共查询到20条相似文献,搜索用时 31 毫秒
1.
Algebraic dynamics approach and algebraic dynamics algorithm for the solution of nonlinear partial differential equations
are applied to the nonlinear advection equation. The results show that the approach is effective for the exact analytical
solution and the algorithm has higher precision than other existing algorithms in numerical computation for the nonlinear
advection equation.
Supported by the National Natural Science Foundation of China (Grant Nos. 90503008 and 10775100), the Doctoral Program Foundation
from the Ministry of Education of China, and the Center of Theoretical Nuclear Physics of HIRFL of China 相似文献
2.
Algebraic dynamics solutions and algebraic dynamics algorithm for nonlinear ordinary differential equations 总被引:2,自引:2,他引:0
The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms
of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary
differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics,
and a new algorithm—algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential
equations by the algebraic dynamics method. In the new algorithm, the time evolution of the ordinary differential system is
described locally by the time translation operator and globally by the time evolution operator. The exact analytical piece-like
solution of the ordinary differential equations is expressed in terms of Taylor series with a local convergent radius, and
its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm
and Symplectic Geometric Algorithm. 相似文献
3.
WANG ShunJin & ZHANG Hua Center of Theoretical Physics Sichuan University Chengdu China 《中国科学G辑(英文版)》2007,50(1):53-69
Based on the exact analytical solution of ordinary differential equations, a truncation of the Taylor series of the exact solution to the Nth order leads to the Nth order algebraic dynamics algorithm. A detailed numerical comparison is presented with Runge-Kutta algorithm and symplectic geometric algorithm for 12 test models. The results show that the algebraic dynamics algorithm can better preserve both geometrical and dynamical fidelity of a dynamical system at a controllable precision, and it can solve the problem of algorithm-induced dissipation for the Runge-Kutta algorithm and the problem of algorithm-induced phase shift for the symplectic geometric algorithm. 相似文献
4.
We investigate an operator renormalization group method to extract and describe the relevant degrees of freedom in the evolution of partial differential equations. The proposed renormalization group approach is formulated as an analytical method providing the fundamental concepts of a numerical algorithm applicable to various dynamical systems. We examine dynamical scaling characteristics in the short-time and the long-time evolution regime providing only a reduced number of degrees of freedom to the evolution process. 相似文献
5.
BAI Cheng-Lin 《理论物理通讯》2003,40(8)
For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability. 相似文献
6.
Huiqun Zhang 《Reports on Mathematical Physics》2007,60(1):97-106
A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained. 相似文献
7.
An extended Fan's algebraic method is used for constructing exact traveling wave solution of nonlinear partial differential equations. The key idea of this method is to introduce an auxiliary ordinary differential equation which is regarded as an extended elliptic equation and
whose degree r is expanded to the case of r>4. The efficiency of the
method is demonstrated by the KdV equation and the variant Boussinesq
equations. The results indicate that the method not only offers all
solutions obtained by using Fu's and Fan's methods, but also some new solutions. 相似文献
8.
9.
We extend techniques developed for the study of turbulent fluid flows to the statistical study of the dynamics of differential delay equations. Because the phase spaces of differential delay equations are infinite dimensional, phase-space densities for these systems are functionals. We derive a Hopf-like functional differential equation governing the evolution of these densities. The functional differential equation is reduced to an infinite chain of linear partial differential equations using perturbation theory. A necessary condition for a measure to be invariant under the action of a nonlinear differential delay equation is given. Finally, we show that the evolution equation for the density functional is the Fourier transform of the infinite-dimensional version of the Kramers-Moyal expansion. 相似文献
10.
A Laplace decomposition algorithm is adopted to investigate numerical solutions of a class of nonlinear partial differential equations with nonlinear term of any order, utt + auxx + bu + cup + du^2p-1 = 0, which contains some important equations of mathematical physics. Three distinct initial conditions are constructed and generalized numerical solutions are thereby obtained, including numerical hyperbolic function solutions and doubly periodic ones. Illustrative figures and comparisons between the numerical and exact solutions with different values of p are used to test the efficiency of the proposed method, which shows good results are azhieved. 相似文献
11.
A lattice Boltzmann model with an amending function forsimulating nonlinear partial differential equations 下载免费PDF全文
This paper proposes a lattice Boltzmann model with an
amending function for one-dimensional nonlinear partial
differential equations (NPDEs) in the form $u_t+\alpha uu_{xx}+\beta u^n u_x+\gamma u_{xxx}+\xi u_{xxxx}=0$. This model is
different from existing models because it lets the time step
be equivalent to the square of the space step and derives higher
accuracy and nonlinear terms in NPDEs. With the Chapman--Enskog
expansion, the governing evolution equation is recovered correctly
from the continuous Boltzmann equation. The numerical results
agree well with the analytical solutions. 相似文献
12.
A New Rational Algebraic Approach to Find Exact Analytical Solutions to a (2+1)-Dimensional System 总被引:1,自引:0,他引:1
BAI Cheng-Jie ZHAO Hong 《理论物理通讯》2007,48(5):801-810
In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on (2+1)-dimensional dispersive long-wave equation. 相似文献
13.
A new expansion method of first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term and its application to mBBM model 下载免费PDF全文
Based on a first order nonlinear ordinary differential equation with at most a sixth-degree nonlinear term which is extended from a type of elliptic equation, and by converting it into a new expansion form, this paper proposes a new algebraic method to construct exact solutions for nonlinear evolution equations. Being concise and straightforward, the method is applied to modified Benjamin-Bona-Mahony (mBBM) model, and some new exact solutions to the system are obtained. The algorithm is of important significance in exploring exact solutions for other nonlinear evolution equations. 相似文献
14.
LI Hua-Mei 《理论物理通讯》2003,39(4):395-400
In this paper, we extend the mapping deformation method proposed by Lou. It is used to find new exacttravelling wave solutions of nonlinear partial differential equation or coupled nonlinear partial differential equations(PDEs). Based on the idea of the homogeneous balance method, we construct the general mapping relation betweenthe solutions of the PDEs and those of the cubic nonlinear Klein-Gordon (NKG) equation. By using this relation andthe abundant solutions of the cubic NKG equation, many explicit and exact travelling wave solutions of three systemsof coupled PDEs, which contain solitary wave solutions, trigonometric function solutions, Jacobian elliptic functionsolutions, and rational solutions, are obtained. 相似文献
15.
Zhi Li & Paul Muir 《advances in applied mathematics and mechanics.》2013,5(4):528-547
In this paper we describe new B-spline Gaussian collocation software for solving two-dimensional
parabolic partial differential equations (PDEs) defined over a rectangular region. The numerical solution is represented as a bi-variate piecewise polynomial (using a tensor product
B-spline basis) with time-dependent unknown coefficients. These coefficients are determined by
imposing collocation conditions: the numerical solution is required to satisfy the PDE
and boundary conditions at images of the Gauss points mapped onto certain subregions of the spatial domain.
This leads to a large system of time-dependent
differential algebraic equations (DAEs) which is solved using the DAE solver, DASPK. We provide numerical results in which we use the new software, called BACOL2D, to solve three test problems. 相似文献
16.
A new variable coefficient algebraic method and non-travelling wave solutions of nonlinear equations 下载免费PDF全文
In this paper, a new auxiliary equation method is presented of constructing more new non-travelling wave solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the validity and the advantages of the method, (2+1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation is employed and many new double periodic non-travelling wave solutions are obtained. This algorithm can also be applied to other nonlinear differential equations. 相似文献
17.
Motivated by the widely used ansätz method and starting from the modified Riemann-Liouville derivative together with a fractional complex transformation that can be utilized to transform nonlinear fractional partial differential equations to nonlinear ordinary differential equations, new types of exact traveling wave solutions to three important nonlinear space- and time-fractional partial differential equations are obtained simultaneously in terms of solutions of a Riccati equation. The results are new and first reported in this paper. 相似文献
18.
The Method of Fundamental Solutions for Solving Convection-Diffusion Equations with Variable Coefficients 下载免费PDF全文
A meshless method based on the method of fundamental solutions (MFS)
is proposed to solve the time-dependent partial differential equations with variable
coefficients. The proposed method combines the time discretization and the one-stage
MFS for spatial discretization. In contrast to the traditional two-stage process,
the one-stage MFS approach is capable of solving a broad spectrum of partial differential
equations. The numerical implementation is simple since both closed-form
approximate particular solution and fundamental solution are easier to find than the
traditional approach. The numerical results show that the one-stage approach is
robust and stable. 相似文献
19.
P. Thum T. Clees G. Weyns G. Nelissen J. Deconinck 《Journal of computational physics》2010,229(19):7260-7276
The article discusses components and performance of an algebraic multigrid (AMG) preconditioner for the fully coupled multi-ion transport and reaction model (MITReM) with nonlinear boundary conditions, important for electrochemical modeling. The governing partial differential equations (PDEs) are discretized in space by a combined finite element and residual distribution method. Solution of the discrete system is obtained by means of a Newton-based nonlinear solver, and an AMG-preconditioned BICGSTAB Krylov linear solver. The presented AMG preconditioner is based on so-called point-based classical AMG. The linear solver is compared to a standard direct and several one-level iterative solvers for a range of geometries and chemical systems with scientific and industrial relevance. The results indicate that point-based AMG methods, carefully designed, are an attractive alternative to more commonly employed numerical methods for the simulation of complex electrochemical processes. 相似文献
20.
In this paper, extended projective Riccati equation method is presented for constructing more new exact solutions of nonlinear differential equations in mathematical physics, which is direct and more powerful than projective Riccati equation method. In order to illustrate the effect of the method, Broer-Kaup-Kupershmidt
system is employed and Jacobi doubly periodic solutions are obtained.
This algorithm can also be applied to other nonlinear differential equations. 相似文献