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1.
Rohollah Bakhshandeh‐Chamazkoti 《Mathematical Methods in the Applied Sciences》2016,39(12):3163-3172
In this research article, a complete analysis of symmetries and conservation laws for the charged squashed Kaluza–Klein black hole space‐time in a Riemannian space is discussed. First, a comprehensive group analysis of the underlying space‐time metric using Lie point symmetries is presented, and then the n‐dimensional optimal system of this space‐time metric, for n = 1,…,4, are computed. It is shown that there is no any n‐dimensional optimal system of Lie symmetry subalgebra associated to the system of geodesic for n≥5. Then the point symmetries of the one‐parameter Lie groups of transformations that leave invariant the action integral corresponding to the Lagrangian that means Noether symmetries are found, and then the conservation laws associated to the system of geodesic equations are calculated via Noether's theorem. Copyright © 2015 John Wiley & Sons, Ltd. 相似文献
2.
Bruno Després 《Numerische Mathematik》2001,89(1):99-134
Summary. We study the mathematical structure of 1D systems of conservation laws written in the Lagrange variable. Modifying the symmetrization
proof of systems of conservation laws with three hypothesis, we prove that these models have a canonical formalism. These
hypothesis are i) the entropy flux is zero, ii) Galilean invariance, iii) reversibility for smooth solutions. Then we study
a family of numerical schemes for the solution of these systems. We prove that they are entropy consistent. We also prove
from general considerations the symmetry of the spectrum of the Jacobian matrix.
Received December 15, 1999 / Published online February 5, 2001 相似文献
3.
We study higher-order conservation laws of the nonlinearizable elliptic Poisson equation as elements of the characteristic
cohomology of the associated exterior differential system. The theory of characteristic cohomology determines a normal form
for differentiated conservation laws by realizing them as elements of the kernel of a linear differential operator. We show
that the
\mathbbS1{\mathbb{S}^1} -symmetry of the PDE leads to a normal form for the undifferentiated conservation laws. Zhiber and Shabat (in Sov Phys Dokl
Akad 24(8):607–609, 1979) determine which potentials of nonlinearizable Poisson equations admit nontrivial Lie–B?cklund transformations. In the case
that such transformations exist, they introduce a pseudo-differential operator that can be used to generate infinitely many
such transformations. We obtain similar results using the theory of characteristic cohomology: we show that for higher-order
conservation laws to exist, it is necessary that the potential satisfies a linear second-order ODE. In this case, at most
two new conservation laws in normal form appear at each even prolongation. By using a recursion motivated by Killing fields,
we show that, for the simplest class of potentials, this upper bound is attained. The recursion circumvents the use of pseudo-differential
operators. We relate higher-order conservation laws to generalized symmetries of the exterior differential system by identifying
their generating functions. This Noether correspondence provides the connection between conservation laws and the canonical
Jacobi fields of Pinkall and Sterling. 相似文献
4.
We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional
second-order parabolic equations. The group classification of this class is revised by employing admissible transformations,
the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation
laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class
the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria
for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different
representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation
of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the
tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate
our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated
with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series
of potential symmetry algebras are studied in detail. 相似文献
5.
A class of partial differential equations (a conservation law and four balance laws), with four independent variables and involving sixteen arbitrary continuously differentiable functions, is considered in the framework of equivalence transformations. These are point transformations of differential equations involving arbitrary elements and live in an augmented space of independent, dependent and additional variables representing values taken by the arbitrary elements. Projecting the admitted infinitesimal equivalence transformations into the space of independent and dependent variables, we determine some finite transformations mapping the system of balance laws to an equivalent one with the same differential structure but involving different arbitrary elements; in particular, the target system we want to recover is an autonomous system of conservation laws. An application to a physical problem is considered. 相似文献
6.
We deal with a Hamilton-Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf-Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton-Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton-Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex-concave case we cannot expect always an existence of a solution satisfying Rankine-Hugoniot condition across the interface. Therefore the concept of generalised Rankine-Hugoniot condition is introduced and prove existence and uniqueness. 相似文献
7.
Arash Yavari Jerrold E. Marsden 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,133(1):723-738
This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles
under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case
of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However,
the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws.
These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case
of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends
on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial
diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics
at their end points. One also obtains a discrete version of the Doyle–Ericksen formula, which relates the magnitude of internal
forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background
metric. 相似文献
8.
Hua-zhong Tang Hua-mu Wu 《计算数学(英文版)》2001,(3)
1. IntroductionWe are illterested in the numerical approximation of viscosity solution of the following lirstorder Hamilton-Jacobi eqllationopt + H(&.,, rk..' ...) gb'.) - 0' (1'1)with initial data ac(~, 0) = ado(x). It is well known that the solutions to problem (1.1) typicallyare continuous (typically they are locally Lipschitz continuous) but with discoatinuous derivatives, even though the initial data ado E Coo. The nonuniqueness of such solutions to (1.1) alsonecessitates the introducti… 相似文献
9.
We classify zeroth-order conservation laws of systems from the class of two-dimensional shallow water equations with variable bottom topography using an optimized version of the method of furcate splitting. The classification is carried out up to equivalence generated by the equivalence group of this class. We find additional point equivalences between some of the listed cases of extensions of the space of zeroth-order conservation laws, which are inequivalent up to transformations from the equivalence group. Hamiltonian structures of systems of shallow water equations are used for relating the classification of zeroth-order conservation laws of these systems to the classification of their Lie symmetries. We also construct generating sets of such conservation laws under action of Lie symmetries. 相似文献
10.
Arash Yavari Jerrold E. Marsden 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2009,60(4):723-738
This paper studies the principle of invariance of balance of energy and its consequences for a system of interacting particles
under groups of transformations. Balance of energy and its invariance is first examined in Euclidean space. Unlike the case
of continuous media, it is shown that conservation and balance laws do not follow from the assumption of invariance of balance of energy under time-dependent isometries of the ambient space. However,
the postulate of invariance of balance of energy under arbitrary diffeomorphisms of the ambient (Euclidean) space, does yield the correct conservation and balance laws.
These ideas are then extended to the case when the ambient space is a Riemannian manifold. Pairwise interactions in the case
of geodesically complete Riemannian ambient manifolds are defined by assuming that the interaction potential explicitly depends
on the pairwise distances of particles. Postulating balance of energy and its invariance under arbitrary time-dependent spatial
diffeomorphisms yields balance of linear momentum. It is seen that pairwise forces are directed along tangents to geodesics
at their end points. One also obtains a discrete version of the Doyle–Ericksen formula, which relates the magnitude of internal
forces to the rate of change of the interatomic energy with respect to a discrete metric that is related to the background
metric.
Dedicated to the memory of Shahram Kavianpour (1975-2007).
Jerrold E. Marsden: Research partially supported by the California Institute of Technology and NSF-ITR Grant ACI-0204932.
Arash Yavari: Research supported by the Georgia Institute of Technology. 相似文献
11.
Pengzi Miao Luen-Fai Tam 《Calculus of Variations and Partial Differential Equations》2009,36(2):141-171
We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive
a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on
which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary
can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of
a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two
volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation
formula and apply it to show that, on Euclidean balls and “small” hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional. 相似文献
12.
Truyen Nguyen 《偏微分方程通讯》2015,40(9):1619-1665
A general global existence result for one-dimensional pressureless Euler/Euler-Poisson systems with or without viscosity is obtained by employing the “sticky particles” model. We first construct entropy solutions for some appropriate scalar conservation laws, then we show that these solutions encode all the information necessary to obtain solutions for the pressureless systems. Another novel contribution is the stability and uniqueness of solutions, which is obtained via a contraction principle in the Wasserstein metric. 相似文献
13.
集值度量广义逆的存在性 总被引:2,自引:2,他引:0
设X,Y为Banach空间,T∈L(X,Y)为从X到Y的线性算子,D(T),N(T),R(T)分别为T的定义域,核空间与值域,使用算子T的自身性质,给出T具有集值度量广义逆T和R(T)D(T)的充分必要条件. 相似文献
14.
<正> 序言.在同倫論中,常常需要考慮滿足這種性質的拓撲空間X設Y為任意的一個正規空間,B為Y的任何一個非空閉集,任何一個由B×(0,1)+Y×(0)到X的映像都可以扩充為一個由Y×(0,1)到X的映像,我們稱這種性質為絕對同倫扩充性質,具有這種性質的空間以及用AHE表示.Borsuk曾經介紹這樣一個重要的定理: 相似文献
15.
A. I. Bashtanov 《Mathematical Notes》2013,93(1-2):209-216
In 2007, S. V. Tikhonov introduced a complete metric on the space of mixing transformations. This metric generates a topology called the leash topology. Tikhonov posed the following problem: what conditions should be satisfied by a mixing transformation T for its conjugacy class to be dense in the space of mixing transformations equipped with the leash topology. We show the conjugacy class to be dense for every mixing transformation T. As a corollary, we find that a generic mixing transformation is rank 1. 相似文献
16.
We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods. 相似文献
17.
In this paper, we introduce a G metric on the G-cone metric space and then prove that a complete G-cone metric space is always a complete G metric space and verify that a contractive mapping on the G-cone metric space is a contractive mapping on the G metric space. At last, we also give a new way to obtain the unique fixed point on G-cone metric space. 相似文献
18.
《Nonlinear Analysis: Theory, Methods & Applications》2010,72(12):e138-e146
We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387–409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods. 相似文献
19.
Mapundi Banda Mohammed Seaïd 《Numerical Methods for Partial Differential Equations》2007,23(5):1211-1234
We present a class of high‐order weighted essentially nonoscillatory (WENO) reconstructions based on relaxation approximation of hyperbolic systems of conservation laws. The main advantage of combining the WENO schemes with relaxation approximation is the fact that the presented schemes avoid solution of the Riemann problems due to the relaxation approach and high‐resolution is obtained by applying the WENO approach. The emphasis is on a fifth‐order scheme and its performance for solving a wide class of systems of conservation laws. To show the effectiveness of these methods, we present numerical results for different test problems on multidimensional hyperbolic systems of conservation laws. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 相似文献
20.
V. R. Krym 《Theoretical and Mathematical Physics》1999,119(3):811-820
A new model of gravitational and electromagnetic interactions is constructed as a version of the classical Kaluza-Klein theory
based on a five-dimensional manifold as the physical space-time. The velocity space of moving particles in the model remains
four-dimensional as in the standard relativity theory. The spaces of particle velocities constitute a four-dimensional distribution
over a smooth five-dimensional manifold. This distribution depends only on the electromagnetic field and is independent of
the metric tensor field. We prove that the equations for the geodesics whose velocity vectors always belong to this distribution
are the same as the charged particle equations of motion in the general relativity theory. The gauge transformations are interpreted
in geometric terms as a particular form of coordinate transformations on the five-dimensional manifold.
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 517–528, June, 1999. 相似文献