the Extended Kalman Filter and the control equations are formulated using two different definitions of a nontemporal independent variable. The paper motivates the need for pivoting action in real-life settings by reporting extensively on the abilities and limitations of estimation-based teach-and-repeat action where pivoting and near-pivoting action is disallowed. Following formulation of the equations in the near-pivot mode, the paper reports upon experiments where taught trajectories which entail a seamless mix of near-straight and near-pivot action are tracked.

Adiabatic decay of internal solitons due to Earth's rotation within the framework of the Gardner-Ostrovsky equation

Science.gov (United States)

Obregon, Maria; Raj, Nawin; Stepanyants, Yury

2018-03-01

The adiabatic decay of different types of internal wave solitons caused by the Earth's rotation is studied within the framework of the Gardner-Ostrovsky equation. The governing equation describing such processes includes quadratic and cubic nonlinear terms, as well as the Boussinesq and Coriolis dispersions: (ut + c ux + α u ux + α1 u2 ux + β uxxx)x = γ u. It is shown that at the early stage of evolution solitons gradually decay under the influence of weak Earth's rotation described by the parameter γ. The characteristic decay time is derived for different types of solitons for positive and negative coefficients of cubic nonlinearity α1 (both signs of that parameter may occur in the oceans). The coefficient of quadratic nonlinearity α determines only a polarity of solitary wave when α1 0. It is found that the adiabatic theory describes well the decay of solitons having bell-shaped profiles. In contrast to that, large amplitude table-top solitons, which can exist when α1 is negative, are structurally unstable. Under the influence of Earth's rotation, they transfer first to the bell-shaped solitons, which decay then adiabatically. Estimates of the characteristic decay time of internal solitons are presented for the real oceanographic conditions.

Improved Durand-equation for multiple application

International Nuclear Information System (INIS)

Weber, M.

1986-01-01

The applicability of Durand's equation could be improved for general use by applying suitable parameters representing the grain-size distribution. Thus, the Durand equation cannot only describe polydisperse (pseudo)-homogeneous or heterogeneous transportation, but also solid-fluid mixtures containing a certain amount of fine particles. Even non-Newtonian influences can be taken into account. The applicability of the extended Durand equation for polydisperse mixtures will be demonstrated by measurement data. With respect to this, the transition between pseudohomogeneous and heterogeneous transport has been considered on the basis of measured concentration profiles

Improving broiler lifestyle: a CFD approach

CSIR Research Space (South Africa)

Humbert, U

2014-01-01

Full Text Available . Boussinesq approximation: ( ) (1) Momentum Equation: ( ) ( ) (2) Temperature Equation: ( ) (3) OpenFOAM, an open...-source, multi-physics toolbox, will be used. OpenFOAM's library of solvers contains a myriad of different applications, supplying solutions to different problems. In this library we find solvers with the ability to model buoyancy flow with different types...

Growth of meromorphic solutions of higher-order linear differential equations

Directory of Open Access Journals (Sweden)

Wenjuan Chen

2009-01-01

Full Text Available In this paper, we investigate the higher-order linear differential equations with meromorphic coefficients. We improve and extend a result of M.S. Liu and C.L. Yuan, by using the estimates for the logarithmic derivative of a transcendental meromorphic function due to Gundersen, and the extended Winman-Valiron theory which proved by J. Wang and H.X. Yi. In addition, we also consider the nonhomogeneous linear differential equations.

Nambu brackets in fluid mechanics and magnetohydrodynamics

International Nuclear Information System (INIS)

Salazar, Roberto; Kurgansky, Michael V

2010-01-01

Concrete examples of the construction of Nambu brackets for equations of motion (both 3D and 2D) of Boussinesq stratified fluids and also for magnetohydrodynamical equations are given. It serves a generalization of Hamiltonian formulation for the considered equations of motion. Two alternative Nambu formulations are proposed, first by using fluid dynamical (kinetic) helicity and/or enstrophy as constitutive elements and second, by using the existing conservation laws of the governing equation.

Construction of a Roe linearization for the ideal MHD equations

International Nuclear Information System (INIS)

Cargo, P.; Gallice, G.; Raviart, P.A.

1996-01-01

In [3], Munz has constructed a Roe linearization for the equations of gas dynamics in Lagrangian coordinates. We extend this construction to the case of the ideal magnetohydrodynamics equations again in Lagrangian coordinates. As a consequence we obtain a Roe linearization for the MHD equations in Eulerian coordinates. (author)

On the existence of solutions for functional differential equations

International Nuclear Information System (INIS)

Walo Omana, R.

1994-12-01

The aim of the paper is to extend the Granas Topological Transversality Method used in boundary value problems for functional differential equations for first and second order, to the case of n-th order functional differential equations. 15 refs

Wavelet diagnostics for detection of coherent structures in ...

Indian Academy of Sciences (India)

The problem addressed is the one of inferring the type and intensity of any spatial ..... Navier–Stokes equations under the Boussinesq approximation for the ... helically organized. ... After prolonged heating, this leads to a complete breakdown.

A new formulation of equations of compressible fluids by analogy with Maxwell's equations

International Nuclear Information System (INIS)

Kambe, Tsutomu

2010-01-01

A compressible ideal fluid is governed by Euler's equation of motion and equations of continuity, entropy and vorticity. This system can be reformulated in a form analogous to that of electromagnetism governed by Maxwell's equations with source terms. The vorticity plays the role of magnetic field, while the velocity field plays the part of a vector potential and the enthalpy (of isentropic flows) plays the part of a scalar potential in electromagnetism. The evolution of source terms of fluid Maxwell equations is determined by solving the equations of motion and continuity. The equation of sound waves can be derived from this formulation, where time evolution of the sound source is determined by the equation of motion. The theory of vortex sound of aeroacoustics is included in this formulation. It is remarkable that the forces acting on a point mass moving in a velocity field of an inviscid fluid are analogous in their form to the electric force and Lorentz force in electromagnetism. The significance of the reformulation is interpreted by examples taken from fluid mechanics. This formulation can be extended to viscous fluids without difficulty. The Maxwell-type equations are unchanged by the viscosity effect, although the source terms have additional terms due to viscosities.

Lipschitz Regularity of Solutions for Mixed Integro-Differential Equations

OpenAIRE

Barles, Guy; Chasseigne, Emmanuel; Ciomaga, Adina; Imbert, Cyril

2011-01-01

We establish new Hoelder and Lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro-differential equations, by the classical Ishii-Lions's method. We thus extend the Hoelder regularity results recently obtained by Barles, Chasseigne and Imbert (2011). In addition, we deal with a new class of nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the loc...

A limit of the confluent Heun equation and the Schroedinger equation for an inverted potential and for an electric dipole

International Nuclear Information System (INIS)

El-Jaick, Lea Jaccoud; Figueiredo, Bartolomeu D.B.

2009-01-01

We reexamine and extend a group of solutions in series of Bessel functions for a limiting case of the confluent Heun equation and, then, apply such solutions to the one-dimensional Schroedinger equation with an inverted quasi-exactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole. For the first problem we find finite and infinite-series solutions which are convergent and bounded for any value of the independent variable. For the angular equation, we also find expansions in series of Jacobi polynomials. (author)

Sum rules in extended RPA theories

International Nuclear Information System (INIS)

Adachi, S.; Lipparini, E.

1988-01-01

Different moments m k of the excitation strength function are studied in the framework of the second RPA and of the extended RPA in which 2p2h correlations are explicitly introduced into the ground state by using first-order perturbation theory. Formal properties of the equations of motion concerning sum rules are derived and compared with those exhibited by the usual 1p1h RPA. The problem of the separation of the spurious solutions in extended RPA calculations is also discussed. (orig.)

Turbulent Convection in an Anelastic Rotating Sphere: A Model for the Circulation on the Giant Planets

National Research Council Canada - National Science Library

Kaspi, Yohai

2008-01-01

... (including the strong variations in gravity and the equation of state). Different from most previous 3D convection models, this model is anelastic rather than Boussinesq and thereby incorporates the full density variation of the planet...

Exp-function method for solving fractional partial differential equations.

Science.gov (United States)

Zheng, Bin

2013-01-01

We extend the Exp-function method to fractional partial differential equations in the sense of modified Riemann-Liouville derivative based on nonlinear fractional complex transformation. For illustrating the validity of this method, we apply it to the space-time fractional Fokas equation and the nonlinear fractional Sharma-Tasso-Olver (STO) equation. As a result, some new exact solutions for them are successfully established.

On geometric approach to Lie symmetries of differential-difference equations

International Nuclear Information System (INIS)

Li Hongjing; Wang Dengshan; Wang Shikun; Wu Ke; Zhao Weizhong

2008-01-01

Based upon Cartan's geometric formulation of differential equations, Harrison and Estabrook proposed a geometric approach for the symmetries of differential equations. In this Letter, we extend Harrison and Estabrook's approach to analyze the symmetries of differential-difference equations. The discrete exterior differential technique is applied in our approach. The Lie symmetry of (2+1)-dimensional Toda equation is investigated by means of our approach

Some New Lie Symmetry Groups of Differential-Difference Equations Obtained from a Simple Direct Method

International Nuclear Information System (INIS)

Zhi Hongyan

2009-01-01

In this paper, based on the symbolic computing system Maple, the direct method for Lie symmetry groups presented by Sen-Yue Lou [J. Phys. A: Math. Gen. 38 (2005) L129] is extended from the continuous differential equations to the differential-difference equations. With the extended method, we study the well-known differential-difference KP equation, KZ equation and (2+1)-dimensional ANNV system, and both the Lie point symmetry groups and the non-Lie symmetry groups are obtained.

New exact solutions for two nonlinear equations

International Nuclear Information System (INIS)

Wang Quandi; Tang Minying

2008-01-01

In this Letter, we investigate two nonlinear equations given by u t -u xxt +3u 2 u x =2u x u xx +uu xxx and u t -u xxt +4u 2 u x =3u x u xx +uu xxx . Through some special phase orbits we obtain four new exact solutions for each equation above. Some previous results are extended

Subsurface offset behaviour in velocity analysis with extended reflectivity images

NARCIS (Netherlands)

Mulder, W.A.

2013-01-01

Migration velocity analysis with the constant-density acoustic wave equation can be accomplished by the focusing of extended migration images, obtained by introducing a subsurface shift in the imaging condition. A reflector in a wrong velocity model will show up as a curve in the extended image. In

Generalized ordinary differential equations not absolutely continuous solutions

CERN Document Server

Kurzweil, Jaroslav

2012-01-01

This book provides a systematic treatment of the Volterra integral equation by means of a modern integration theory which extends considerably the field of differential equations. It contains many new concepts and results in the framework of a unifying theory. In particular, this new approach is suitable in situations where fast oscillations occur.

Pseudodifferential equations over non-Archimedean spaces

CERN Document Server

Zúñiga-Galindo, W A

2016-01-01

Focusing on p-adic and adelic analogues of pseudodifferential equations, this monograph presents a very general theory of parabolic-type equations and their Markov processes motivated by their connection with models of complex hierarchic systems. The Gelfand-Shilov method for constructing fundamental solutions using local zeta functions is developed in a p-adic setting and several particular equations are studied, such as the p-adic analogues of the Klein-Gordon equation. Pseudodifferential equations for complex-valued functions on non-Archimedean local fields are central to contemporary harmonic analysis and mathematical physics and their theory reveals a deep connection with probability and number theory. The results of this book extend and complement the material presented by Vladimirov, Volovich and Zelenov (1994) and Kochubei (2001), which emphasize spectral theory and evolution equations in a single variable, and Albeverio, Khrennikov and Shelkovich (2010), which deals mainly with the theory and applica...

Discrete q-derivatives and symmetries of q-difference equations

Energy Technology Data Exchange (ETDEWEB)

Levi, D [Dipartimento di Fisica, Universita Roma Tre and INFN-Sezione di Roma Tre, Via della Vasca Navale 84, 00146 Rome (Italy); Negro, J [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain); Olmo, M A del [Departamento de FIsica Teorica, Universidad de Valladolid, E-47011, Valladolid (Spain)

2004-03-12

In this paper we extend the umbral calculus, developed to deal with difference equations on uniform lattices, to q-difference equations. We show that many properties considered for shift invariant difference operators satisfying the umbral calculus can be implemented to the case of the q-difference operators. This q-umbral calculus can be used to provide solutions to linear q-difference equations and q-differential delay equations. To illustrate the method, we will apply the obtained results to the construction of symmetry solutions for the q-heat equation.

Multi-fidelity Gaussian process regression for prediction of random fields

International Nuclear Information System (INIS)

Parussini, L.; Venturi, D.; Perdikaris, P.; Karniadakis, G.E.

2017-01-01

We propose a new multi-fidelity Gaussian process regression (GPR) approach for prediction of random fields based on observations of surrogate models or hierarchies of surrogate models. Our method builds upon recent work on recursive Bayesian techniques, in particular recursive co-kriging, and extends it to vector-valued fields and various types of covariances, including separable and non-separable ones. The framework we propose is general and can be used to perform uncertainty propagation and quantification in model-based simulations, multi-fidelity data fusion, and surrogate-based optimization. We demonstrate the effectiveness of the proposed recursive GPR techniques through various examples. Specifically, we study the stochastic Burgers equation and the stochastic Oberbeck–Boussinesq equations describing natural convection within a square enclosure. In both cases we find that the standard deviation of the Gaussian predictors as well as the absolute errors relative to benchmark stochastic solutions are very small, suggesting that the proposed multi-fidelity GPR approaches can yield highly accurate results.

Multi-fidelity Gaussian process regression for prediction of random fields

Energy Technology Data Exchange (ETDEWEB)

Parussini, L. [Department of Engineering and Architecture, University of Trieste (Italy); Venturi, D., E-mail: venturi@ucsc.edu [Department of Applied Mathematics and Statistics, University of California Santa Cruz (United States); Perdikaris, P. [Department of Mechanical Engineering, Massachusetts Institute of Technology (United States); Karniadakis, G.E. [Division of Applied Mathematics, Brown University (United States)

2017-05-01

We propose a new multi-fidelity Gaussian process regression (GPR) approach for prediction of random fields based on observations of surrogate models or hierarchies of surrogate models. Our method builds upon recent work on recursive Bayesian techniques, in particular recursive co-kriging, and extends it to vector-valued fields and various types of covariances, including separable and non-separable ones. The framework we propose is general and can be used to perform uncertainty propagation and quantification in model-based simulations, multi-fidelity data fusion, and surrogate-based optimization. We demonstrate the effectiveness of the proposed recursive GPR techniques through various examples. Specifically, we study the stochastic Burgers equation and the stochastic Oberbeck–Boussinesq equations describing natural convection within a square enclosure. In both cases we find that the standard deviation of the Gaussian predictors as well as the absolute errors relative to benchmark stochastic solutions are very small, suggesting that the proposed multi-fidelity GPR approaches can yield highly accurate results.

Integral equation hierarchy for continuum percolation

International Nuclear Information System (INIS)

Given, J.A.

1988-01-01

In this thesis a projection operator technique is presented that yields hierarchies of integral equations satisfied exactly by the n-point connectedness functions in a continuum version of the site-bond percolation problem. The n-point connectedness functions carry the same structural information for a percolation problem as then-point correlation functions do for a thermal problem. This method extends the Potts model mapping of Fortuin and Kastelyn to the continuum by exploiting an s-state generalization of the Widom-Rowlinson model, a continuum model for phase separation. The projection operator technique is used to produce an integral equation hierarchy for percolation similar to the Born-Green heirarchy. The Kirkwood superposition approximation (SA) is extended to percolation in order to close this hierarchy and yield a nonlinear integral equation for the two-point connectedness function. The fact that this function, in the SA, is the analytic continuation to negative density of the two-point correlation function in a corresponding thermal problem is discussed. The BGY equation for percolation is solved numerically, both by an expansion in powers of the density, and by an iterative technique due to Kirkwood. It is argued both analytically and numerically, that the BYG equation for percolation, unlike its thermal counterpart, shows non-classical critical behavior, with η = 1 and γ = 0.05 ± .1. Finally a sequence of refinements to the superposition approximations based in the theory of fluids by Rice and Lekner is discussed

Single-peak solitary wave solutions for the variant Boussinesq ...

Indian Academy of Sciences (India)

ear dispersive waves in shallow water. This equation has attracted a lot of attention ... which is a model for water waves (a = 0), where u(x, t) is the velocity, H(x, t) is the total depth and the subscripts denote partial ... cusped solitary wave solutions of the osmosis K(2, 2) equation. Zhang and Chen [6] obtained new types of ...

Electron transfer dynamics: Zusman equation versus exact theory

International Nuclear Information System (INIS)

Shi Qiang; Chen Liping; Nan Guangjun; Xu Ruixue; Yan Yijing

2009-01-01

The Zusman equation has been widely used to study the effect of solvent dynamics on electron transfer reactions. However, application of this equation is limited by the classical treatment of the nuclear degrees of freedom. In this paper, we revisit the Zusman equation in the framework of the exact hierarchical equations of motion formalism, and show that a high temperature approximation of the hierarchical theory is equivalent to the Zusman equation in describing electron transfer dynamics. Thus the exact hierarchical formalism naturally extends the Zusman equation to include quantum nuclear dynamics at low temperatures. This new finding has also inspired us to rescale the original hierarchical equations and incorporate a filtering algorithm to efficiently propagate the hierarchical equations. Numerical exact results are also presented for the electron transfer reaction dynamics and rate constant calculations.

Subsurface offset behaviour in velocity analysis with extended reflectivity images

NARCIS (Netherlands)

Mulder, W.A.

2012-01-01

Migration velocity analysis with the wave equation can be accomplished by focusing of extended migration images, obtained by introducing a subsurface offset or shift. A reflector in the wrong velocity model will show up as a curve in the extended image. In the correct model, it should collapse to a

Probabilistic Representations of Solutions to the Heat Equation

Indian Academy of Sciences (India)

In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition , is given by the convolution of with the heat kernel (Gaussian density). Our results also extend the ...

Exact solutions for the cubic-quintic nonlinear Schroedinger equation

International Nuclear Information System (INIS)

Zhu Jiamin; Ma Zhengyi

2007-01-01

In this paper, the cubic-quintic nonlinear Schroedinger equation is solved through the extended elliptic sub-equation method. As a consequence, many types of exact travelling wave solutions are obtained which including bell and kink profile solitary wave solutions, triangular periodic wave solutions and singular solutions

A procedure to construct exact solutions of nonlinear fractional differential equations.

Science.gov (United States)

Güner, Özkan; Cevikel, Adem C

2014-01-01

We use the fractional transformation to convert the nonlinear partial fractional differential equations with the nonlinear ordinary differential equations. The Exp-function method is extended to solve fractional partial differential equations in the sense of the modified Riemann-Liouville derivative. We apply the Exp-function method to the time fractional Sharma-Tasso-Olver equation, the space fractional Burgers equation, and the time fractional fmKdV equation. As a result, we obtain some new exact solutions.

Regularity criteria for the 3D magneto-micropolar fluid equations via ...

Indian Academy of Sciences (India)

3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations. Keywords. Magneto-micropolar fluid equations; regularity criteria; direction of velocity. 2010 Mathematics Subject Classification. 35Q35, 76W05 ...

Smooth, cusped, and discontinuous traveling waves in the periodic fluid resonance equation

Science.gov (United States)

Kruse, Matthew Thomas

The principal motivation for this dissertation is to extend the study of small amplitude high frequency wave propagation in solutions for hyperbolic conservation laws begun by A. Majda and R. Rosales in 1984. It was then that Majda and Rosales obtained equations governing the leading order wave amplitudes of resonantly interacting weakly nonlinear high frequency wave trains in the compressible Euler equations. The equations were obtained through systematic application of multiple scales and result in a pair of nonlinear acoustic wave equations coupled through a convolution operator. The extended solutions satisfy a pair of inviscid Burgers' equations coupled via a spatial convolution operator. Since then, many mathematicians have used this technique to extend the time validity of solutions to systems of equations other than the Euler equations and have arrived at similar nonlinear non-local systems. This work attempts to look at some of the basic features of the linear and nonlinear coupled and decoupled non- local equations, offering some analytic solutions and numerical insight into the phenomena associated with these equations. We do so by examining a single non-local linear equation, and then a single equation coupling a Burgers' nonlinearity with a linear convolution operator. The linear case is completely solvable. Analytic solutions are provided along with numerical results showing the fundamental properties of the linear non- local equations. In the nonlinear case some analytic solutions, including steady state profiles and traveling wave solutions, are provided along with a battery of numerical simulations. Evidence indicates the existence of attractors for solutions of the single equation with a single mode kernel. Provided resonant interaction takes place, the profile of the attractor is uniquely dependent on the kernel alone. Hamiltonian equations are obtained for both the linear and nonlinear equations with the condition that the resonant kernel must

Radar equations for modern radar

CERN Document Server

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

Hojman's theorem of the third-order ordinary differential equation

International Nuclear Information System (INIS)

Hong-Sheng, Lü; Hong-Bin, Zhang; Shu-Long, Gu

2009-01-01

This paper extends Hojman's conservation law to the third-order differential equation. A new conserved quantity is constructed based on the Lie group of transformation generators of the equations of motion. The generators contain variations of the time and generalized coordinates. Two independent non-trivial conserved quantities of the third-order ordinary differential equation are obtained. A simple example is presented to illustrate the applications of the results. (general)

Solution of the stellar structure equations in Eulerian coordinates

International Nuclear Information System (INIS)

Deupree, R.G.

1976-01-01

The equations of hydrostatic and thermal equilibrium, assuming only radiative energy transport and spherical symmetry, are solved in Eulerian coordinates by a suitable modification of the Henyey method. An Eulerian approach may possibly be more suitably extended to more spatial dimensions than the usual Lagrangian procedure. The principle advantage of this method is that the equations of hydrostatic and thermal equilibrium and Poisson's equation may be solved simultaneously

Invariant quantum mechanical equations of motion

Energy Technology Data Exchange (ETDEWEB)

Wigner, E. P. [Princeton University, Princeton, NJ (United States)

1963-01-15

One of the last few years’ most important developments in theoretical physics is the recognition that it is useful to extend to complex numbers the definition domain of intrinsically real variables, such as energy or angular momentum. This leads one to review many subjects which were considered to be closed. It should not have surprised me, therefore, when Dr. Salam asked me to report, at this seminar, on equations for elementary particles which are not believed to exist in nature, such as particles with imaginary mass. Even though the equations which describe such particles will play no role in the theory as long as the variables such as energy or angular momentum have physically meaningful values, that is, as long as they are real, they may play a significant role when the definition domain of these variables is extended.

Large-Eddy Simulation of Flow and Pollutant Transport in Urban Street Canyons with Ground Heating

OpenAIRE

Li, Xian-Xiang; Britter, Rex E.; Koh, Tieh Yong; Norford, Leslie Keith; Liu, Chun-Ho; Entekhabi, Dara; Leung, Dennis Y. C.

2009-01-01

Our study employed large-eddy simulation (LES) based on a one-equation subgrid-scale model to investigate the flow field and pollutant dispersion characteristics inside urban street canyons. Unstable thermal stratification was produced by heating the ground of the street canyon. Using the Boussinesq approximation, thermal buoyancy forces were taken into account in both the Navier–Stokes equations and the transport equation for subgrid-scale turbulent kinetic energy (TKE). The LESs were valida...

The classification of the single travelling wave solutions to the ...

Indian Academy of Sciences (India)

The discrimination system for the polynomial method is applied to variant Boussinesq equations to classify single travelling wave solutions. In particular, we construct corresponding solutions to the concrete parameters to show that each solution in the classification can be realized.

Evolution equations for Killing fields

International Nuclear Information System (INIS)

Coll, B.

1977-01-01

The problem of finding necessary and sufficient conditions on the Cauchy data for Einstein equations which insure the existence of Killing fields in a neighborhood of an initial hypersurface has been considered recently by Berezdivin, Coll, and Moncrief. Nevertheless, it can be shown that the evolution equations obtained in all these cases are of nonstrictly hyperbolic type, and, thus, the Cauchy data must belong to a special class of functions. We prove here that, for the vacuum and Einstein--Maxwell space--times and in a coordinate independent way, one can always choose, as evolution equations for the Killing fields, a strictly hyperbolic system: The above theorems can be thus extended to all Cauchy data for which the Einstein evolution problem has been proved to be well set

Existence and Uniqueness of Solution of Schrodinger equation in extended Colombeau algebra

Directory of Open Access Journals (Sweden)

Fariba Fattahi

2014-09-01

Full Text Available In this paper, we establish the existence and uniquenessresult of the linear Schr¨odinger equation with Marchaudfractional derivative in Colombeau generalized algebra.The purpose of introducing Marchaud fractional derivativeis regularizing it in Colombeau sense.

Modified Darboux transformations with foreign auxiliary equations

International Nuclear Information System (INIS)

Schulze-Halberg, Axel

2011-01-01

We construct a new type of first-order Darboux transformations for the stationary Schroedinger equation. In contrast to the conventional case, our Darboux transformations support arbitrary (foreign) auxiliary equations. We show that among other applications, our formalism can be used to systematically construct Darboux transformations for Schroedinger equations with energy-dependent potentials, including a recent result (Lin et al., 2007) as a special case. -- Highlights: → We generalize the Darboux transformation for the Schroedinger equation. → By admitting arbitrary auxiliary functions, we provide a new tool for generating solutions. → As a special case we recover a recent result on energy-dependent potentials. → We extend the latter result to very general energy-dependence.

An extended Henon-Heiles system

International Nuclear Information System (INIS)

Hone, A.N.W.; Novikov, V.; Verhoeven, C.

2008-01-01

We consider an integrable system of partial differential equations possessing a Lax pair with an energy-dependent Lax operator of second order, of the type described by Antonowicz and Fordy. By taking the travelling wave reduction of this system we show that the integrable case (ii) Henon-Heiles system can be extended by adding an arbitrary number of non-polynomial (rational) terms to the potential

Modelling conjugation with stochastic differential equations

DEFF Research Database (Denmark)

Philipsen, Kirsten Riber; Christiansen, Lasse Engbo; Hasman, Henrik

2010-01-01

Enterococcus faecium strains in a rich exhaustible media. The model contains a new expression for a substrate dependent conjugation rate. A maximum likelihood based method is used to estimate the model parameters. Different models including different noise structure for the system and observations are compared......Conjugation is an important mechanism involved in the transfer of resistance between bacteria. In this article a stochastic differential equation based model consisting of a continuous time state equation and a discrete time measurement equation is introduced to model growth and conjugation of two...... using a likelihood-ratio test and Akaike's information criterion. Experiments indicating conjugation on the agar plates selecting for transconjugants motivates the introduction of an extended model, for which conjugation on the agar plate is described in the measurement equation. This model is compared...

A novel numerical flux for the 3D Euler equations with general equation of state

KAUST Repository

Toro, Eleuterio F.

2015-09-30

Here we extend the flux vector splitting approach recently proposed in (E F Toro and M E Vázquez-Cendón. Flux splitting schemes for the Euler equations. Computers and Fluids. Vol. 70, Pages 1-12, 2012). The scheme was originally presented for the 1D Euler equations for ideal gases and its extension presented in this paper is threefold: (i) we solve the three-dimensional Euler equations on general meshes; (ii) we use a general equation of state; and (iii) we achieve high order of accuracy in both space and time through application of the semi-discrete ADER methodology on general meshes. The resulting methods are systematically assessed for accuracy, robustness and efficiency on a carefully selected suite of test problems. Formal high accuracy is assessed through convergence rates studies for schemes of up to 4th order of accuracy in both space and time on unstructured meshes.

Extending the Constant Coefficient Solution Technique to Variable Coefficient Ordinary Differential Equations

Science.gov (United States)

Mohammed, Ahmed; Zeleke, Aklilu

2015-01-01

We introduce a class of second-order ordinary differential equations (ODEs) with variable coefficients whose closed-form solutions can be obtained by the same method used to solve ODEs with constant coefficients. General solutions for the homogeneous case are discussed.

Conformal gravity, the Einstein equations and spaces of complex null geodesics

Energy Technology Data Exchange (ETDEWEB)

Baston, R.J.; Mason, L.J.

1987-07-01

The aim of the paper is to give a twistorial characterisation of the field equations of conformal gravity and of Einstein spacetimes. Strong evidence is provided for a particularly concise characterisation of these equations in terms of 'formal neighbourhoods'of the space of complex null geodesics. Second-order perturbations of the metric of complexified Minkowski space are considered. These correspond to certain infinitesimal deformations of its space of complex null geodesics, PN. PN has a natural codimension one embedding into a larger space. It is shown that deformations extend automatically to the fourth-order embedding (that is, the fourth formal neighbourhood). They extend to the fifth formal neighbourhood if and only if the corresponding perturbation in the metric has vanishing Bach tensor. Finally, deformations which extend to the sixth formal neighbourhood correspond to perturbations in the metric that are conformally related to ones satisfying the Einstein equations. The authors present arguments which suggest that the results will also hold when spacetime is fully curved.

Conformal gravity, the Einstein equations and spaces of complex null geodesics

International Nuclear Information System (INIS)

Baston, R.J.; Mason, L.J.

1987-01-01

The aim of the paper is to give a twistorial characterisation of the field equations of conformal gravity and of Einstein spacetimes. Strong evidence is provided for a particularly concise characterisation of these equations in terms of 'formal neighbourhoods'of the space of complex null geodesics. Second-order perturbations of the metric of complexified Minkowski space are considered. These correspond to certain infinitesimal deformations of its space of complex null geodesics, PN. PN has a natural codimension one embedding into a larger space. It is shown that deformations extend automatically to the fourth-order embedding (that is, the fourth formal neighbourhood). They extend to the fifth formal neighbourhood if and only if the corresponding perturbation in the metric has vanishing Bach tensor. Finally, deformations which extend to the sixth formal neighbourhood correspond to perturbations in the metric that are conformally related to ones satisfying the Einstein equations. The authors present arguments which suggest that the results will also hold when spacetime is fully curved. (author)

Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation

International Nuclear Information System (INIS)

Hernandez-Heredero, Rafael; Reyes, Enrique G

2009-01-01

We announce two new structures associated with the Camassa-Holm (CH) equation: a Lie algebra of nonlocal symmetries, and a Darboux transformation for this important equation, which we construct using only our symmetries. We also extend our results to the associated Camassa-Holm equation introduced by J Schiff (1998 Physica D 121 24-43). (fast track communication)

Nonlocal symmetries and a Darboux transformation for the Camassa-Holm equation

Energy Technology Data Exchange (ETDEWEB)

Hernandez-Heredero, Rafael [Departamento de Matematica Aplicada, EUIT de Telecomunicacion, Universidad Politecnica de Madrid, Campus Sur Ctra de Valencia Km. 7. 28031, Madrid (Spain); Reyes, Enrique G [Departamento de Matematica y Ciencia de la Computacion, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago (Chile)], E-mail: rafahh@euitt.upm.es, E-mail: ereyes@fermat.usach.cl

2009-05-08

We announce two new structures associated with the Camassa-Holm (CH) equation: a Lie algebra of nonlocal symmetries, and a Darboux transformation for this important equation, which we construct using only our symmetries. We also extend our results to the associated Camassa-Holm equation introduced by J Schiff (1998 Physica D 121 24-43). (fast track communication)

A microscopic derivation of stochastic differential equations

International Nuclear Information System (INIS)

Arimitsu, Toshihico

1996-01-01

With the help of the formulation of Non-Equilibrium Thermo Field Dynamics, a unified canonical operator formalism is constructed for the quantum stochastic differential equations. In the course of its construction, it is found that there are at least two formulations, i.e. one is non-hermitian and the other is hermitian. Having settled which framework should be satisfied by the quantum stochastic differential equations, a microscopic derivation is performed for these stochastic differential equations by extending the projector methods. This investigation may open a new field for quantum systems in order to understand the deeper meaning of dissipation

Teachers and Technology: Development of an Extended Theory of Planned Behavior

Science.gov (United States)

Teo, Timothy; Zhou, Mingming; Noyes, Jan

2016-01-01

This study tests the validity of an extended theory of planned behaviour (TPB) to explain teachers' intention to use technology for teaching and learning. Five hundred and ninety two participants completed a survey questionnaire measuring their responses to eight constructs which form an extended TPB. Using structural equation modelling, the…

ON ENTIRE SOLUTIONS OF TWO TYPES OF SYSTEMS OF COMPLEX DIFFERENTIAL-DIFFERENCE EQUATIONS

Institute of Scientific and Technical Information of China (English)

Lingyun GAO

2017-01-01

In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential (difference) equations to the systems of differential-difference equations.

Speeds of Propagation in Classical and Relativistic Extended Thermodynamics

Directory of Open Access Journals (Sweden)

Müller Ingo

1999-01-01

Full Text Available The Navier-Stokes-Fourier theory of viscous, heat-conducting fluids provides parabolic equations and thus predicts infinite pulse speeds. Naturally this feature has disqualified the theory for relativistic thermodynamics which must insist on finite speeds and, moreover, on speeds smaller than $c$. The attempts at a remedy have proved heuristically important for a new systematic type of thermodynamics: Extended thermodynamics. That new theory has symmetric hyperbolic field equations and thus it provides finite pulse speeds. Extended thermodynamics is a whole hierarchy of theories with an increasing number of fields when gradients and rates of thermodynamic processes become steeper and faster. The first stage in this hierarchy is the 14-field theory which may already be a useful tool for the relativist in many applications. The 14 fields -- and further fields -- are conveniently chosen from the moments of the kinetic theory of gases. The hierarchy is complete only when the number of fields tends to infinity. In that case the pulse speed of non-relativistic extended thermodynamics tends to infinity while the pulse speed of relativistic extended thermodynamics tends to $c$, the speed of light. In extended thermodynamics symmetric hyperbolicity -- and finite speeds -- are implied by the concavity of the entropy density. This is still true in relativistic thermodynamics for a privileged entropy density which is the entropy density of the rest frame for non-degenerate gases.

A Priori Regularity of Parabolic Partial Differential Equations

KAUST Repository

Berkemeier, Francisco

2018-05-13

In this thesis, we consider parabolic partial differential equations such as the heat equation, the Fokker-Planck equation, and the porous media equation. Our aim is to develop methods that provide a priori estimates for solutions with singular initial data. These estimates are obtained by understanding the time decay of norms of solutions. First, we derive regularity results for the heat equation by estimating the decay of Lebesgue norms. Then, we apply similar methods to the Fokker-Planck equation with suitable assumptions on the advection and diffusion. Finally, we conclude by extending our techniques to the porous media equation. The sharpness of our results is confirmed by examining known solutions of these equations. The main contribution of this thesis is the use of functional inequalities to express decay of norms as differential inequalities. These are then combined with ODE methods to deduce estimates for the norms of solutions and their derivatives.

Systems of evolution equations and the singular perturbation method

International Nuclear Information System (INIS)

Mika, J.

Several fundamental theorems are presented important for the solution of linear evolution equations in the Banach space. The algorithm is deduced extending the solution of the system of singularly perturbed evolution equations into an asymptotic series with respect to a small positive parameter. The asymptotic convergence is shown of an approximate solution to the accurate solution. Singularly perturbed evolution equations of the resonance type were analysed. The special role is considered of the asymptotic equivalence of P1 equations obtained as the first order approximation if the spherical harmonics method is applied to the linear Boltzmann equation, and the diffusion equations of the linear transport theory where the small parameter approaches zero. (J.B.)

Semiclassical regularization of Vlasov equations and wavepackets for nonlinear Schrödinger equations

Science.gov (United States)

Athanassoulis, Agissilaos

2018-03-01

We consider the semiclassical limit of nonlinear Schrödinger equations with initial data that are well localized in both position and momentum (non-parametric wavepackets). We recover the Wigner measure (WM) of the problem, a macroscopic phase-space density which controls the propagation of the physical observables such as mass, energy and momentum. WMs have been used to create effective models for wave propagation in: random media, quantum molecular dynamics, mean field limits, and the propagation of electrons in graphene. In nonlinear settings, the Vlasov-type equations obtained for the WM are often ill-posed on the physically interesting spaces of initial data. In this paper we are able to select the measure-valued solution of the 1  +  1 dimensional Vlasov-Poisson equation which correctly captures the semiclassical limit, thus finally resolving the non-uniqueness in the seminal result of Zhang et al (2012 Comm. Pure Appl. Math. 55 582-632). The same approach is also applied to the Vlasov-Dirac-Benney equation with small wavepacket initial data, extending several known results.

Turbulent mixed buoyancy driven flow and heat transfer in lid driven enclosure

International Nuclear Information System (INIS)

Mishra, Ajay Kumar; Sharma, Anil Kumar

2015-01-01

Turbulent mixed buoyancy driven flow and heat transfer of air in lid driven rectangular enclosure has been investigated for Grashof number in the range of 10 8 to 10 11 and for Richardson number 0.1, 1 and 10. Steady two dimensional Reynolds-Averaged-Navier-Stokes equations and conservation equations of mass and energy, coupled with the Boussinesq approximation, are solved. The spatial derivatives in the equations are discretized using the finite-element method. The SIMPLE algorithm is used to resolve pressure-velocity coupling. Turbulence is modeled with the k-ω closure model with physical boundary conditions along with the Boussinesq approximation, for the flow and heat transfer. The predicted results are validated against benchmark solutions reported in literature. The results include stream lines and temperature fields are presented to understand flow and heat transfer characteristics. There is a marked reduction in mean Nusselt number (about 58%) as the Richardson number increases from 0.1 to 10 for the case of Ra=10 10 signifying the effect of reduction of top lid velocity resulting in reduction of turbulent mixing. (author)

Quantum master equation for QED in exact renormalization group

International Nuclear Information System (INIS)

Igarashi, Yuji; Itoh, Katsumi; Sonoda, Hidenori

2007-01-01

Recently, one of us (H. S.) gave an explicit form of the Ward-Takahashi identity for the Wilson action of QED. We first rederive the identity using a functional method. The identity makes it possible to realize the gauge symmetry even in the presence of a momentum cutoff. In the cutoff dependent realization, the nilpotency of the BRS transformation is lost. Using the Batalin-Vilkovisky formalism, we extend the Wilson action by including the antifield contributions. Then, the Ward-Takahashi identity for the Wilson action is lifted to a quantum master equation, and the modified BRS transformation regains nilpotency. We also obtain a flow equation for the extended Wilson action. (author)

Regularity criteria for the 3D magneto-micropolar fluid equations via ...

Indian Academy of Sciences (India)

We consider sufficient conditions to ensure the smoothness of solutions to 3D magneto-micropolar fluid equations. It involves only the direction of the velocity and the magnetic field. Our result extends to the cases of Navier–Stokes and MHD equations.

Extending Newton's law from nonlocal-in-time kinetic energy

International Nuclear Information System (INIS)

Suykens, J.A.K.

2009-01-01

We study a new equation of motion derived from a context of classical Newtonian mechanics by replacing the kinetic energy with a form of nonlocal-in-time kinetic energy. It leads to a hypothetical extension of Newton's second law of motion. In a first stage the obtainable solution form is studied by considering an unknown value for the nonlocality time extent. This is done in relation to higher-order Euler-Lagrange equations and a Hamiltonian framework. In a second stage the free particle case and harmonic oscillator case are studied and compared with quantum mechanical results. For a free particle it is shown that the solution form is a superposition of the classical straight line motion and a Fourier series. We discuss the link with quanta interpretations made in Pais-Uhlenbeck oscillators. The discrete nature emerges from the continuous time setting through application of the least action principle. The harmonic oscillator case leads to energy levels that approximately correspond to the quantum harmonic oscillator levels. The solution to the extended Newton equation also admits a quantization of the nonlocality time extent, which is determined by the classical oscillator frequency. The extended equation suggests a new possible way for understanding the relationship between classical and quantum mechanics

Integrodifferential equation approach. Pt. 1

International Nuclear Information System (INIS)

Oehm, W.; Sofianos, S.A.; Fiedeldey, H.; South Africa Univ., Pretoria. Dept. of Physics); Fabre de la Ripelle, M.; South Africa Univ., Pretoria. Dept. of Physics)

1990-02-01

A single integrodifferential equation in two variables, valid for A nucleons interacting by pure Wigner forces, which has previously only been solved in the extreme and uncoupled adiabatic approximations is now solved exactly for three- and four-nucleon systems. The results are in good agreement with the values obtained for the binding energies by means of an empirical interpolation formula. This validates all our previous conclusions, in particular that the omission of higher (than two) order correlations in our four-body equation only produces a rather small underbinding. The integrodifferential equation approach (IDEA) is here also extended to spin-dependent forces of the Malfliet-Tjon type, resulting in two coupled integrodifferential equations in two variables. The exact solution and the interpolated adiabatic approximation are again in good agreement. The inclusion of the hypercentral part of the two-body interaction in the definition of the Faddeev-type components again leads to substantial improvement for fully local potentials, acting in all partial waves. (orig.)

A fractional Dirac equation and its solution

International Nuclear Information System (INIS)

Muslih, Sami I; Agrawal, Om P; Baleanu, Dumitru

2010-01-01

This paper presents a fractional Dirac equation and its solution. The fractional Dirac equation may be obtained using a fractional variational principle and a fractional Klein-Gordon equation; both methods are considered here. We extend the variational formulations for fractional discrete systems to fractional field systems defined in terms of Caputo derivatives. By applying the variational principle to a fractional action S, we obtain the fractional Euler-Lagrange equations of motion. We present a Lagrangian and a Hamiltonian for the fractional Dirac equation of order α. We also use a fractional Klein-Gordon equation to obtain the fractional Dirac equation which is the same as that obtained using the fractional variational principle. Eigensolutions of this equation are presented which follow the same approach as that for the solution of the standard Dirac equation. We also provide expressions for the path integral quantization for the fractional Dirac field which, in the limit α → 1, approaches to the path integral for the regular Dirac field. It is hoped that the fractional Dirac equation and the path integral quantization of the fractional field will allow further development of fractional relativistic quantum mechanics.

Mathematical Modelling of the Evaporating Liquid Films on the Basis of the Generalized Interface Conditions

Directory of Open Access Journals (Sweden)

Goncharova Olga

2016-01-01

Full Text Available The two-dimensional films, flowing down an inclined, non-uniformly heated substrate are studied. The results contain the new mathematical models developed with the help of the long-wave approximation of the Navier-Stokes and heat transfer equations or Oberbeck-Boussinesq equations in the case, when the generalized conditions are formulated at thermocapillary interface. The evolution equations for the film thickness include the effects of gravity, viscosity, capillarity, thermocapillarity, additional stress effects and evaporation.

Calculation of propellant gas pressure by simple extended corresponding state principle

Directory of Open Access Journals (Sweden)

Bin Xu

2016-04-01

Full Text Available The virial equation can well describe gas state at high temperature and pressure, but the difficulties in virial coefficient calculation limit the use of virial equation. Simple extended corresponding state principle (SE-CSP is introduced in virial equation. Based on a corresponding state equation, including three characteristic parameters, an extended parameter is introduced to describe the second virial coefficient expressions of main products of propellant gas. The modified SE-CSP second virial coefficient expression was extrapolated based on the virial coefficients experimental temperature, and the second virial coefficients obtained are in good agreement with the experimental data at a low temperature and the theoretical values at high temperature. The maximum pressure in the closed bomb test was calculated with modified SE-CSP virial coefficient expressions with the calculated error of less than 2%, and the error was smaller than the result calculated with the reported values under the same calculation conditions. The modified SE-CSP virial coefficient expression provides a convenient and efficient method for practical virial coefficient calculation without resorting to complicated molecular model design and integral calculation.

Application of a Lorentz transformation in six dimensions to an extension of dirac equation

International Nuclear Information System (INIS)

Sabry, A.A.

2000-01-01

On applying a six dimensional Lorentz transformation (by adding three time components to the usual 3 space components) a covariant extended Dirac equation in this six dimensional space is suggested. From the free field solution of the extended equation we obtained four solutions for E , the first component of energy which depends on very big masses M and M 0 . The observed masses of the leptons and their corresponding neutrinos satisfying the same free field equation, can then be obtained on using second quantisation procedures on the field operators

Modelling performance of a small array of Wave Energy Converters: Comparison of Spectral and Boussinesq models

International Nuclear Information System (INIS)

Greenwood, Charles; Christie, David; Venugopal, Vengatesan; Morrison, James; Vogler, Arne

2016-01-01

This paper presents results from numerical simulations of three Oscillating Wave Surge Converters (OWSC) using two different computational models, Boussinesq wave (BW) and Spectral wave (SW) of the commercial software suite MIKE. The simulation of a shallow water wave farm applies alternative methods for implementing a frequency dependent absorption in both the BW and SW models, where energy extraction is based on experimental data from a scaled Oyster device. The effects of including wave diffraction within the SW model is tested by using diffraction smoothing steps and various directional wave conditions. The results of this study reveal important information on the models realms of validity that is heavily dependent on the incident sea state and the removal of diffraction for the SW model. This yields an increase in simulation accuracy for far-field disturbances when diffraction is entirely removed. This highlights specific conditions where the BW and SW model may thrive but also regions where reduced performance is observed. The results presented in this paper have not been validated with real sea site wave device array performance, however, the methodology described would be useful to device developers to arrive at preliminary decisions on array configurations and to minimise negative environmental impacts.

An extended integrable fractional-order KP soliton hierarchy

International Nuclear Information System (INIS)

Li Li

2011-01-01

In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.

An extended integrable fractional-order KP soliton hierarchy

Energy Technology Data Exchange (ETDEWEB)

Li Li, E-mail: li07099@163.co [College of Maths and Systematic Science, Shenyang Normal University, Shenyang 110034 (China)

2011-01-17

In this Letter, we consider the modified derivatives and integrals of fractional-order pseudo-differential operators. A sequence of Lax KP equations hierarchy and extended fractional KP (fKP) hierarchy are introduced, and the fKP hierarchy has Lax presentations with the extended Lax operators. In the case of the extension with the half-order pseudo-differential operators, a new integrable fKP hierarchy is obtained. A few particular examples of fractional order will be listed, together with their Lax pairs.

Solutions of hyperbolic equations with the CIP-BS method

International Nuclear Information System (INIS)

Utsumi, Takayuki; Koga, James; Yamagiwa, Mitsuru; Yabe, Takashi; Aoki, Takayuki

2004-01-01

In this paper, we show that a new numerical method, the Constrained Interpolation Profile - Basis Set (CIP-BS) method, can solve general hyperbolic equations efficiently. This method uses a simple polynomial basis set that is easily extendable to any desired higher-order accuracy. The interpolating profile is chosen so that the subgrid scale solution approaches the local real solution owing to the constraints from the spatial derivatives of the master equations. Then, introducing scalar products, the linear and nonlinear partial differential equations are uniquely reduced to the ordinary differential equations for values and spatial derivatives at the grid points. The method gives stable, less diffusive, and accurate results. It is successfully applied to the continuity equation, the Burgers equation, the Korteweg-de Vries equation, and one-dimensional shock tube problems. (author)

On the invariance properties of the Klein–Gordon equation with ...

Indian Academy of Sciences (India)

Abstract. Here we attempt to find the nature of the external electromagnetic field such that the KG equation with external electromagnetic field is invariant. Lie's extended group method is applied to obtain the class of external electromagnetic field which admits the invariance of the KG equation. Though, the field potential ...

On the solvability of initial-value problems for nonlinear implicit difference equations

Directory of Open Access Journals (Sweden)

Ha Thi Ngoc Yen

2004-07-01

Full Text Available Our aim is twofold. First, we propose a natural definition of index for linear nonautonomous implicit difference equations, which is similar to that of linear differential-algebraic equations. Then we extend this index notion to a class of nonlinear implicit difference equations and prove some existence theorems for their initial-value problems.

Generalized variational formulations for extended exponentially fractional integral

Directory of Open Access Journals (Sweden)

Zuo-Jun Wang

2016-01-01

Full Text Available Recently, the fractional variational principles as well as their applications yield a special attention. For a fractional variational problem based on different types of fractional integral and derivatives operators, corresponding fractional Lagrangian and Hamiltonian formulation and relevant Euler–Lagrange type equations are already presented by scholars. The formulations of fractional variational principles still can be developed more. We make an attempt to generalize the formulations for fractional variational principles. As a result we obtain generalized and complementary fractional variational formulations for extended exponentially fractional integral for example and corresponding Euler–Lagrange equations. Two illustrative examples are presented. It is observed that the formulations are in exact agreement with the Euler–Lagrange equations.

Unraveling gravity beyond Einstein with extended test bodies

International Nuclear Information System (INIS)

Puetzfeld, Dirk; Obukhov, Yuri N.

2013-01-01

The motion of test bodies in gravity is tightly linked to the conservation laws. This well-known fact in the context of General Relativity is also valid for gravitational theories which go beyond Einstein's theory. Here we derive the equations of motion for test bodies for a very large class of gravitational theories with a general nonminimal coupling to matter. These equations form the basis for future systematic tests of alternative gravity theories. Our treatment is covariant and generalizes the classic Mathisson–Papapetrou–Dixon result for spinning (extended) test bodies. The equations of motion for structureless test bodies turn out to be surprisingly simple, despite the very general nature of the theories considered.

On energy conservation in extended magnetohydrodynamics

International Nuclear Information System (INIS)

Kimura, Keiji; Morrison, P. J.

2014-01-01

A systematic study of energy conservation for extended magnetohydrodynamic models that include Hall terms and electron inertia is performed. It is observed that commonly used models do not conserve energy in the ideal limit, i.e., when viscosity and resistivity are neglected. In particular, a term in the momentum equation that is often neglected is seen to be needed for conservation of energy

The relationship among the solutions of two auxiliary ordinary differential equations

International Nuclear Information System (INIS)

Liu Xiaoping; Liu Chunping

2009-01-01

In a recent article [Phys. Lett. A 356 (2006) 124], Sirendaoreji extended their auxiliary equation method by introducing a new auxiliary ordinary differential equation (NAODE) and its 14 solutions. Then the author studied some nonlinear evolution equations (NLEEs) and got more exact travelling wave solutions. In this paper, we will show that the 14 solutions of the NAODE are actually the same as the solutions obtained by original auxiliary equation method, and they are only different in the form.

A NEW OSCILLATION CRITERION FOR FIRST ORDER NEUTRAL DELAY DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2006-01-01

In this paper, a new sufficient condition for the oscillation of all solutions of first order neutral delay differential equations is obtained. Secondly, the result can also be extended to a general neutral differential equation, and many known results in the literatures are improved.

Thermodynamic admissibility of the extended Pom-Pom model for branched polymers

NARCIS (Netherlands)

Soulages, J.; Hütter, M.; Öttinger, H.C.

2006-01-01

The thermodynamic consistency of the eXtended Pom-Pom (XPP) model for branched polymers of Verbeeten et al. [W.M.H. Verbeeten, G.W.M. Peters, F.P.T. Baaijens, Differential constitutive equations for polymer melts: the extended pom-pom model, J. Rheol. 45 (4) (2001) 823–843; W.M.H. Verbeeten, G.W.M.

Hamiltonian structure of linearly extended Virasoro algebra

International Nuclear Information System (INIS)

Arakelyan, T.A.; Savvidi, G.K.

1991-01-01

The Hamiltonian structure of linearly extended Virasoro algebra which admits free bosonic field representation is described. An example of a non-trivial extension is found. The hierarchy of integrable non-linear equations corresponding to this Hamiltonian structure is constructed. This hierarchy admits the Lax representation by matrix Lax operator of second order

Interacting fields of arbitrary spin and N > 4 supersymmetric self-dual Yang-Mills equations

International Nuclear Information System (INIS)

Devchand, Ch.; Ogievetsky, V.

1996-06-01

We show that the self-dual Yang-Mills equations afford supersymmetrization to systems of equations invariant under global N-extended super-Poincare transformations for arbitrary values of N, without the limitation (N ≤ 4) applicable to standard non-self-dual Yang-Mills theories. These systems of equations provide novel classically consistent interactions for vector supermultiplets containing fields of spin up to N-2/2. The equations of motion of the component fields of spin greater than 1/2 are interacting variants of the first-order Dirac-Fierz equations for zero rest-mass fields of arbitrary spin. The interactions are governed by conserved currents which are constructed by an iterative procedure. In (arbitrarily extended) chiral superspace, the equations of motion for the (arbitrarily large) self-dual supermultiplet are shown to be completely equivalent to the set of algebraic supercurvature defining the self-dual superconnection. (author). 25 refs

Equations for formally real meadows

NARCIS (Netherlands)

Bergstra, J.A.; Bethke, I.; Ponse, A.

2015-01-01

We consider the signatures Σm = (0,1,−,+,⋅,−1)  of meadows and (Σm,s)  of signed meadows. We give two complete axiomatizations of the equational theories of the real numbers with respect to these signatures. In the first case, we extend the axiomatization of zero-totalized fields by a single axiom

New solitary wave solutions to the modified Kawahara equation

International Nuclear Information System (INIS)

Wazwaz, Abdul-Majid

2007-01-01

In this work we use the sine-cosine method, the tanh method, the extended tanh method, and ansatze of hyperbolic functions for analytic treatment for the modified Kawahara equation. New solitons solutions and periodic solutions are formally derived. The change of the parameters, that will drastically change the characteristics of the equation, is examined. The employed approaches are reliable and manageable

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

Directory of Open Access Journals (Sweden)

Aly R. Seadawy

2018-03-01

Full Text Available This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM in exactly solving a well-known nonlinear equation of partial differential equations (PDEs. In this respect, the longitudinal wave equation (LWE that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method. Keywords: Extended trial equation method, Longitudinal wave equation in a MEE circular rod, Dark solitons, Bright solitons, Solitary wave, Periodic solitary wave

Wave-equation dispersion inversion

KAUST Repository

Li, Jing

2016-12-08

We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.

New Formulation of the Governing Equations for Analyzing Outrigger Structures

International Nuclear Information System (INIS)

Er, G.-K.

2010-01-01

In this paper, an easily comprehensible solution procedure is proposed for the analysis of outrigger-braced structures. The idea is based on the compatibility of the columns' axial deformation. The unknowns are selected to be the axial forces in the columns. The resulted governing equations and the equations for the optimum analysis of the outrigger locations are different from the conventional ones, but numerical analysis shows that the results obtained with the new equations are same as those obtained with conventional equations. The relations between the new equations and the conventional ones are also figured out. The new procedure of formulating the governing equations can be easily extended to more complicated cases of outrigger-braced structures.

Building Context with Tumor Growth Modeling Projects in Differential Equations

Science.gov (United States)

Beier, Julie C.; Gevertz, Jana L.; Howard, Keith E.

2015-01-01

The use of modeling projects serves to integrate, reinforce, and extend student knowledge. Here we present two projects related to tumor growth appropriate for a first course in differential equations. They illustrate the use of problem-based learning to reinforce and extend course content via a writing or research experience. Here we discuss…

A quantum extended Kalman filter

International Nuclear Information System (INIS)

Emzir, Muhammad F; Woolley, Matthew J; Petersen, Ian R

2017-01-01

In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements. (paper)

A quantum extended Kalman filter

Science.gov (United States)

Emzir, Muhammad F.; Woolley, Matthew J.; Petersen, Ian R.

2017-06-01

In quantum physics, a stochastic master equation (SME) estimates the state (density operator) of a quantum system in the Schrödinger picture based on a record of measurements made on the system. In the Heisenberg picture, the SME is a quantum filter. For a linear quantum system subject to linear measurements and Gaussian noise, the dynamics may be described by quantum stochastic differential equations (QSDEs), also known as quantum Langevin equations, and the quantum filter reduces to a so-called quantum Kalman filter. In this article, we introduce a quantum extended Kalman filter (quantum EKF), which applies a commutative approximation and a time-varying linearization to systems of nonlinear QSDEs. We will show that there are conditions under which a filter similar to a classical EKF can be implemented for quantum systems. The boundedness of estimation errors and the filtering problem with ‘state-dependent’ covariances for process and measurement noises are also discussed. We demonstrate the effectiveness of the quantum EKF by applying it to systems that involve multiple modes, nonlinear Hamiltonians, and simultaneous jump-diffusive measurements.

The Camassa-Holm equation as an incompressible Euler equation: A geometric point of view

Science.gov (United States)

Gallouët, Thomas; Vialard, François-Xavier

2018-04-01

The group of diffeomorphisms of a compact manifold endowed with the L2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein-Fisher-Rao distance is a natural extension of the classical L2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa-Holm (CH) equation in one dimension. Geometrically, we present an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S1, solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on R2 which preserves a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler-Arnold equation for the Hdiv right-invariant metric are length minimizing geodesics for sufficiently short times.

Poincaré-MacMillan Equations of Motion for a Nonlinear Nonholonomic Dynamical System

Science.gov (United States)

Amjad, Hussain; Syed Tauseef, Mohyud-Din; Ahmet, Yildirim

2012-03-01

MacMillan's equations are extended to Poincaré's formalism, and MacMillan's equations for nonlinear nonholonomic systems are obtained in terms of Poincaré parameters. The equivalence of the results obtained here with other forms of equations of motion is demonstrated. An illustrative example of the theory is provided as well.

EXISTENCE AND UNIQUENESS OF SOLUTIONS TO A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION

Institute of Scientific and Technical Information of China (English)

无

2011-01-01

The initial value problem of a nonlinear fractional differential equation is discussed in this paper. Using the nonlinear alternative of Leray-Schauder type and the contraction mapping principle,we obtain the existence and uniqueness of solutions to the fractional differential equation,which extend some results of the previous papers.

Traveling solitary wave solutions to evolution equations with nonlinear terms of any order

International Nuclear Information System (INIS)

Feng Zhaosheng

2003-01-01

Many physical phenomena in one- or higher-dimensional space can be described by nonlinear evolution equations, which can be reduced to ordinary differential equations such as the Lienard equation. Thus, to study those ordinary differential equations is of significance not only in mathematics itself, but also in physics. In this paper, a kind of explicit exact solutions to the Lienard equation is obtained. The applications of the solutions to the nonlinear RR-equation and the compound KdV-type equation are presented, which extend the results obtained in the previous literature

Early-Time Solution of the Horizontal Unconfined Aquifer in the Buildup Phase

Science.gov (United States)

Gravanis, Elias; Akylas, Evangelos

2017-10-01

We derive the early-time solution of the Boussinesq equation for the horizontal unconfined aquifer in the buildup phase under constant recharge and zero inflow. The solution is expressed as a power series of a suitable similarity variable, which is constructed so that to satisfy the boundary conditions at both ends of the aquifer, that is, it is a polynomial approximation of the exact solution. The series turns out to be asymptotic and it is regularized by resummation techniques that are used to define divergent series. The outflow rate in this regime is linear in time, and the (dimensionless) coefficient is calculated to eight significant figures. The local error of the series is quantified by its deviation from satisfying the self-similar Boussinesq equation at every point. The local error turns out to be everywhere positive, hence, so is the integrated error, which in turn quantifies the degree of convergence of the series to the exact solution.

Multiphase averaging of periodic soliton equations

International Nuclear Information System (INIS)

Forest, M.G.

1979-01-01

The multiphase averaging of periodic soliton equations is considered. Particular attention is given to the periodic sine-Gordon and Korteweg-deVries (KdV) equations. The periodic sine-Gordon equation and its associated inverse spectral theory are analyzed, including a discussion of the spectral representations of exact, N-phase sine-Gordon solutions. The emphasis is on physical characteristics of the periodic waves, with a motivation from the well-known whole-line solitons. A canonical Hamiltonian approach for the modulational theory of N-phase waves is prescribed. A concrete illustration of this averaging method is provided with the periodic sine-Gordon equation; explicit averaging results are given only for the N = 1 case, laying a foundation for a more thorough treatment of the general N-phase problem. For the KdV equation, very general results are given for multiphase averaging of the N-phase waves. The single-phase results of Whitham are extended to general N phases, and more importantly, an invariant representation in terms of Abelian differentials on a Riemann surface is provided. Several consequences of this invariant representation are deduced, including strong evidence for the Hamiltonian structure of N-phase modulational equations

Numerical Investigation of the Liquid Film Flows with Evaporation at Thermocapillary Interface

Directory of Open Access Journals (Sweden)

Rezanova Ekaterina

2016-01-01

Full Text Available Flows of the thin liquid layers on an inclined non-uniformly heated substrate are investigated numerically. The evaporation at the thermocapillary interface is taking into account. The Oberbeck-Boussinesq equations and the generalized kinematic, dynamic and energy conditions on a thermocapillary boundary are used for governing equations. The evolution equation, which determines the position of the interface, is obtained on the basis of the long-wave approximation of the equations for moderate Reynolds numbers. The numerical algorithm for solving of this evolution equation is presented. Comparison of the numerical results of flows of various liquids is presented.

Solving Fuzzy Fractional Differential Equations Using Zadeh's Extension Principle

Science.gov (United States)

Ahmad, M. Z.; Hasan, M. K.; Abbasbandy, S.

2013-01-01

We study a fuzzy fractional differential equation (FFDE) and present its solution using Zadeh's extension principle. The proposed study extends the case of fuzzy differential equations of integer order. We also propose a numerical method to approximate the solution of FFDEs. To solve nonlinear problems, the proposed numerical method is then incorporated into an unconstrained optimisation technique. Several numerical examples are provided. PMID:24082853

Novel loop-like solitons for the generalized Vakhnenko equation

International Nuclear Information System (INIS)

Zhang Min; Ma Yu-Lan; Li Bang-Qing

2013-01-01

A non-traveling wave solution of a generalized Vakhnenko equation arising from the high-frequent wave motion in a relaxing medium is derived via the extended Riccati mapping method. The solution includes an arbitrary function of an independent variable. Based on the solution, two hyperbolic functions are chosen to construct new solitons. Novel single-loop-like and double-loop-like solitons are found for the equation

Extended Deterministic Mean-Field Games

KAUST Repository

Gomes, Diogo A.

2016-04-21

In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.

Extended Deterministic Mean-Field Games

KAUST Repository

Gomes, Diogo A.; Voskanyan, Vardan K.

2016-01-01

In this paper, we consider mean-field games where the interaction of each player with the mean field takes into account not only the states of the players but also their collective behavior. To do so, we develop a random variable framework that is particularly convenient for these problems. We prove an existence result for extended mean-field games and establish uniqueness conditions. In the last section, we consider the Master Equation and discuss properties of its solutions.

Extended use of a computation method for electrical conductivity-case of non-Saha-equilibrium plasmas

International Nuclear Information System (INIS)

Chen, R.L.W.

1981-01-01

The use of an equation obtained earlier for computing electrical conductivity may be extended to partially ionized gases which depart from the Saha equation, provided the electron velocity distributions do not deviate from the Maxwellian distribution. (author)

Extended hamiltonian formalism and Lorentz-violating lagrangians

Directory of Open Access Journals (Sweden)

Don Colladay

2017-09-01

Full Text Available A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler–Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.

Extended hamiltonian formalism and Lorentz-violating lagrangians

Science.gov (United States)

Colladay, Don

2017-09-01

A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler-Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.

Determination of source terms in a degenerate parabolic equation

International Nuclear Information System (INIS)

Cannarsa, P; Tort, J; Yamamoto, M

2010-01-01

In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation

Gaussian solitary waves for the logarithmic-KdV and the logarithmic-KP equations

International Nuclear Information System (INIS)

Wazwaz, Abdul-Majid

2014-01-01

We investigate the logarithmic-KdV equation for more Gaussian solitary waves. We extend this work to derive the logarithmic-KP (Kadomtsev–Petviashvili) equation. We show that both logarithmic models are characterized by their Gaussian solitons. (paper)

Extended resolvent and inverse scattering with an application to KPI

International Nuclear Information System (INIS)

Boiti, M.; Pempinelli, F.; Pogrebkov, A.K.; Prinari, B.

2003-01-01

We present in detail an extended resolvent approach for investigating linear problems associated to 2+1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schroedinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given

Approximate Solution of LR Fuzzy Sylvester Matrix Equations

Directory of Open Access Journals (Sweden)

Xiaobin Guo

2013-01-01

Full Text Available The fuzzy Sylvester matrix equation AX~+X~B=C~ in which A,B are m×m and n×n crisp matrices, respectively, and C~ is an m×n LR fuzzy numbers matrix is investigated. Based on the Kronecker product of matrices, we convert the fuzzy Sylvester matrix equation into an LR fuzzy linear system. Then we extend the fuzzy linear system into two systems of linear equations according to the arithmetic operations of LR fuzzy numbers. The fuzzy approximate solution of the original fuzzy matrix equation is obtained by solving the crisp linear systems. The existence condition of the LR fuzzy solution is also discussed. Some examples are given to illustrate the proposed method.

Generalized differential transform method to differential-difference equation

International Nuclear Information System (INIS)

Zou Li; Wang Zhen; Zong Zhi

2009-01-01

In this Letter, we generalize the differential transform method to solve differential-difference equation for the first time. Two simple but typical examples are applied to illustrate the validity and the great potential of the generalized differential transform method in solving differential-difference equation. A Pade technique is also introduced and combined with GDTM in aim of extending the convergence area of presented series solutions. Comparisons are made between the results of the proposed method and exact solutions. Then we apply the differential transform method to the discrete KdV equation and the discrete mKdV equation, and successfully obtain solitary wave solutions. The results reveal that the proposed method is very effective and simple. We should point out that generalized differential transform method is also easy to be applied to other nonlinear differential-difference equation.

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

Science.gov (United States)

Diethelm, Kai; Ford, Neville J.; Freed, Alan D.; Gray, Hugh R. (Technical Monitor)

2002-01-01

We discuss an Adams-type predictor-corrector method for the numerical solution of fractional differential equations. The method may be used both for linear and for nonlinear problems, and it may be extended to multi-term equations (involving more than one differential operator) too.

The periodic wave solutions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations

International Nuclear Information System (INIS)

Sheng Zhang

2006-01-01

More periodic wave solutions expressed by Jacobi elliptic functions for the (2 + 1)-dimensional Konopelchenko-Dubrovsky equations are obtained by using the extended F-expansion method. In the limit cases, the solitary wave solutions and trigonometric function solutions for the equations are also obtained

Periodicity and positivity of a class of fractional differential equations.

Science.gov (United States)

Ibrahim, Rabha W; Ahmad, M Z; Mohammed, M Jasim

2016-01-01

Fractional differential equations have been discussed in this study. We utilize the Riemann-Liouville fractional calculus to implement it within the generalization of the well known class of differential equations. The Rayleigh differential equation has been generalized of fractional second order. The existence of periodic and positive outcome is established in a new method. The solution is described in a fractional periodic Sobolev space. Positivity of outcomes is considered under certain requirements. We develop and extend some recent works. An example is constructed.

Hamiltonian dynamics of extended objects

Science.gov (United States)

Capovilla, R.; Guven, J.; Rojas, E.

2004-12-01

We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler Lagrange equations.

Hamiltonian dynamics of extended objects

International Nuclear Information System (INIS)

Capovilla, R; Guven, J; Rojas, E

2004-01-01

We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler-Lagrange equations

Interference-exact radiative transfer equation

DEFF Research Database (Denmark)

Partanen, Mikko; Haÿrynen, Teppo; Oksanen, Jani

2017-01-01

Maxwell's equations with stochastic or quantum optical source terms accounting for the quantum nature of light. We show that both the nonlocal wave and local particle features associated with interference and emission of propagating fields in stratified geometries can be fully captured by local damping...... and scattering coefficients derived from the recently introduced quantized fluctuational electrodynamics (QFED) framework. In addition to describing the nonlocal optical interference processes as local directionally resolved effects, this allows reformulating the well known and widely used radiative transfer...... equation (RTE) as a physically transparent interference-exact model that extends the useful range of computationally efficient and quantum optically accurate interference-aware optical models from simple structures to full optical devices....

Extended exploding reflector concept for computing prestack traveltimes for waves of different type in the DSR framework

KAUST Repository

Duchkov, Anton A.

2013-09-22

The double-square-root (DSR) equation can be viewed as a Hamilton-Jacobi equation describing kinematics of downward data continuation in depth. It describes simultaneous propagation of source and receiver rays which allows computing reflection wave prestack traveltimes (for multiple sources) in a one run thus speeding up solution of the forward problem. Here we give and overview of different alternative forms of the DSR equation which allows stepping in two-way time and subsurface offset instead of depth. Different forms of the DSR equation are suitable for computing different types of waves including reflected, head and diving waves. We develop a WENO-RK numerical scheme for solving all mentioned forms of the DSR equation. Finally the extended exploding reflector concept can be used for computing prestack traveltimes while initiating the numerical solver as if a reflector was exploding in extended imaging space.

Differential geometry techniques for sets of nonlinear partial differential equations

Science.gov (United States)

Estabrook, Frank B.

1990-01-01

An attempt is made to show that the Cartan theory of partial differential equations can be a useful technique for applied mathematics. Techniques for finding consistent subfamilies of solutions that are generically rich and well-posed and for introducing potentials or other usefully consistent auxiliary fields are introduced. An extended sample calculation involving the Korteweg-de Vries equation is given.

An extended rational thermodynamics model for surface excess fluxes

NARCIS (Netherlands)

Sagis, L.M.C.

2012-01-01

In this paper, we derive constitutive equations for the surface excess fluxes in multiphase systems, in the context of an extended rational thermodynamics formalism. This formalism allows us to derive Maxwell–Cattaneo type constitutive laws for the surface extra stress tensor, the surface thermal

Extended resolvent and inverse scattering with an application to KPI

Science.gov (United States)

Boiti, M.; Pempinelli, F.; Pogrebkov, A. K.; Prinari, B.

2003-08-01

We present in detail an extended resolvent approach for investigating linear problems associated to 2+1 dimensional integrable equations. Our presentation is based as an example on the nonstationary Schrödinger equation with potential being a perturbation of the one-soliton potential by means of a decaying two-dimensional function. Modification of the inverse scattering theory as well as properties of the Jost solutions and spectral data as follows from the resolvent approach are given.

Extension of the Gladstone-Dale equation for flame flow field diagnosis by optical computerized tomography

International Nuclear Information System (INIS)

Chen Yunyun; Li Zhenhua; Song Yang; He Anzhi

2009-01-01

An extended model of the original Gladstone-Dale (G-D) equation is proposed for optical computerized tomography (OCT) diagnosis of flame flow fields. For the purpose of verifying the newly established model, propane combustion is used as a practical example for experiment, and moire deflection tomography is introduced with the probe wavelength 808 nm. The results indicate that the temperature based on the extended model is more accurate than that based on the original G-D equation. In a word, the extended model can be suitable for all kinds of flame flow fields whatever the components, temperature, and ionization are.

Continuous symmetric reductions of the Adler-Bobenko-Suris equations

International Nuclear Information System (INIS)

Tsoubelis, D; Xenitidis, P

2009-01-01

Continuously symmetric solutions of the Adler-Bobenko-Suris class of discrete integrable equations are presented. Initially defined by their invariance under the action of both of the extended three-point generalized symmetries admitted by the corresponding equations, these solutions are shown to be determined by an integrable system of partial differential equations. The connection of this system to the Nijhoff-Hone-Joshi 'generating partial differential equations' is established and an auto-Baecklund transformation and a Lax pair for it are constructed. Applied to the H1 and Q1 δ=0 members of the Adler-Bobenko-Suris family, the method of continuously symmetric reductions yields explicit solutions determined by the Painleve trancendents

Study on state equation for hydrogen storage measurement by volumetric method

International Nuclear Information System (INIS)

Dai Wei; Xu Jiajing; Wang Chaoyang; Tang Yongjian

2014-01-01

Volumetric measurement technique is one of the most popular methods for determining the amount of hydrogen storage. A new state equation was established which extended the limitations from the ideal gas state equation, the van der Waals equation and the Gou equation. The new state equation was then employed to describe the p-V-T character of hydrogen and investigate the adsorption quantity of hydrogen storage in resorcin-formaldehyde aerogel under different temperatures and pressures. The new equation was used to describe the density of hydrogen under different temperatures and pressures. The results are in good agreement with the experimental data. The differences arising from various underlying physics were carefully analyzed. (authors)

Baicklund transformation and multiple soliton solutions for the (3+1)-dimensional Jimbo-Miwa equation

Institute of Scientific and Technical Information of China (English)

张解放; 吴锋民

2002-01-01

We study an approach to constructing multiple soliton solutions of the (3+1)-dimensional nonlinear evolution equation. We take the (3+1)-dimensional Jimbo-Miwa (JM) equation as an example. Using the extended homogeneous balance method, one can find a Backlund transformation to decompose the (3+1)-dimensional JM equation into a linear partial differential equation and two bilinear partial differential equations. Starting from these linear and bilinear partial differential equations, some multiple soliton solutions for the (3+1)-dimensional JM equation are obtained by introducing a class of formal solutions.

Differential equations and integrable models: the SU(3) case

International Nuclear Information System (INIS)

Dorey, Patrick; Tateo, Roberto

2000-01-01

We exhibit a relationship between the massless a 2 (2) integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schroedinger equation. This forms part of a more general correspondence involving A 2 -related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the non-linear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators phi 12 , phi 21 and phi 15 . This is checked against previous results obtained using the thermodynamic Bethe ansatz

An extended step characteristic method for solving the transport equation in general geometries

International Nuclear Information System (INIS)

DeHart, M.D.; Pevey, R.E.; Parish, T.A.

1994-01-01

A method for applying the discrete ordinates method to solve the Boltzmann transport equation on arbitrary two-dimensional meshes has been developed. The finite difference approach normally used to approximate spatial derivatives in extrapolating angular fluxes across a cell is replaced by direct solution of the characteristic form of the transport equation for each discrete direction. Thus, computational cells are not restricted to the geometrical shape of a mesh element characteristic of a given coordinate system. However, in terms of the treatment of energy and angular dependencies, this method resembles traditional discrete ordinates techniques. By using the method developed here, a general two-dimensional space can be approximated by an irregular mesh comprised of arbitrary polygons. Results for a number of test problems have been compared with solutions obtained from traditional methods, with good agreement. Comparisons include benchmarks against analytical results for problems with simple geometry, as well as numerical results obtained from traditional discrete ordinates methods by applying the ANISN and TWOTRAN-II computer programs

Novel algorithm of large-scale simultaneous linear equations

International Nuclear Information System (INIS)

Fujiwara, T; Hoshi, T; Yamamoto, S; Sogabe, T; Zhang, S-L

2010-01-01

We review our recently developed methods of solving large-scale simultaneous linear equations and applications to electronic structure calculations both in one-electron theory and many-electron theory. This is the shifted COCG (conjugate orthogonal conjugate gradient) method based on the Krylov subspace, and the most important issue for applications is the shift equation and the seed switching method, which greatly reduce the computational cost. The applications to nano-scale Si crystals and the double orbital extended Hubbard model are presented.

Third order differential equations with delay

Directory of Open Access Journals (Sweden)

Petr Liška

2015-05-01

Full Text Available In this paper, we study the oscillation and asymptotic properties of solutions of certain nonlinear third order differential equations with delay. In particular, we extend results of I. Mojsej (Nonlinear Analysis 68, 2008 and we improve conditions on the property B of N. Parhi and S. Padhi (Indian J. Pure Appl. Math., 33, 2002.

Symbolic dynamics of the Lorenz equations

International Nuclear Information System (INIS)

Fang Hai-ping; Hao Bailin.

1994-07-01

The Lorenz equations are investigated in a wide range of parameters by using the method of symbolic dynamics. First, the systematics of stable periodic orbits in the Lorenz equations is compared with that of the one-dimensional cubic map, which shares the same discrete symmetry with the Lorenz model. The systematics is then ''corrected'' in such a way as to encompass all the known periodic windows of the Lorenz equations with only one exception. Second, in order to justify the above approach and to understand the exceptions, another 1D map with a discontinuity is extracted from an extension of the geometric Lorenz attractor and its symbolic dynamics is constructed. All this has to be done in light of symbolic dynamics of two-dimensional maps. Finally, symbolic dynamics for the actual Poincare return map of the Lorenz equations is constructed in a heuristic way. New periodic windows of the Lorenz equations and their parameters can be predicted from this symbolic dynamics in combination with the 1D cubic map. The extended geometric 2D Lorenz map and the 1D antisymmetric map with a discontinuity describe the topological aspects of the Lorenz equations to high accuracy. (author). 44 refs, 17 figs, 8 tabs

Oscillating particle-like solutions of nonlinear Klein-Gordon equation

International Nuclear Information System (INIS)

Bogolubsky, I.L.

1976-01-01

A denumerable set of oscillating spherically-symmetric particle-like solutions of the Klein-Gordon equation with cubic nonlinearity is found. Extended particles modelled by them turn out to be slightly radiating and long-lived

Transport equation solving methods

International Nuclear Information System (INIS)

Granjean, P.M.

1984-06-01

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 [fr

A Unified Introduction to Ordinary Differential Equations

Science.gov (United States)

Lutzer, Carl V.

2006-01-01

This article describes how a presentation from the point of view of differential operators can be used to (partially) unify the myriad techniques in an introductory course in ordinary differential equations by providing students with a powerful, flexible paradigm that extends into (or from) linear algebra. (Contains 1 footnote.)

Singularities of Type-Q ABS Equations

Directory of Open Access Journals (Sweden)

James Atkinson

2011-07-01

Full Text Available The type-Q equations lie on the top level of the hierarchy introduced by Adler, Bobenko and Suris (ABS in their classification of discrete counterparts of KdV-type integrable partial differential equations. We ask what singularities are possible in the solutions of these equations, and examine the relationship between the singularities and the principal integrability feature of multidimensional consistency. These questions are considered in the global setting and therefore extend previous considerations of singularities which have been local. What emerges are some simple geometric criteria that determine the allowed singularities, and the interesting discovery that generically the presence of singularities leads to a breakdown in the global consistency of such systems despite their local consistency property. This failure to be globally consistent is quantified by introducing a natural notion of monodromy for isolated singularities.

Optimal control of stochastic difference Volterra equations an introduction

CERN Document Server

Shaikhet, Leonid

2015-01-01

This book showcases a subclass of hereditary systems, that is, systems with behaviour depending not only on their current state but also on their past history; it is an introduction to the mathematical theory of optimal control for stochastic difference Volterra equations of neutral type. As such, it will be of much interest to researchers interested in modelling processes in physics, mechanics, automatic regulation, economics and finance, biology, sociology and medicine for all of which such equations are very popular tools. The text deals with problems of optimal control such as meeting given performance criteria, and stabilization, extending them to neutral stochastic difference Volterra equations. In particular, it contrasts the difference analogues of solutions to optimal control and optimal estimation problems for stochastic integral Volterra equations with optimal solutions for corresponding problems in stochastic difference Volterra equations. Optimal Control of Stochastic Difference Volterra Equation...

Elliptic Euler–Poisson–Darboux equation, critical points and integrable systems

International Nuclear Information System (INIS)

Konopelchenko, B G; Ortenzi, G

2013-01-01

The structure and properties of families of critical points for classes of functions W(z, z-bar ) obeying the elliptic Euler–Poisson–Darboux equation E(1/2, 1/2) are studied. General variational and differential equations governing the dependence of critical points in variational (deformation) parameters are found. Explicit examples of the corresponding integrable quasi-linear differential systems and hierarchies are presented. There are the extended dispersionless Toda/nonlinear Schrödinger hierarchies, the ‘inverse’ hierarchy and equations associated with the real-analytic Eisenstein series E(β, β-bar ;1/2) among them. The specific bi-Hamiltonian structure of these equations is also discussed. (paper)

A difference-equation formalism for the nodal domains of separable billiards

Energy Technology Data Exchange (ETDEWEB)

Manjunath, Naren; Samajdar, Rhine [Indian Institute of Science, Bangalore 560012 (India); Jain, Sudhir R., E-mail: srjain@barc.gov.in [Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085 (India)

2016-09-15

Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in Samajdar and Jain (2014). The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

An extended model based on the modified Nernst-Planck equation for describing transdermal iontophoresis of weak electrolytes.

Science.gov (United States)

Imanidis, Georgios; Luetolf, Peter

2006-07-01

An extended model for iontophoretic enhancement of transdermal drug permeation under constant voltage is described based on the previously modified Nernst-Planck equation, which included the effect of convective solvent flow. This model resulted in an analytical expression for the enhancement factor as a function of applied voltage, convective flow velocity due to electroosmosis, ratio of lipid to aqueous pathway passive permeability, and weighted average net ionic valence of the permeant in the aqueous epidermis domain. The shift of pH in the epidermis compared to bulk caused by the electrical double layer at the lipid-aqueous domain interface was evaluated using the Poisson-Boltzmann equation. This was solved numerically for representative surface charge densities and yielded pH differences between bulk and epidermal aqueous domain between 0.05 and 0.4 pH units. The developed model was used to analyze the experimental enhancement of an amphoteric weak electrolyte measured in vitro using human cadaver epidermis and a voltage of 250 mV at different pH values. Parameter values characterizing the involved factors were determined that yielded the experimental enhancement factors and passive permeability coefficients at all pH values. The model provided a very good agreement between experimental and calculated enhancement and passive permeability. The deduced parameters showed (i) that the pH shift in the aqueous permeation pathway had a notable effect on the ionic valence and the partitioning of the drug in this domain for a high surface charge density and depending on the pK(a) and pI of the drug in relation to the bulk pH; (ii) the magnitude and the direction of convective transport due to electroosmosis typically reflected the density and sign, respectively, of surface charge of the tissue and its effect on enhancement was substantial for bulk pH values differing from the pI of epidermal tissue; (iii) the aqueous pathway predominantly determined passive

Extended Lagrangian formalism for rheonomic systems with variable mass

Directory of Open Access Journals (Sweden)

Mušicki Đorđe

2017-01-01

Full Text Available In this paper the extended Lagrangian formalism for the rheonomic systems (Dj. Mušicki, 2004, which began with the modification of the mechanics of such systems (V. Vujičić, 1987, is extended to the systems with variable mass, with emphasis on the corresponding energy relations. This extended Lagrangian formalism is based on the extension of the set of chosen generalized coordinates by new quantities, suggested by the form of nonstationary constraints, which determine the position of the frame of reference in respect to which these generalized coordinates refer. As a consequence, an extended system of the Lagrangian equations is formulated, accommodated to the variability of the masses of particles, where the additional ones correspond to the additional generalized coordinates. By means of these equations, the energy relations of such systems have been studied, where it is demonstrated that here there are four types of energy conservation laws. The obtained energy laws are more complete and natural than the corresponding ones in the usual Lagrangian formulation for such systems. It is demonstrated that the obtained energy laws, are in full accordance with the energy laws in the corresponding vector formulation, if they are expressed in terms of the quantities introduced in this formulation of mechanics. The obtained results are illustrated by an example: the motion of a rocket, which ejects the gasses backwards, while this rocket moves up a straight line on an oblique plane, which glides uniformly in a horizontal direction.

Asymptotics for Estimating Equations in Hidden Markov Models

DEFF Research Database (Denmark)

Hansen, Jørgen Vinsløv; Jensen, Jens Ledet

Results on asymptotic normality for the maximum likelihood estimate in hidden Markov models are extended in two directions. The stationarity assumption is relaxed, which allows for a covariate process influencing the hidden Markov process. Furthermore a class of estimating equations is considered...

Hamiltonian dynamics of extended objects

Energy Technology Data Exchange (ETDEWEB)

Capovilla, R [Departamento de FIsica, Centro de Investigacion y de Estudios Avanzados del IPN, Apdo Postal 14-740, 07000 Mexico, DF (Mexico); Guven, J [School of Theoretical Physics, Dublin Institute for Advanced Studies, 10 Burlington Road, Dublin 4 (Ireland); Rojas, E [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, Apdo Postal 70-543, 04510 Mexico, DF (Mexico)

2004-12-07

We consider relativistic extended objects described by a reparametrization-invariant local action that depends on the extrinsic curvature of the worldvolume swept out by the object as it evolves. We provide a Hamiltonian formulation of the dynamics of such higher derivative models which is motivated by the ADM formulation of general relativity. The canonical momenta are identified by looking at boundary behaviour under small deformations of the action; the relationship between the momentum conjugate to the embedding functions and the conserved momentum density is established. The canonical Hamiltonian is constructed explicitly; the constraints on the phase space, both primary and secondary, are identified and the role they play in the theory is described. The multipliers implementing the primary constraints are identified in terms of the ADM lapse and shift variables and Hamilton's equations are shown to be consistent with the Euler-Lagrange equations.

The extended (G/G)-expansion method and travelling wave ...

Indian Academy of Sciences (India)

In this paper, we construct the travelling wave solutions to the perturbed nonlinear Schrödinger's equation (NLSE) with Kerr law non-linearity by the extended (′/)-expansion method. Based on this method, we obtain abundant exact travelling wave solutions of NLSE with Kerr law nonlinearity with arbitrary parameters.

Extending generalized Kubelka-Munk to three-dimensional radiative transfer.

Science.gov (United States)

Sandoval, Christopher; Kim, Arnold D

2015-08-10

The generalized Kubelka-Munk (gKM) approximation is a linear transformation of the double spherical harmonics of order one (DP1) approximation of the radiative transfer equation. Here, we extend the gKM approximation to study problems in three-dimensional radiative transfer. In particular, we derive the gKM approximation for the problem of collimated beam propagation and scattering in a plane-parallel slab composed of a uniform absorbing and scattering medium. The result is an 8×8 system of partial differential equations that is much easier to solve than the radiative transfer equation. We compare the solutions of the gKM approximation with Monte Carlo simulations of the radiative transfer equation to identify the range of validity for this approximation. We find that the gKM approximation is accurate for isotropic scattering media that are sufficiently thick and much less accurate for anisotropic, forward-peaked scattering media.

Hamiltonian formalism of two-dimensional Vlasov kinetic equation.

Science.gov (United States)

Pavlov, Maxim V

2014-12-08

In this paper, the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo-Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for the one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D. J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case, a Hamiltonian hydrodynamic lattice for the Russo-Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.

Equations for transient flow-boiling in a duct

International Nuclear Information System (INIS)

Mathers, W.G.; Ferch, R.L.; Hancox, W.T.; McDonald, B.H.

1981-01-01

In this paper we derive a separated phase model for weakly coupled flows which extends a model presented elsewhere (BANERJEE, FERCH, MATHERS and McDONALD, 1978). A hyperbolic system of seven partial differential equations results with ensemble-averaged phase velocities, enthalpies and pressures, and void fraction as dependent variables (UVUTUP model). The required constitutive equations for mass, momentum and energy transfer between phases and between the phases and the boundaries are discussed. The relationship of the UVUTUP model to other existing models is also presented

A high order solver for the unbounded Poisson equation

DEFF Research Database (Denmark)

Hejlesen, Mads Mølholm; Rasmussen, Johannes Tophøj; Chatelain, Philippe

2013-01-01

. The method is extended to directly solve the derivatives of the solution to Poissonʼs equation. In this way differential operators such as the divergence or curl of the solution field can be solved to the same high order convergence without additional computational effort. The method, is applied......A high order converging Poisson solver is presented, based on the Greenʼs function solution to Poissonʼs equation subject to free-space boundary conditions. The high order convergence is achieved by formulating regularised integration kernels, analogous to a smoothing of the solution field...... and validated, however not restricted, to the equations of fluid mechanics, and can be used in many applications to solve Poissonʼs equation on a rectangular unbounded domain....

Rational extended thermodynamics of a rarefied polyatomic gas with molecular relaxation processes

Science.gov (United States)

Arima, Takashi; Ruggeri, Tommaso; Sugiyama, Masaru

2017-10-01

We present a more refined version of rational extended thermodynamics of rarefied polyatomic gases in which molecular rotational and vibrational relaxation processes are treated individually. In this case, we need a triple hierarchy of the moment system and the system of balance equations is closed via the maximum entropy principle. Three different types of the production terms in the system, which are suggested by a generalized BGK-type collision term in the Boltzmann equation, are adopted. In particular, the rational extended thermodynamic theory with seven independent fields (ET7) is analyzed in detail. Finally, the dispersion relation of ultrasonic wave derived from the ET7 theory is confirmed by the experimental data for CO2, Cl2, and Br2 gases.

Jacobian elliptic function expansion solutions of nonlinear stochastic equations

International Nuclear Information System (INIS)

Wei Caimin; Xia Zunquan; Tian Naishuo

2005-01-01

Jacobian elliptic function expansion method is extended and applied to construct the exact solutions of the nonlinear Wick-type stochastic partial differential equations (SPDEs) and some new exact solutions are obtained via this method and Hermite transformation

Quaternion wave equations in curved space-time

Science.gov (United States)

Edmonds, J. D., Jr.

1974-01-01

The quaternion formulation of relativistic quantum theory is extended to include curvilinear coordinates and curved space-time in order to provide a framework for a unified quantum/gravity theory. Six basic quaternion fields are identified in curved space-time, the four-vector basis quaternions are identified, and the necessary covariant derivatives are obtained. Invariant field equations are derived, and a general invertable coordinate transformation is developed. The results yield a way of writing quaternion wave equations in curvilinear coordinates and curved space-time as well as a natural framework for solving the problem of second quantization for gravity.

Extended Range of a Gun Launched Smart Projectile Using Controllable Canards

OpenAIRE

Mark Costello

2001-01-01

This effort investigates the extent to which moveable canards can extend the range of indirect fire munitions using both projectile body and canard lift. Implications on terminal velocity and time of flight using this mechanism to extend range are examined for various canard configurations. Performance predictions are conducted using a six-degree-of-freedom simulation model that has previously been validated against range data. The projectile dynamic equations are formed in the body frame and...

A New Pseudoinverse Matrix Method For Balancing Chemical Equations And Their Stability

International Nuclear Information System (INIS)

Risteski, Ice B.

2008-01-01

In this work is given a new pseudoniverse matrix method for balancing chemical equations. Here offered method is founded on virtue of the solution of a Diophantine matrix equation by using of a Moore-Penrose pseudoinverse matrix. The method has been tested on several typical chemical equations and found to be very successful for the all equations in our extensive balancing research. This method, which works successfully without any limitations, also has the capability to determine the feasibility of a new chemical reaction, and if it is feasible, then it will balance the equation. Chemical equations treated here possess atoms with fractional oxidation numbers. Also, in the present work are introduced necessary and sufficient criteria for stability of chemical equations over stability of their extended matrices

The extended (G/G)-expansion method and travelling wave ...

Indian Academy of Sciences (India)

Home; Journals; Pramana – Journal of Physics; Volume 82; Issue 6. The extended (′/)-expansion method and travelling wave solutions for the perturbed nonlinear Schrödinger's equation with Kerr law nonlinearity. Zaiyun Zhang Jianhua Huang Juan Zhong Sha-Sha Dou Jiao Liu Dan Peng Ting Gao. Research Articles ...

An infinite-dimensional model of free convection

Energy Technology Data Exchange (ETDEWEB)

Iudovich, V.I. (Rostovskii Gosudarstvennyi Universitet, Rostov-on-Don (USSR))

1990-12-01

An infinite-dimensional model is derived from the equations of free convection in the Boussinesq-Oberbeck approximation. The velocity field is approximated by a single mode, while the heat-conduction equation is conserved fully. It is shown that, for all supercritical Rayleigh numbers, there exist exactly two secondary convective regimes. The case of ideal convection with zero viscosity and thermal conductivity is examined. The averaging method is used to study convection regimes at high Reynolds numbers. 10 refs.

Magnetohydrodynamic Kelvin-Helmholtz instability; Magnetohydrodynamische Kelvin-Helmholtz-Instabilitaet

Energy Technology Data Exchange (ETDEWEB)

Brett, Walter

2014-07-21

In the presented work the Kelvin-Helmholtz-Instability in magnetohydrodynamic flows is analyzed with the methods of Multiple Scales. The concerned fluids are incompressible or have a varying density perpendicular to the vortex sheet, which is taken into account using a Boussinesq-Approximation and constant Brunt-Vaeisaelae-Frequencies. The Multiple Scale Analysis leads to nonlinear evolution equations for the amplitude of the perturbations. Special solutions to these equations are presented and the effects of the magnetic fields are discussed.

Physical Controls on Copepod Aggregations in the Gulf of Maine

Science.gov (United States)

2013-06-01

Boussinesq, and within each layer the velocity is vertically uniform. The model solves the vorticity- Bernoulli form of the momentum equations: ∂u ∂t + (f...waves. The river source remains on for the entire model run. As the fluid enters the domain, it not only has mass, but some momentum as well (analogous...the momentum equations just behind the nose. We can use these profiles to try and understand the dynamics of this overturning circulation. It is well

BHR equations re-derived with immiscible particle effects

Energy Technology Data Exchange (ETDEWEB)

Schwarzkopf, John Dennis [Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Horwitz, Jeremy A. [Stanford Univ., CA (United States)

2015-05-01

Compressible and variable density turbulent flows with dispersed phase effects are found in many applications ranging from combustion to cloud formation. These types of flows are among the most challenging to simulate. While the exact equations governing a system of particles and fluid are known, computational resources limit the scale and detail that can be simulated in this type of problem. Therefore, a common method is to simulate averaged versions of the flow equations, which still capture salient physics and is relatively less computationally expensive. Besnard developed such a model for variable density miscible turbulence, where ensemble-averaging was applied to the flow equations to yield a set of filtered equations. Besnard further derived transport equations for the Reynolds stresses, the turbulent mass flux, and the density-specific volume covariance, to help close the filtered momentum and continuity equations. We re-derive the exact BHR closure equations which include integral terms owing to immiscible effects. Physical interpretations of the additional terms are proposed along with simple models. The goal of this work is to extend the BHR model to allow for the simulation of turbulent flows where an immiscible dispersed phase is non-trivially coupled with the carrier phase.

Separation of massive field equation of arbitrary spin in Robertson-Walker space-time

International Nuclear Information System (INIS)

Zecca, A.

2006-01-01

The massive spin-(3/2) field equation is explicitly integrated in the Robertson-Walker space-time by the Newman Penrose formalism. The solution is obtained by extending a separation procedure previously used to solve the spin-1 equation. The separated time dependence results in two coupled equations depending on the cosmological background evolution. The separated angular equations are explicitly integrated and the eigenvalues determined. The separated radial equations are integrated in the flat space-time case. The separation method of solution is then generalized, by induction, to prove the main result, that is the separability of the massive field equations of arbitrary spin in the Robertson-Walker space-time

Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions

Directory of Open Access Journals (Sweden)

Jose Luiz Boldrini

2003-11-01

Full Text Available We study the existence and regularity of weak solutions of a phase field type model for pure material solidification in presence of natural convection. We assume that the non-stationary solidification process occurs in a two dimensional bounded domain. The governing equations of the model are the phase field equation coupled with a nonlinear heat equation and a modified Navier-Stokes equation. These equations include buoyancy forces modelled by Boussinesq approximation and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations only hold in the non-solid regions, which are not known a priori, we have a free boundary-value problem.

Generalized latent variable modeling multilevel, longitudinal, and structural equation models

CERN Document Server

Skrondal, Anders; Rabe-Hesketh, Sophia

2004-01-01

This book unifies and extends latent variable models, including multilevel or generalized linear mixed models, longitudinal or panel models, item response or factor models, latent class or finite mixture models, and structural equation models.

Minimal parameter solution of the orthogonal matrix differential equation

Science.gov (United States)

Bar-Itzhack, Itzhack Y.; Markley, F. Landis

1990-01-01

As demonstrated in this work, all orthogonal matrices solve a first order differential equation. The straightforward solution of this equation requires n sup 2 integrations to obtain the element of the nth order matrix. There are, however, only n(n-1)/2 independent parameters which determine an orthogonal matrix. The questions of choosing them, finding their differential equation and expressing the orthogonal matrix in terms of these parameters are considered. Several possibilities which are based on attitude determination in three dimensions are examined. It is shown that not all 3-D methods have useful extensions to higher dimensions. It is also shown why the rate of change of the matrix elements, which are the elements of the angular rate vector in 3-D, are the elements of a tensor of the second rank (dyadic) in spaces other than three dimensional. It is proven that the 3-D Gibbs vector (or Cayley Parameters) are extendable to other dimensions. An algorithm is developed emplying the resulting parameters, which are termed Extended Rodrigues Parameters, and numerical results are presented of the application of the algorithm to a fourth order matrix.

Real gas equation-of-state capability at Sandia Livermore

International Nuclear Information System (INIS)

Clark, G.L.

1978-03-01

A library of FORTRAN routines is described which model the equations-of-state for gases commonly encountered in compressible gas dynamics applications. The present library includes thermodynamic properties packages as well as compressibility models, both for pure gases and mixtures. Tables are included to allow hand calculation of equation-of-state problems for the gases hydrogen, helium, neon, argon, oxygen, air, and nitrogen. Generally the tables extend to several thousand atmospheres, with temperatures from the cryogenic realm to 400 K

Holomorphic Vector Bundles Corresponding to some Soliton Solutions of the Ward Equation

Energy Technology Data Exchange (ETDEWEB)

Zhu, Xiujuan, E-mail: yzzhuxiujuan@sina.com [Jiangsu Second Normal University, School of Mathematics and Information Technology (China)

2015-12-15

Holomorphic vector bundles corresponding to the static soliton solution of the Ward equation were explicitly presented by Ward in terms of a meromorphic framing. Bundles (for simplicity, “bundle” is to be taken throughout to mean “holomorphic vector bundle”) corresponding to all Ward k-soliton solutions whose extended solutions have only simple poles, and some Ward 2-soliton solutions whose extended solutions have only a second-order pole, were explicitly described by us in a previous paper. In this paper, we go on to present some bundles corresponding to soliton-antisoliton solutions of the Ward equation, and Ward 3-soliton solutions whose extended solutions have a simple pole and a double pole. To give some more interpretation of the bundles, we study the second Chern number of the corresponded bundles and find that it can be obtained directly from the patching matrices. We also point out some information about bundles corresponding to Ward soliton solutions whose extended solutions have general pole data at the end of the paper.

True amplitude wave equation migration arising from true amplitude one-way wave equations

Science.gov (United States)

Zhang, Yu; Zhang, Guanquan; Bleistein, Norman

2003-10-01

to these newly defined wavefields in heterogeneous media leads to the Kirchhoff inversion formula for common-shot data when the one-way wavefields are replaced by their ray theoretic approximations. This extension enhances the original WEM method. The objective of that technique was a reflector map, only. The underlying theory did not address amplitude issues. Computer output obtained using numerically generated data confirms the accuracy of this inversion method. However, there are practical limitations. The observed data must be a solution of the wave equation. Therefore, the data over the entire survey area must be collected from a single common-shot experiment. Multi-experiment data, such as common-offset data, cannot be used with this method as currently formulated. Research on extending the method is ongoing at this time.

Author Details

African Journals Online (AJOL)

Tladi, Maleafisha Joseph Pekwa Stephen. Vol 40, No 6 (2017) - Articles On the qualitative theory of the rotating boussinesq and quasi-geostrophic equations. Abstract. ISSN: 1727-933X. AJOL African Journals Online. HOW TO USE AJOL... for Researchers · for Librarians · for Authors · FAQ's · More about AJOL · AJOL's ...

Reduced equations for finite beta tearing modes in tokamaks

International Nuclear Information System (INIS)

Izzo, R.; Monticello, D.A.; DeLucia, J.; Park, W.; Ryu, C.M.

1984-10-01

The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have nabla vector.B vector = O satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio epsilon of the torus, and for β approx. epsilon or smaller. We demonstrate this by deriving a reduced set of MHD equations that are correct to 5th order in epsilon. These equations contain the exact equilibrium relation and as such can be used to find 3-D stellarator equilibria. In addition, if a subsidiary ordering in eta, the resistivity, is made, the equations of Glasser, Greene, and Johnson are recovered. This set of reduced equations has been coded by extending the initial value code, HILO. Results obtained, for both ideal and resistive linear stability, from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. The agreement is shown to be excellent for both zero and finite beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter

Equations of motion for a rotor blade, including gravity, pitch action and rotor speed variations

DEFF Research Database (Denmark)

Kallesøe, Bjarne Skovmose

2007-01-01

This paper extends Hodges-Dowell's partial differential equations of blade motion, by including the effects from gravity, pitch action and varying rotor speed. New equations describing the pitch action and rotor speeds are also derived. The physical interpretation of the individual terms...... in the equations is discussed. The partial differential equations of motion are approximated by ordinary differential equations of motion using an assumed mode method. The ordinary differential equations are used to simulate a sudden pitch change of a rotating blade. This work is a part of a project on pitch blade...

Extended I-Love relations for slowly rotating neutron stars

Science.gov (United States)

Gagnon-Bischoff, Jérémie; Green, Stephen R.; Landry, Philippe; Ortiz, Néstor

2018-03-01

Observations of gravitational waves from inspiralling neutron star binaries—such as GW170817—can be used to constrain the nuclear equation of state by placing bounds on stellar tidal deformability. For slowly rotating neutron stars, the response to a weak quadrupolar tidal field is characterized by four internal-structure-dependent constants called "Love numbers." The tidal Love numbers k2el and k2mag measure the tides raised by the gravitoelectric and gravitomagnetic components of the applied field, and the rotational-tidal Love numbers fo and ko measure those raised by couplings between the applied field and the neutron star spin. In this work, we compute these four Love numbers for perfect fluid neutron stars with realistic equations of state. We discover (nearly) equation-of-state independent relations between the rotational-tidal Love numbers and the moment of inertia, thereby extending the scope of I-Love-Q universality. We find that similar relations hold among the tidal and rotational-tidal Love numbers. These relations extend the applications of I-Love universality in gravitational-wave astronomy. As our findings differ from those reported in the literature, we derive general formulas for the rotational-tidal Love numbers in post-Newtonian theory and confirm numerically that they agree with our general-relativistic computations in the weak-field limit.

Optimal Operation of Radial Distribution Systems Using Extended Dynamic Programming

DEFF Research Database (Denmark)

Lopez, Juan Camilo; Vergara, Pedro P.; Lyra, Christiano

2018-01-01

An extended dynamic programming (EDP) approach is developed to optimize the ac steady-state operation of radial electrical distribution systems (EDS). Based on the optimality principle of the recursive Hamilton-Jacobi-Bellman equations, the proposed EDP approach determines the optimal operation o...... approach is illustrated using real-scale systems and comparisons with commercial programming solvers. Finally, generalizations to consider other EDS operation problems are also discussed.......An extended dynamic programming (EDP) approach is developed to optimize the ac steady-state operation of radial electrical distribution systems (EDS). Based on the optimality principle of the recursive Hamilton-Jacobi-Bellman equations, the proposed EDP approach determines the optimal operation...... of the EDS by setting the values of the controllable variables at each time period. A suitable definition for the stages of the problem makes it possible to represent the optimal ac power flow of radial EDS as a dynamic programming problem, wherein the 'curse of dimensionality' is a minor concern, since...

The behavior of steady quasisolitons near the limit cases of third-order nonlinear Schrödinger equation

DEFF Research Database (Denmark)

Karpman, V.I.; Shagalov, A.G.; Juul Rasmussen, J.

2002-01-01

The behavior of steady quasisoliton solutions to the extended third-order nonlinear Schrodinger (NLS) equation is studied in two cases: (i) when the coefficients in the equation approach the Hirota conditions, and (ii) near the limit of the regular NLS equation. (C) 2002 Published by Elsevier...

A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

Directory of Open Access Journals (Sweden)

Mehmet Ali Akinlar

2013-01-01

Full Text Available A new solution technique for analytical solutions of fractional partial differential equations (FPDEs is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as well as a more general fractional differential equation.

Parametric Borel summability for some semilinear system of partial differential equations

Directory of Open Access Journals (Sweden)

Hiroshi Yamazawa

2015-01-01

Full Text Available In this paper we study the Borel summability of formal solutions with a parameter of first order semilinear system of partial differential equations with \\(n\\ independent variables. In [Singular perturbation of linear systems with a regular singularity, J. Dynam. Control. Syst. 8 (2002, 313-322], Balser and Kostov proved the Borel summability of formal solutions with respect to a singular perturbation parameter for a linear equation with one independent variable. We shall extend their results to a semilinear system of equations with general independent variables.

A New Auto-Baecklund Transformation and Two-Soliton Solution for (3+1)-Dimensional Jimbo-Miwa Equation

International Nuclear Information System (INIS)

Liu Chunping; Zhou Ling

2011-01-01

By improving the extended homogeneous balance method, a general method is suggested to derive a new auto-Baecklund transformation (BT) for (3+1)-Dimensional Jimbo-Miwa (JM) equation. The auto-BT obtained by using our method only involves one quadratic homogeneity equation written as a bilinear equation. Based on the auto-BT, two-soliton solution of the (3+1)-Dimensional JM equation is obtained. (general)

Minimal length, Friedmann equations and maximum density

Energy Technology Data Exchange (ETDEWEB)

Awad, Adel [Center for Theoretical Physics, British University of Egypt,Sherouk City 11837, P.O. Box 43 (Egypt); Department of Physics, Faculty of Science, Ain Shams University,Cairo, 11566 (Egypt); Ali, Ahmed Farag [Centre for Fundamental Physics, Zewail City of Science and Technology,Sheikh Zayed, 12588, Giza (Egypt); Department of Physics, Faculty of Science, Benha University,Benha, 13518 (Egypt)

2014-06-16

Inspired by Jacobson’s thermodynamic approach, Cai et al. have shown the emergence of Friedmann equations from the first law of thermodynamics. We extend Akbar-Cai derivation http://dx.doi.org/10.1103/PhysRevD.75.084003 of Friedmann equations to accommodate a general entropy-area law. Studying the resulted Friedmann equations using a specific entropy-area law, which is motivated by the generalized uncertainty principle (GUP), reveals the existence of a maximum energy density closed to Planck density. Allowing for a general continuous pressure p(ρ,a) leads to bounded curvature invariants and a general nonsingular evolution. In this case, the maximum energy density is reached in a finite time and there is no cosmological evolution beyond this point which leaves the big bang singularity inaccessible from a spacetime prospective. The existence of maximum energy density and a general nonsingular evolution is independent of the equation of state and the spacial curvature k. As an example we study the evolution of the equation of state p=ωρ through its phase-space diagram to show the existence of a maximum energy which is reachable in a finite time.

Thermal convection around a heat source embedded in a box containing a saturated porous medium

Energy Technology Data Exchange (ETDEWEB)

Himasekhar, K.; Bau, H.H. (Univ. of Pennsylvania, Philadelphia (USA))

1988-08-01

A study of the thermal convection around a uniform flux cylinder embedded in a box containing a saturated porous medium is carried out experimentally and theoretically. The experimental work includes heat transfer and temperature field measurements. It is observed that for low Rayleigh numbers, the flow is two dimensional and time independent. Once a critical Rayleigh number is exceeded, the flow undergoes a Hopf bifurcation and becomes three dimensional and time dependent. The theoretical study involves the numerical solution of the two-dimensional Darcy-Oberbeck-Boussinesq equations. The complicated geometry is conveniently handled by mapping the physical domain onto a rectangle via the use of boundary-fitted coordinates. The numerical code can easily be extended to handle diverse geometric configurations. For low Rayleigh numbers, the theoretical results agree favorably with the experimental observations. However, the appearance of three-dimensional flow phenomena limits the range of utility of the numerical code.

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....

The Liquid Film Flow with Evaporation: Numerical Modelling

Directory of Open Access Journals (Sweden)

Rezanova Ekaterina

2016-01-01

Full Text Available The flow of thin liquid layer on an inclined substrate is investigated numerically. The mathematical modelling is based on the Oberbeck-Boussinesq equations and the generalized conditions on the thermocapillary boundary simplified during the parametrical analysis. In the framework of the long-wave approximation the evolution equation which determines the thickness of the liquid layer in the case of moderate Reynolds numbers is derived. The results of numerical modelling of the liquid flow with evaporation at the interface are obtained.

Thermohydraulic analysis in pipelines using the finite element method

International Nuclear Information System (INIS)

Costa, L.E.; Idelsohn, S.R.

1984-01-01

The Finite Element Method (FEM) is employed for the numerical solution of fluid flow problems with combined heat transfer mechanisms. Boussinesq approximations are used for the solution of the governing equations. The application of the FEM leads to a set of simultaneous nonlinear equations. The development of the method, for the solution of bidimensional and axisymmetric problems, is presented. Examples of fluid flow in pipes, including natural and forced convection, are solved with the proposed method and discussed in the paper. (Author) [pt

High-order quantum algorithm for solving linear differential equations

International Nuclear Information System (INIS)

Berry, Dominic W

2014-01-01

Linear differential equations are ubiquitous in science and engineering. Quantum computers can simulate quantum systems, which are described by a restricted type of linear differential equations. Here we extend quantum simulation algorithms to general inhomogeneous sparse linear differential equations, which describe many classical physical systems. We examine the use of high-order methods (where the error over a time step is a high power of the size of the time step) to improve the efficiency. These provide scaling close to Δt 2 in the evolution time Δt. As with other algorithms of this type, the solution is encoded in amplitudes of the quantum state, and it is possible to extract global features of the solution. (paper)

Iterative Solutions of Nonlinear Integral Equations of Hammerstein Type

Directory of Open Access Journals (Sweden)

Abebe R. Tufa

2015-11-01

Full Text Available Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1and L2, respectively. Suppose that the Hammerstein type equation u + KFu = 0 has a solution in H. It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein type equation. The results obtained in this paper improve and extend known results in the literature.

Invariant Lagrangians, mechanical connections and the Lagrange-Poincare equations

International Nuclear Information System (INIS)

Mestdag, T; Crampin, M

2008-01-01

We deal with Lagrangian systems that are invariant under the action of a symmetry group. The mechanical connection is a principal connection that is associated with Lagrangians which have a kinetic energy function that is defined by a Riemannian metric. In this paper, we extend this notion to arbitrary Lagrangians. We then derive the reduced Lagrange-Poincare equations in a new fashion and we show how solutions of the Euler-Lagrange equations can be reconstructed with the help of the mechanical connection. Illustrative examples confirm the theory

Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

International Nuclear Information System (INIS)

FAN, WESLEY C.; DRUMM, CLIFTON R.; POWELL, JENNIFER L. email wcfan@sandia.gov

2002-01-01

The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations

Sufficient Descent Conjugate Gradient Methods for Solving Convex Constrained Nonlinear Monotone Equations

Directory of Open Access Journals (Sweden)

San-Yang Liu

2014-01-01

Full Text Available Two unified frameworks of some sufficient descent conjugate gradient methods are considered. Combined with the hyperplane projection method of Solodov and Svaiter, they are extended to solve convex constrained nonlinear monotone equations. Their global convergence is proven under some mild conditions. Numerical results illustrate that these methods are efficient and can be applied to solve large-scale nonsmooth equations.

Discrete Ordinates Approximations to the First- and Second-Order Radiation Transport Equations

CERN Document Server

Fan, W C; Powell, J L

2002-01-01

The conventional discrete ordinates approximation to the Boltzmann transport equation can be described in a matrix form. Specifically, the within-group scattering integral can be represented by three components: a moment-to-discrete matrix, a scattering cross-section matrix and a discrete-to-moment matrix. Using and extending these entities, we derive and summarize the matrix representations of the second-order transport equations.

Using Automated Essay Scores as an Anchor When Equating Constructed Response Writing Tests

Science.gov (United States)

Almond, Russell G.

2014-01-01

Assessments consisting of only a few extended constructed response items (essays) are not typically equated using anchor test designs as there are typically too few essay prompts in each form to allow for meaningful equating. This article explores the idea that output from an automated scoring program designed to measure writing fluency (a common…

The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials

International Nuclear Information System (INIS)

Contreras-Astorga, Alonso; Schulze-Halberg, Axel

2014-01-01

We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)

The confluent supersymmetry algorithm for Dirac equations with pseudoscalar potentials

Energy Technology Data Exchange (ETDEWEB)

Contreras-Astorga, Alonso, E-mail: aloncont@iun.edu; Schulze-Halberg, Axel, E-mail: axgeschu@iun.edu, E-mail: xbataxel@gmail.com [Department of Mathematics and Actuarial Science and Department of Physics, Indiana University Northwest, 3400 Broadway, Gary, Indiana 46408 (United States)

2014-10-15

We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, “Intertwining technique for the one-dimensional stationary Dirac equation,” Ann. Phys. 305, 151–189 (2003)].

Extended Galilean symmetries of non-relativistic strings

Energy Technology Data Exchange (ETDEWEB)

Batlle, Carles [Departament de Matemàtiques and IOC, Universitat Politècnica de Catalunya, EPSEVG,Av. V. Balaguer 1, E-08808 Vilanova i la Geltrú (Spain); Gomis, Joaquim; Not, Daniel [Departament de Física Quàntica i Astrofísica and Institut de Ciències del Cosmos (ICCUB),Universitat de Barcelona,Martí i Franquès 1, E-08028 Barcelona (Spain)

2017-02-09

We consider two non-relativistic strings and their Galilean symmetries. These strings are obtained as the two possible non-relativistic (NR) limits of a relativistic string. One of them is non-vibrating and represents a continuum of non-relativistic massless particles, and the other one is a non-relativistic vibrating string. For both cases we write the generator of the most general point transformation and impose the condition of Noether symmetry. As a result we obtain two sets of non-relativistic Killing equations for the vector fields that generate the symmetry transformations. Solving these equations shows that NR strings exhibit two extended, infinite dimensional space-time symmetries which contain, as a subset, the Galilean symmetries. For each case, we compute the associated conserved charges and discuss the existence of non-central extensions.

A novel noncommutative KdV-type equation, its recursion operator, and solitons

Science.gov (United States)

Carillo, Sandra; Lo Schiavo, Mauro; Porten, Egmont; Schiebold, Cornelia

2018-04-01

A noncommutative KdV-type equation is introduced extending the Bäcklund chart in Carillo et al. [Symmetry Integrability Geom.: Methods Appl. 12, 087 (2016)]. This equation, called meta-mKdV here, is linked by Cole-Hopf transformations to the two noncommutative versions of the mKdV equations listed in Olver and Sokolov [Commun. Math. Phys. 193, 245 (1998), Theorem 3.6]. For this meta-mKdV, and its mirror counterpart, recursion operators, hierarchies, and an explicit solution class are derived.

On the number of polynomial solutions of Bernoulli and Abel polynomial differential equations

Science.gov (United States)

Cima, A.; Gasull, A.; Mañosas, F.

2017-12-01

In this paper we determine the maximum number of polynomial solutions of Bernoulli differential equations and of some integrable polynomial Abel differential equations. As far as we know, the tools used to prove our results have not been utilized before for studying this type of questions. We show that the addressed problems can be reduced to know the number of polynomial solutions of a related polynomial equation of arbitrary degree. Then we approach to these equations either applying several tools developed to study extended Fermat problems for polynomial equations, or reducing the question to the computation of the genus of some associated planar algebraic curves.

The Wertheim integral equation theory with the ideal chain approximation and a dimer equation of state: Generalization to mixtures of hard-sphere chain fluids

International Nuclear Information System (INIS)

Chang, J.; Sandler, S.I.

1995-01-01

We have extended the Wertheim integral equation theory to mixtures of hard spheres with two attraction sites in order to model homonuclear hard-sphere chain fluids, and then solved these equations with the polymer-Percus--Yevick closure and the ideal chain approximation to obtain the average intermolecular and overall radial distribution functions. We obtain explicit expressions for the contact values of these distribution functions and a set of one-dimensional integral equations from which the distribution functions can be calculated without iteration or numerical Fourier transformation. We compare the resulting predictions for the distribution functions with Monte Carlo simulation results we report here for five selected binary mixtures. It is found that the accuracy of the prediction of the structure is the best for dimer mixtures and declines with increasing chain length and chain-length asymmetry. For the equation of state, we have extended the dimer version of the thermodynamic perturbation theory to the hard-sphere chain mixture by introducing the dimer mixture as an intermediate reference system. The Helmholtz free energy of chain fluids is then expressed in terms of the free energy of the hard-sphere mixture and the contact values of the correlation functions of monomer and dimer mixtures. We compared with the simulation results, the resulting equation of state is found to be the most accurate among existing theories with a relative average error of 1.79% for 4-mer/8-mer mixtures, which is the worst case studied in this work. copyright 1995 American Institute of Physics

On the analytical solution of Fornberg–Whitham equation with the ...

Indian Academy of Sciences (India)

This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence ..... The authors are grateful to the financial support extended by the King Fahd University of. Petroleum ...

Semi-continuous and multigroup models in extended kinetic theory

International Nuclear Information System (INIS)

Koller, W.

2000-01-01

The aim of this thesis is to study energy discretization of the Boltzmann equation in the framework of extended kinetic theory. In case that external fields can be neglected, the semi- continuous Boltzmann equation yields a sound basis for various generalizations. Semi-continuous kinetic equations describing a three component gas mixture interacting with monochromatic photons as well as a four component gas mixture undergoing chemical reactions are established and investigated. These equations reflect all major aspects (conservation laws, equilibria, H-theorem) of the full continuous kinetic description. For the treatment of the spatial dependence, an expansion of the distribution function in terms of Legendre polynomials is carried out. An implicit finite differencing scheme is combined with the operator splitting method. The obtained numerical schemes are applied to the space homogeneous study of binary chemical reactions and to spatially one-dimensional laser-induced acoustic waves. In the presence of external fields, the developed overlapping multigroup approach (with the spline-interpolation as its extension) is well suited for numerical studies. Furthermore, two formulations of consistent multigroup approaches to the non-linear Boltzmann equation are presented. (author)

Variational principle for nonlinear gyrokinetic Vlasov--Maxwell equations

International Nuclear Information System (INIS)

Brizard, Alain J.

2000-01-01

A new variational principle for the nonlinear gyrokinetic Vlasov--Maxwell equations is presented. This Eulerian variational principle uses constrained variations for the gyrocenter Vlasov distribution in eight-dimensional extended phase space and turns out to be simpler than the Lagrangian variational principle recently presented by H. Sugama [Phys. Plasmas 7, 466 (2000)]. A local energy conservation law is then derived explicitly by the Noether method. In future work, this new variational principle will be used to derive self-consistent, nonlinear, low-frequency Vlasov--Maxwell bounce-gyrokinetic equations, in which the fast gyromotion and bounce-motion time scales have been eliminated

A two-component generalization of the reduced Ostrovsky equation and its integrable semi-discrete analogue

International Nuclear Information System (INIS)

Feng, Bao-Feng; Maruno, Ken-ichi; Ohta, Yasuhiro

2017-01-01

In the present paper, we propose a two-component generalization of the reduced Ostrovsky (Vakhnenko) equation, whose differential form can be viewed as the short-wave limit of a two-component Degasperis–Procesi (DP) equation. They are integrable due to the existence of Lax pairs. Moreover, we have shown that the two-component reduced Ostrovsky equation can be reduced from an extended BKP hierarchy with negative flow through a pseudo 3-reduction and a hodograph (reciprocal) transform. As a by-product, its bilinear form and N -soliton solution in terms of pfaffians are presented. One- and two-soliton solutions are provided and analyzed. In the second part of the paper, we start with a modified BKP hierarchy, which is a Bäcklund transformation of the above extended BKP hierarchy, an integrable semi-discrete analogue of the two-component reduced Ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations. In particular, the backward difference form of above semi-discrete two-component reduced Ostrovsky equation gives rise to the integrable semi-discretization of the short wave limit of a two-component DP equation. Their N -soliton solutions in terms of pffafians are also provided. (paper)

On the relation between the Einstein field equations and the Jacobi–Ricci–Bianchi system

International Nuclear Information System (INIS)

Van den Bergh, N

2013-01-01

The 1 + 3 covariant equations, embedded in an extended tetrad formalism and describing a spacetime with an arbitrary energy–momentum distribution, are reconsidered. It is shown that, provided the 1 + 3 splitting is performed with respect to a generic time-like congruence with a tangent vector u, the Einstein field equations can be regarded as the integrability conditions for the Jacobi and Bianchi equations together with the Ricci equations for u. The same conclusion holds for a generic null congruence in the Newman–Penrose framework. (paper)

Translational control of a graphically simulated robot arm by kinematic rate equations that overcome elbow joint singularity

Science.gov (United States)

Barker, L. K.; Houck, J. A.; Carzoo, S. W.

1984-01-01

An operator commands a robot hand to move in a certain direction relative to its own axis system by specifying a velocity in that direction. This velocity command is then resolved into individual joint rotational velocities in the robot arm to effect the motion. However, the usual resolved-rate equations become singular when the robot arm is straightened. To overcome this elbow joint singularity, equations were developed which allow continued translational control of the robot hand even though the robot arm is (or is nearly) fully extended. A feature of the equations near full arm extension is that an operator simply extends and retracts the robot arm to reverse the direction of the elbow bend (difficult maneuver for the usual resolved-rate equations). Results show successful movement of a graphically simulated robot arm.

Babenko’s Approach to Abel’s Integral Equations

Directory of Open Access Journals (Sweden)

Chenkuan Li

2018-03-01

Full Text Available The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t + λ Γ ( α ∫ 0 t ( t − τ α − 1 y ( τ d τ = f ( t , ( t > 0 and its variants by fractional calculus. Applying Babenko’s approach and fractional integrals, we provide a general method for solving Abel’s integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary α ∈ R by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.

Approximate Analytic and Numerical Solutions to Lane-Emden Equation via Fuzzy Modeling Method

Directory of Open Access Journals (Sweden)

De-Gang Wang

2012-01-01

Full Text Available A novel algorithm, called variable weight fuzzy marginal linearization (VWFML method, is proposed. This method can supply approximate analytic and numerical solutions to Lane-Emden equations. And it is easy to be implemented and extended for solving other nonlinear differential equations. Numerical examples are included to demonstrate the validity and applicability of the developed technique.

Reduced equations for finite beta tearing modes in tokamaks

International Nuclear Information System (INIS)

Izzo, R.; Monticello, D.A.; DeLucia, J.; Park, W.; Ryu, C.M.

1985-01-01

The equations of resistive magnetohydrodynamics (MHD) are recast in a form that is useful for studying the evolution of those toroidal systems where the fast magnetosonic wave plays no important role. The equations are exact and have del x B = 0 satisfied explicitly. From this set of equations it is a simple matter to derive the equations of reduced MHD to any order in the inverse aspect ratio epsilon of the torus and for βapprox.epsilon or smaller. This is demonstrated by deriving a reduced set of MHD equations that are correct to fifth order in epsilon. These equations contain the exact equilibrium relation and, as such, can be used to find three-dimensional stellarator equilibria. In addition, if a subsidiary ordering in eta, the resistivity, is made, the equations of Glasser, Greene, and Johnson [Phys. Fluids 8, 875 (1967); 19, 567 (1967)] are recovered. This set of reduced equations has been coded by extending the initial value code hIlo [Phys. Fluids 26, 3066 (1983)]. Results obtained for both ideal and resistive linear stability from the reduced equations are compared with those obtained by solving the full set of MHD equations in a cylinder. Good agreement is shown for both zero and finite-beta calculations. Comparisons are also made with analytic theory illuminating the present limitations of the latter

Extended biorthogonal matrix polynomials

Directory of Open Access Journals (Sweden)

Ayman Shehata

2017-01-01

Full Text Available The pair of biorthogonal matrix polynomials for commutative matrices were first introduced by Varma and Tasdelen in [22]. The main aim of this paper is to extend the properties of the pair of biorthogonal matrix polynomials of Varma and Tasdelen and certain generating matrix functions, finite series, some matrix recurrence relations, several important properties of matrix differential recurrence relations, biorthogonality relations and matrix differential equation for the pair of biorthogonal matrix polynomials J(A,B n (x, k and K(A,B n (x, k are discussed. For the matrix polynomials J(A,B n (x, k, various families of bilinear and bilateral generating matrix functions are constructed in the sequel.

Basic quantum mechanics for three Dirac equations in a curved spacetime

International Nuclear Information System (INIS)

Arminjon, Mayeul

2010-01-01

We study the basic quantum mechanics for a fully general set of Dirac matrices in a curved spacetime by extending Pauli's method. We further extend this study to three versions of the Dirac equation: the standard (Dirac-Fock-Weyl or DFW) equation, and two alternative versions, both of which are based on the recently proposed linear tensor representations of the Dirac field (TRD). We begin with the current conservation: we show that the latter applies to any solution of the Dirac equation, if the field of Dirac matrices γμ satisfies a specific PDE. This equation is always satisfied for DFW with its restricted choice for the γμ matrices. It similarly restricts the choice of the γμ matrices for TRD. However, this restriction can be achieved. The frame dependence of a general Hamiltonian operator is studied. We show that in any given reference frame with minor restrictions on the spacetime metric, the axioms of quantum mechanics impose a unique form for the Hilbert space scalar product. Finally, the condition for the general Dirac Hamiltonian operator to be Hermitian is derived in a general curved spacetime. For DFW, the validity of this hermeticity condition depends on the choice of the γμ matrices. (author)

The relationship between vapour pressure, vaporization enthalpy, and enthalpy of transfer from solution to gas: An extension of the Martin equation

International Nuclear Information System (INIS)

Srisaipet, A.; Aryusuk, K.; Lilitchan, S.; Krisnangkura, K.

2007-01-01

Martin's equation, Δ sln g G=Δ sln g G o +zδ sln g G, is extended to cover vaporization free energy (Δ l g G). The extended equation is further expanded in terms of enthalpy and entropy and then used to correlate vaporization enthalpy (Δ l g H) and enthalpy of transfer from solution to gas (Δ sln g H). Data available in the literatures are used to validate and support the speculations derived from the proposed equation

Exact traveling wave solution of nonlinear variants of the RLW and the PHI-four equations

Energy Technology Data Exchange (ETDEWEB)

Soliman, A.A. [Department of Mathematics, Faculty of Education (AL-Arish), Suez Canal University, AL-Arish 45111 (Egypt); Department of Mathematics, Teacher' s College, Bisha, P.O. Box 551 (Saudi Arabia)], E-mail: asoliman_99@yahoo.com

2007-08-27

By means of the modified extended tanh-function (METF) method the multiple traveling wave solutions of some different kinds of nonlinear partial differential equations are presented and implemented in a computer algebraic system. The solutions for the nonlinear equations such as variants of the RLW and variant of the PHI-four equations are exactly obtained and so the efficiency of the method can be demonstrated.

Group-theoretical aspects of the discrete sine-Gordon equation

International Nuclear Information System (INIS)

Orfanidis, S.J.

1980-01-01

The group-theoretical interpretation of the sine-Gordon equation in terms of connection forms on fiber bundles is extended to the discrete case. Solutions of the discrete sine-Gordon equation induce surfaces on a lattice in the SU(2) group space. The inverse scattering representation, expressing the parallel transport of fibers, is implemented by means of finite rotations. Discrete Baecklund transformations are realized as gauge transformations. The three-dimensional inverse scattering representation is used to derive a discrete nonlinear sigma model, and the corresponding Baecklund transformation and Pohlmeyer's R transformation are constructed

Spinodal equation of state for rutile TiO2

DEFF Research Database (Denmark)

Francisco, E.; Bermejo, M.; Baonza, V. Garcia

2003-01-01

We present a general computational scheme to extend the spinodal equation of state [Garcia Baonza , Phys. Rev. B 51, 28 (1995)] to the interpretation of the cell parameters response to hydrostatic pressure in orthogonal lattices. As an important example, we analyze the pressure (p)-volume (V)-tem...

Dimerization of Carboxylic Acids: An Equation of State Approach

DEFF Research Database (Denmark)

Tsivintzelis, Ioannis; Kontogeorgis, Georgios; Panayiotou, Costas

2017-01-01

The association term of the nonrandom hydrogen bonding theory, which is an equation of state model, is extended to describe the dimerization of carboxylic acids in binary mixtures with inert solvents and in systems of two different acids. Subsequently, the model is applied to describe the excess...

Using "Tracker" to Prove the Simple Harmonic Motion Equation

Science.gov (United States)

Kinchin, John

2016-01-01

Simple harmonic motion (SHM) is a common topic for many students to study. Using the free, though versatile, motion tracking software; "Tracker", we can extend the students experience and show that the general equation for SHM does lead to the correct period of a simple pendulum.

Partner symmetries of the complex Monge-Ampere equation yield hyper-Kaehler metrics without continuous symmetries

International Nuclear Information System (INIS)

Malykh, A A; Nutku, Y; Sheftel, M B

2003-01-01

We extend the Mason-Newman Lax pair for the elliptic complex Monge-Ampere equation so that this equation itself emerges as an algebraic consequence. We regard the function in the extended Lax equations as a complex potential. Their differential compatibility condition coincides with the determining equation for the symmetries of the complex Monge-Ampere equation. We shall identify the real and imaginary parts of the potential, which we call partner symmetries, with the translational and dilatational symmetry characteristics, respectively. Then we choose the dilatational symmetry characteristic as the new unknown replacing the Kaehler potential. This directly leads to a Legendre transformation. Studying the integrability conditions of the Legendre-transformed system we arrive at a set of linear equations satisfied by a single real potential. This enables us to construct non-invariant solutions of the Legendre transform of the complex Monge-Ampere equation. Using these solutions we obtained explicit Legendre-transformed hyper-Kaehler metrics with a anti-self-dual Riemann curvature 2-form that admit no Killing vectors. They satisfy the Einstein field equations with Euclidean signature. We give the detailed derivation of the solution announced earlier and present a new solution with an added parameter. We compare our method of partner symmetries for finding non-invariant solutions to that of Dunajski and Mason who use 'hidden' symmetries for the same purpose

The roots to an equation in particle slowing down theory

International Nuclear Information System (INIS)

Sjoestrand, N.G.

1979-08-01

Previous work on the roots to an equation arising in studies on anisotropic neutron scattering has been extended to include parameter values of interest for a problem put forward by M.M.R. Williams. Detailed numerical results are given. (author)

Analytical approach for the Floquet theory of delay differential equations.

Science.gov (United States)

Simmendinger, C; Wunderlin, A; Pelster, A

1999-05-01

We present an analytical approach to deal with nonlinear delay differential equations close to instabilities of time periodic reference states. To this end we start with approximately determining such reference states by extending the Poincaré-Lindstedt and the Shohat expansions, which were originally developed for ordinary differential equations. Then we systematically elaborate a linear stability analysis around a time periodic reference state. This allows us to approximately calculate the Floquet eigenvalues and their corresponding eigensolutions by using matrix valued continued fractions.

Numerical solution of plasma fluid equations using locally refined grids

International Nuclear Information System (INIS)

Colella, P.

1997-01-01

This paper describes a numerical method for the solution of plasma fluid equations on block-structured, locally refined grids. The plasma under consideration is typical of those used for the processing of semiconductors. The governing equations consist of a drift-diffusion model of the electrons and an isothermal model of the ions coupled by Poisson's equation. A discretization of the equations is given for a uniform spatial grid, and a time-split integration scheme is developed. The algorithm is then extended to accommodate locally refined grids. This extension involves the advancement of the discrete system on a hierarchy of levels, each of which represents a degree of refinement, together with synchronization steps to ensure consistency across levels. A brief discussion of a software implementation is followed by a presentation of numerical results

On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Science.gov (United States)

Gunzburger, M. D.; Nicolaides, R. A.

1986-01-01

Substructuring methods are in common use in mechanics problems where typically the associated linear systems of algebraic equations are positive definite. Here these methods are extended to problems which lead to nonpositive definite, nonsymmetric matrices. The extension is based on an algorithm which carries out the block Gauss elimination procedure without the need for interchanges even when a pivot matrix is singular. Examples are provided wherein the method is used in connection with finite element solutions of the stationary Stokes equations and the Helmholtz equation, and dual methods for second-order elliptic equations.

Energy analysis of four dimensional extended hyperbolic Scarf I plus three dimensional separable trigonometric noncentral potentials using SUSY QM approach

International Nuclear Information System (INIS)

Suparmi, A.; Cari, C.; Deta, U. A.; Handhika, J.

2016-01-01

The non-relativistic energies and wave functions of extended hyperbolic Scarf I plus separable non-central shape invariant potential in four dimensions are investigated using Supersymmetric Quantum Mechanics (SUSY QM) Approach. The three dimensional separable non-central shape invariant angular potential consists of trigonometric Scarf II, Manning Rosen and Poschl-Teller potentials. The four dimensional Schrodinger equation with separable shape invariant non-central potential is reduced into four one dimensional Schrodinger equations through variable separation method. By using SUSY QM, the non-relativistic energies and radial wave functions are obtained from radial Schrodinger equation, the orbital quantum numbers and angular wave functions are obtained from angular Schrodinger equations. The extended potential means there is perturbation terms in potential and cause the decrease in energy spectra of Scarf I potential. (paper)

A nondissipative simulation method for the drift kinetic equation

International Nuclear Information System (INIS)

Watanabe, Tomo-Hiko; Sugama, Hideo; Sato, Tetsuya

2001-07-01

With the aim to study the ion temperature gradient (ITG) driven turbulence, a nondissipative kinetic simulation scheme is developed and comprehensively benchmarked. The new simulation method preserving the time-reversibility of basic kinetic equations can successfully reproduce the analytical solutions of asymmetric three-mode ITG equations which are extended to provide a more general reference for benchmarking than the previous work [T.-H. Watanabe, H. Sugama, and T. Sato: Phys. Plasmas 7 (2000) 984]. It is also applied to a dissipative three-mode system, and shows a good agreement with the analytical solution. The nondissipative simulation result of the ITG turbulence accurately satisfies the entropy balance equation. Usefulness of the nondissipative method for the drift kinetic simulations is confirmed in comparisons with other dissipative schemes. (author)

A relaxation-projection method for compressible flows. Part I: The numerical equation of state for the Euler equations

International Nuclear Information System (INIS)

Saurel, Richard; Franquet, Erwin; Daniel, Eric; Le Metayer, Olivier

2007-01-01

A new projection method is developed for the Euler equations to determine the thermodynamic state in computational cells. It consists in the resolution of a mechanical relaxation problem between the various sub-volumes present in a computational cell. These sub-volumes correspond to the ones traveled by the various waves that produce states with different pressures, velocities, densities and temperatures. Contrarily to Godunov type schemes the relaxed state corresponds to mechanical equilibrium only and remains out of thermal equilibrium. The pressure computation with this relaxation process replaces the use of the conventional equation of state (EOS). A simplified relaxation method is also derived and provides a specific EOS (named the Numerical EOS). The use of the Numerical EOS gives a cure to spurious pressure oscillations that appear at contact discontinuities for fluids governed by real gas EOS. It is then extended to the computation of interface problems separating fluids with different EOS (liquid-gas interface for example) with the Euler equations. The resulting method is very robust, accurate, oscillation free and conservative. For the sake of simplicity and efficiency the method is developed in a Lagrange-projection context and is validated over exact solutions. In a companion paper [F. Petitpas, E. Franquet, R. Saurel, A relaxation-projection method for compressible flows. Part II: computation of interfaces and multiphase mixtures with stiff mechanical relaxation. J. Comput. Phys. (submitted for publication)], the method is extended to the numerical approximation of a non-conservative hyperbolic multiphase flow model for interface computation and shock propagation into mixtures

Second order time evolution of the multigroup diffusion and P1 equations for radiation transport

International Nuclear Information System (INIS)

Olson, Gordon L.

2011-01-01

Highlights: → An existing multigroup transport algorithm is extended to be second-order in time. → A new algorithm is presented that does not require a grey acceleration solution. → The two algorithms are tested with 2D, multi-material problems. → The two algorithms have comparable computational requirements. - Abstract: An existing solution method for solving the multigroup radiation equations, linear multifrequency-grey acceleration, is here extended to be second order in time. This method works for simple diffusion and for flux-limited diffusion, with or without material conduction. A new method is developed that does not require the solution of an averaged grey transport equation. It is effective solving both the diffusion and P 1 forms of the transport equation. Two dimensional, multi-material test problems are used to compare the solution methods.

Further improved F-expansion method and new exact solutions of Konopelchenko-Dubrovsky equation

International Nuclear Information System (INIS)

Wang Dengshan; Zhang Hongqing

2005-01-01

In this paper, with the aid of the symbolic computation we improve the extended F-expansion method in [Chaos, Solitons and Fractals 2004; 22:111] and propose the further improved F-expansion method. Using this method, we have gotten many new exact solutions which we have never seen before within our knowledge of the (2 + 1)-dimensional Konopelchenko-Dubrovsky equation. In addition,the solutions we get are more general than the solutions that the extended F-expansion method gets.The solutions we get include Jacobi elliptic function solutions, soliton-like solutions, trigonometric function solutions and so on. Our method can also apply to other partial differential equations and can also get many new exact solutions

Flat structure and potential vector fields related with algebraic solutions to Painlevé VI equation

Directory of Open Access Journals (Sweden)

Mitsuo Kato

2018-01-01

Full Text Available A potential vector field is a solution of an extended WDVV equation which is a generalization of a WDVV equation. It is expected that potential vector fields corresponding to algebraic solutions of Painlevé VI equation can be written by using polynomials or algebraic functions explicitly. The purpose of this paper is to construct potential vector fields corresponding to more than thirty non-equivalent algebraic solutions.

Formal derivation of a 6 equation macro scale model for two-phase flows - link with the 4 equation macro scale model implemented in Flica 4; Etablissement formel d'un modele diphasique macroscopique a 6 equations - lien avec le modele macroscopique a 4 equations flica 4

Energy Technology Data Exchange (ETDEWEB)

Gregoire, O

2008-07-01

In order to simulate nuclear reactor cores, we presently use the 4 equation model implemented within FLICA4 code. This model is complemented with 2 algebraic closures for thermal disequilibrium and relative velocity between phases. Using such closures, means an 'a priori' knowledge of flows calculated in order to ensure that modelling assumptions apply. In order to improve the degree of universality to our macroscopic modelling, we propose in the report to derive a more general 6 equation model (balance equations for mass, momentum and enthalpy for each phase) for 2-phase flows. We apply the up-scaling procedure (Whitaker, 1999) classically used in porous media analysis to the statistically averaged equations (Aniel-Buchheit et al., 2003). By doing this, we apply the double-averaging procedure (Pedras and De Lemos, 2001 and Pinson et al. 2006): statistical and spatial averages. Then, using weighted averages (analogous to Favre's average) we extend the spatial averaging concept to variable density and 2-phase flows. This approach allows the global recovering of the structure of the systems of equations implemented in industrial codes. Supplementary contributions, such as dispersion, are also highlighted. Mechanical and thermal exchanges between solids and fluid are formally derived. Then, thanks to realistic simplifying assumptions, we show how it is possible to derive the original 4 equation model from the full 6 equation model. (author)

Interval oscillation criteria for second-order forced impulsive delay differential equations with damping term.

Science.gov (United States)

Thandapani, Ethiraju; Kannan, Manju; Pinelas, Sandra

2016-01-01

In this paper, we present some sufficient conditions for the oscillation of all solutions of a second order forced impulsive delay differential equation with damping term. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. The results obtained in this paper extend and generalize some of the the known results for forced impulsive differential equations. An example is provided to illustrate the main result.

Spectrum of the ballooning Schroedinger equation

International Nuclear Information System (INIS)

Dewar, R.L.

1997-01-01

The ballooning Schroedinger equation (BSE) is a model equation for investigating global modes that can, when approximated by a Wentzel-Kramers-Brillouin (WKB) ansatz, be described by a ballooning formalism locally to a field line. This second order differential equation with coefficients periodic in the independent variable θ k is assumed to apply even in cases where simple WKB quantization conditions break down, thus providing an alternative to semiclassical quantization. Also, it provides a test bed for developing more advanced WKB methods: e.g. the apparent discontinuity between quantization formulae for open-quotes trappedclose quotes and open-quotes passingclose quotes modes, whose ray paths have different topologies, is removed by extending the WKB method to include the phenomena of tunnelling and reflection. The BSE is applied to instabilities with shear in the real part of the local frequency, so that the dispersion relation is inherently complex. As the frequency shear is increased, it is found that trapped modes go over to passing modes, reducing the maximum growth rate by averaging over θ k

An Efficient Series Solution for Nonlinear Multiterm Fractional Differential Equations

Directory of Open Access Journals (Sweden)

Moh’d Khier Al-Srihin

2017-01-01

Full Text Available In this paper, we introduce an efficient series solution for a class of nonlinear multiterm fractional differential equations of Caputo type. The approach is a generalization to our recent work for single fractional differential equations. We extend the idea of the Taylor series expansion method to multiterm fractional differential equations, where we overcome the difficulty of computing iterated fractional derivatives, which are difficult to be computed in general. The terms of the series are obtained sequentially using a closed formula, where only integer derivatives have to be computed. Several examples are presented to illustrate the efficiency of the new approach and comparison with the Adomian decomposition method is performed.

Experimental and numerical analyses of different extended surfaces

International Nuclear Information System (INIS)

Diani, A; Mancin, S; Zilio, C; Rossetto, L

2012-01-01

Air is a cheap and safe fluid, widely used in electronic, aerospace and air conditioning applications. Because of its poor heat transfer properties, it always flows through extended surfaces, such as finned surfaces, to enhance the convective heat transfer. In this paper, experimental results are reviewed and numerical studies during air forced convection through extended surfaces are presented. The thermal and hydraulic behaviours of a reference trapezoidal finned surface, experimentally evaluated by present authors in an open-circuit wind tunnel, has been compared with numerical simulations carried out by using the commercial CFD software COMSOL Multiphysics. Once the model has been validated, numerical simulations have been extended to other rectangular finned configurations, in order to study the effects of the fin thickness, fin pitch and fin height on the thermo-hydraulic behaviour of the extended surfaces. Moreover, several pin fin surfaces have been simulated in the same range of operating conditions previously analyzed. Numerical results about heat transfer and pressure drop, for both plain finned and pin fin surfaces, have been compared with empirical correlations from the open literature, and more accurate equations have been developed, proposed, and validated.

Planck constant as spectral parameter in integrable systems and KZB equations

OpenAIRE

Levin, A.NRU HSE, Department of Mathematics, Myasnitskaya str. 20, Moscow, 101000, Russia; Olshanetsky, M.(ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218, Russia); Zotov, A.(ITEP, B. Cheremushkinskaya str. 25, Moscow, 117218, Russia)

2014-01-01

We construct special rational ${\\rm gl}_N$ Knizhnik-Zamolodchikov-Bernard (KZB) equations with $\\tilde N$ punctures by deformation of the corresponding quantum ${\\rm gl}_N$ rational $R$-matrix. They have two parameters. The limit of the first one brings the model to the ordinary rational KZ equation. Another one is $\\tau$. At the level of classical mechanics the deformation parameter $\\tau$ allows to extend the previously obtained modified Gaudin models to the modified Schlesinger systems. Ne...

A Novel Method for Analytical Solutions of Fractional Partial Differential Equations

OpenAIRE

Mehmet Ali Akinlar; Muhammet Kurulay

2013-01-01

A new solution technique for analytical solutions of fractional partial differential equations (FPDEs) is presented. The solutions are expressed as a finite sum of a vector type functional. By employing MAPLE software, it is shown that the solutions might be extended to an arbitrary degree which makes the present method not only different from the others in the literature but also quite efficient. The method is applied to special Bagley-Torvik and Diethelm fractional differential equations as...

A regularization of the Burgers equation using a filtered convective velocity

International Nuclear Information System (INIS)

Norgard, Greg; Mohseni, Kamran

2008-01-01

This paper examines the properties of a regularization of the Burgers equation in one and multiple dimensions using a filtered convective velocity, which we have dubbed as the convectively filtered Burgers (CFB) equation. A physical motivation behind the filtering technique is presented. An existence and uniqueness theorem for multiple dimensions and a general class of filters is proven. Multiple invariants of motion are found for the CFB equation which are shown to be shared with the viscous and inviscid Burgers equations. Traveling wave solutions are found for a general class of filters and are shown to converge to weak solutions of the inviscid Burgers equation with the correct wave speed. Numerical simulations are conducted in 1D and 2D cases where the shock behavior, shock thickness and kinetic energy decay are examined. Energy spectra are also examined and are shown to be related to the smoothness of the solutions. This approach is presented with the hope of being extended to shock regularization of compressible Euler equations

Solitary wave and periodic wave solutions for the thermally forced gravity waves in atmosphere

International Nuclear Information System (INIS)

Li Ziliang

2008-01-01

By introducing a new transformation, a new direct and unified algebraic method for constructing multiple travelling wave solutions of general nonlinear evolution equations is presented and implemented in a computer algebraic system, which extends Fan's direct algebraic method to the case when r > 4. The solutions of a first-order nonlinear ordinary differential equation with a higher degree nonlinear term and Fan's direct algebraic method of obtaining exact solutions to nonlinear partial differential equations are applied to the combined KdV-mKdV-GKdV equation, which is derived from a simple incompressible non-hydrostatic Boussinesq equation with the influence of thermal forcing and is applied to investigate internal gravity waves in the atmosphere. As a result, by taking advantage of the new first-order nonlinear ordinary differential equation with a fifth-degree nonlinear term and an eighth-degree nonlinear term, periodic wave solutions associated with the Jacobin elliptic function and the bell and kink profile solitary wave solutions are obtained under the effect of thermal forcing. Most importantly, the mechanism of propagation and generation of the periodic waves and the solitary waves is analysed in detail according to the values of the heating parameter, which show that the effect of heating in atmosphere helps to excite westerly or easterly propagating periodic internal gravity waves and internal solitary waves in atmosphere, which are affected by the local excitation structures in atmosphere. In addition, as an illustrative sample, the properties of the solitary wave solution and Jacobin periodic solution are shown by some figures under the consideration of heating interaction

Global existence of a generalized solution for the radiative transfer equations

International Nuclear Information System (INIS)

Golse, F.; Perthame, B.

1984-01-01

We prove global existence of a generalized solution of the radiative transfer equations, extending Mercier's result to the case of a layer with an initially cold area. Our Theorem relies on the results of Crandall and Ligett [fr

Dynamics of electrically charged extended bodies: classical and quantum systems

International Nuclear Information System (INIS)

Aaberge, T.

1987-01-01

The author present generalizations of classical mechanics and quantum mechanics that make it possible to describe N charged extended bodies.In particular, we are able to write down a set of coupled equations for the system of N bodies plus field. The theory is based on a theory for the description of N charged chemical fluid components

Classical dynamics of brane-world extended objects

International Nuclear Information System (INIS)

Vasilic, Milovan

2010-01-01

We make use of the universally valid stress-energy conservation law to study the motion of various branelike extended objects in a generic brane-world. Without specifying any particular action, we are able to derive the world-sheet equations that govern the dynamics of brane-world test branes. In particular, the brane-world test particles are shown to follow geodesics with respect to the brane-world induced metric. At the same time, the presence of extended objects is shown to influence the brane-world geometry. It is demonstrated that codimension-1 branes necessarily violate the brane-world smooth structure, while lower-dimensional branes violate the very continuity. In particular, the truly zero-size massive particles are shown not to exist in a continuous brane-world. As an example, static, axially symmetric membrane-world in 4d Minkowski background is analyzed.

On the qualitative theory of the rotating boussinesq and quasi ...

African Journals Online (AJOL)

The equations are derived from conservation laws in continuum physics followed by the formulation of the problem as initial value problem on Hilbert spaces. By using Faedo-Galerkin approximation and semigroup technique, existence and uniqueness of solutions are proved. Additionally, the manuscript outlines how ...

Extended exploding reflector concept for computing prestack traveltimes for waves of different type in the DSR framework

KAUST Repository

Duchkov, Anton A.; Serdyukov, Alexander S.; Alkhalifah, Tariq Ali

2013-01-01

including reflected, head and diving waves. We develop a WENO-RK numerical scheme for solving all mentioned forms of the DSR equation. Finally the extended exploding reflector concept can be used for computing prestack traveltimes while initiating the numerical solver as if a reflector was exploding in extended imaging space.

On extension of solutions of a simultaneous system of iterative functional equations

Directory of Open Access Journals (Sweden)

Janusz Matkowski

2009-01-01

Full Text Available Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form \\[ \\varphi(x = h (x, \\varphi[f_1(x],\\ldots,\\varphi[f_m(x],\\] \\[\\varphi(x = H (x, \\varphi[F_1(x],\\ldots,\\varphi[F_m(x],\\] to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, Iteration groups, commuting functions and simultaneous systems of linear functional equations, Opuscula Math. 28 (2008 4, 531-541].

Differential equation of exospheric lateral transport and its application to terrestrial hydrogen

Science.gov (United States)

Hodges, R. R., Jr.

1973-01-01

The differential equation description of exospheric lateral transport of Hodges and Johnson is reformulated to extend its utility to light gases. Accuracy of the revised equation is established by applying it to terrestrial hydrogen. The resulting global distributions for several static exobase models are shown to be essentially the same as those that have been computed by Quessette using an integral equation approach. The present theory is subsequently used to elucidate the effects of nonzero lateral flow, exobase rotation, and diurnal tidal winds on the hydrogen distribution. Finally it is shown that the differential equation of exospheric transport is analogous to a diffusion equation. Hence it is practical to consider exospheric transport as a continuation of thermospheric diffusion, a concept that alleviates the need for an artificial exobase dividing thermosphere and exosphere.

Extending the Rayleigh equation to allow competing isotope fractionating pathways to improve quantification of biodegradation

NARCIS (Netherlands)

van Breukelen, B.M.

2007-01-01

The Rayleigh equation relates the change in isotope ratio of an element in a substrate to the extent of substrate consumption via a single kinetic isotopic fractionation factor (α). Substrate consumption is, however, commonly distributed over several metabolic pathways each potentially having a

The spectral transform as a tool for solving nonlinear discrete evolution equations

International Nuclear Information System (INIS)

Levi, D.

1979-01-01

In this contribution we study nonlinear differential difference equations which became important to the description of an increasing number of problems in natural science. Difference equations arise for instance in the study of electrical networks, in statistical problems, in queueing problems, in ecological problems, as computer models for differential equations and as models for wave excitation in plasma or vibrations of particles in an anharmonic lattice. We shall first review the passages necessary to solve linear discrete evolution equations by the discrete Fourier transfrom, then, starting from the Zakharov-Shabat discretized eigenvalue, problem, we shall introduce the spectral transform. In the following part we obtain the correlation between the evolution of the potentials and scattering data through the Wronskian technique, giving at the same time many other properties as, for example, the Baecklund transformations. Finally we recover some of the important equations belonging to this class of nonlinear discrete evolution equations and extend the method to equations with n-dependent coefficients. (HJ)

Stochastic population dynamics in spatially extended predator-prey systems

Science.gov (United States)

Dobramysl, Ulrich; Mobilia, Mauro; Pleimling, Michel; Täuber, Uwe C.

2018-02-01

Spatially extended population dynamics models that incorporate demographic noise serve as case studies for the crucial role of fluctuations and correlations in biological systems. Numerical and analytic tools from non-equilibrium statistical physics capture the stochastic kinetics of these complex interacting many-particle systems beyond rate equation approximations. Including spatial structure and stochastic noise in models for predator-prey competition invalidates the neutral Lotka-Volterra population cycles. Stochastic models yield long-lived erratic oscillations stemming from a resonant amplification mechanism. Spatially extended predator-prey systems display noise-stabilized activity fronts that generate persistent correlations. Fluctuation-induced renormalizations of the oscillation parameters can be analyzed perturbatively via a Doi-Peliti field theory mapping of the master equation; related tools allow detailed characterization of extinction pathways. The critical steady-state and non-equilibrium relaxation dynamics at the predator extinction threshold are governed by the directed percolation universality class. Spatial predation rate variability results in more localized clusters, enhancing both competing species’ population densities. Affixing variable interaction rates to individual particles and allowing for trait inheritance subject to mutations induces fast evolutionary dynamics for the rate distributions. Stochastic spatial variants of three-species competition with ‘rock-paper-scissors’ interactions metaphorically describe cyclic dominance. These models illustrate intimate connections between population dynamics and evolutionary game theory, underscore the role of fluctuations to drive populations toward extinction, and demonstrate how space can support species diversity. Two-dimensional cyclic three-species May-Leonard models are characterized by the emergence of spiraling patterns whose properties are elucidated by a mapping onto a complex

3-D resistive MHD calculations for tokamak plasmas: beyond the simple reduced set of equations

International Nuclear Information System (INIS)

Carreras, B.A.; Garcia, L.; Hender, T.C.; Hicks, H.R.; Holmes, J.A.; Lynch, V.E.; Masden, B.F.

1983-01-01

Numerical studies of the resistive stability of tokamak plasmas in cylindrical geometry have been performed using: (1) the full set of resistive Magnetohydrodynamic (MHD) equations and (2) an extended version of the reduced set of resistive MHD equations including diamagnetic and electron temperature effects. In particular, the nonlinear interaction of tearing modes of many helicities has been investigated. The numerical results confirm many of the features uncovered previously using the simple reduced equations. (author)

Numerical integration of the Langevin equation: Monte Carlo simulation

International Nuclear Information System (INIS)

Ermak, D.L.; Buckholz, H.

1980-01-01

Monte Carlo simulation techniques are derived for solving the ordinary Langevin equation of motion for a Brownian particle in the presence of an external force. These methods allow considerable freedom in selecting the size of the time step, which is restricted only by the rate of change in the external force. This approach is extended to the generalized Langevin equation which uses a memory function in the friction force term. General simulation techniques are derived which are independent of the form of the memory function. A special method requiring less storage space is presented for the case of the exponential memory function

An efficient technique for higher order fractional differential equation.

Science.gov (United States)

Ali, Ayyaz; Iqbal, Muhammad Asad; Ul-Hassan, Qazi Mahmood; Ahmad, Jamshad; Mohyud-Din, Syed Tauseef

2016-01-01

In this study, we establish exact solutions of fractional Kawahara equation by using the idea of [Formula: see text]-expansion method. The results of different studies show that the method is very effective and can be used as an alternative for finding exact solutions of nonlinear evolution equations (NLEEs) in mathematical physics. The solitary wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. Graphical representations along with the numerical data reinforce the efficacy of the used procedure. The specified idea is very effective, expedient for fractional PDEs, and could be extended to other physical problems.

An equations of motion approach for open shell systems

International Nuclear Information System (INIS)

Yeager, D.L.; McKoy, V.

1975-01-01

A straightforward scheme is developed for extending the equations of motion formalism to systems with simple open shell ground states. Equations for open shell random phase approximation (RPA) are given for the cases of one electron outside of a closed shell in a nondegenerate molecular orbital and for the triplet ground state with two electrons outside of a closed shell in degenerate molecular orbitals. Applications to other open shells and extension of the open shell EOM to higher orders are both straightforward. Results for the open shell RPA for lithium atom and oxygen molecule are given

Investigation of two and three parameter equations of state for cryogenic fluids

International Nuclear Information System (INIS)

Jenkins, S.L.; Majumdar, A.K.; Hendricks, R.C.