共查询到20条相似文献,搜索用时 11 毫秒
1.
Tao Zhou 《advances in applied mathematics and mechanics.》2014,6(4):403-418
In this work, we concern with the numerical approach for delay differential equations
with random coefficients. We first show that the exact solution of the problem considered admits good regularity in the random space, provided that the given data satisfy some reasonable assumptions. A stochastic collocation method is proposed to approximate the solution in the random space, and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations. Convergence property of the proposed method is analyzed. It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space. Numerical examples are given to illustrate the theoretical results. 相似文献
2.
Manfred Opper 《Annalen der Physik》2019,531(3)
The statistical inference of the state variable and the drift function of stochastic differential equations (SDE) from sparsely sampled observations are discussed herein. A variational approach is used to approximate the distribution over the unknown path of the SDE conditioned on the observations. This approach also provides approximations for the intractable likelihood of the drift. The method is combined with a nonparametric Bayesian approach which is based on a Gaussian process prior over drift functions. 相似文献
3.
We consider a physical system which is coupled indirectly to a Markovian resevoir through an oscillator mode. This is the
case, for example, in the usual model of an atomic sample in a leaky optical cavity which is ubiquitous in quantum optics.
In the strong coupling limit the oscillator can be eliminated entirely from the model, leaving an effective direct coupling
between the system an the resevoir. Here we provide a mathematically rigorous treatment of this limit as a weak limit of the
time evolution and observables on a suitably chosen exponential domain in Fock space. The resulting effective model may contain
emission and absorption as well as scattering interactions.
R.v.H. is supported by the Army Research Office under Grant W911NF-06-1-0378. 相似文献
4.
We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example. 相似文献
5.
In this paper, the asymptotical p-moment stabifity of stochastic impulsive differential equations is studied, and a comparison theory to ensure the asymptotieal p-moment stability for trivial solution of this system is established, from which we can find out whether a stochastic impulsive differential system is stable just from a deterministic comparison system. As an application of this theory, we control the chaos of stochastic Chen system using impulsive method, and a stable region is deduced too. Finally, numerical simulations verify the feasibility of our method. 相似文献
6.
A general and easy-to-code numerical method based on radial basis functions
(RBFs) collocation is proposed for the solution of delay differential equations
(DDEs). It relies on the interpolation properties of infinitely smooth RBFs, which allow
for a large accuracy over a scattered and relatively small discretization support.
Hardy's multiquadric is chosen as RBF and combined with the Residual Subsampling
Algorithm of Driscoll and Heryudono for support adaptivity. The performance
of the method is very satisfactory, as demonstrated over a cross-section of
benchmark DDEs, and by comparison with existing general-purpose and specialized
numerical schemes for DDEs. 相似文献
7.
R. Sakthivel 《Reports on Mathematical Physics》2011,68(2):163-174
8.
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
9.
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
10.
In this work, a unified lattice Boltzmann model is proposed for the fourth order partial differential equation with time-dependent variable coefficients, which has the form . A compensation function is added to the evolution equation to recover the macroscopic equation. Applying Chapman-Enskog expansion and the Taylor expansion method, we recover the macroscopic equation correctly. Through analyzing the error, our model reaches second-order accuracy in time. A series of constant-coefficient and variable-coefficient partial differential equations are successfully simulated, which tests the effectiveness and stability of the present model. 相似文献
11.
More new exact solutions for a class of nonlinear coupled
differential equations are obtained by using a direct and efficient
hyperbola function transform method based on the idea of the extended
homogeneous balance method. 相似文献
12.
FENG Qing-Hua 《理论物理通讯》2014,62(2):167-172
In this paper, a new fractional projective Riccati equation method is proposed to establish exact solutions for fractional partial differential equations in the sense of modified Riemann—Liouville derivative. This method can be seen as the fractional version of the known projective Riccati equation method. For illustrating the validity of this method, we apply this method to solve the space—time fractional Whitham—Broer—Kaup (WBK) equations and the nonlinear fractional Sharma—Tasso—Olever (STO) equation, and as a result, some new exact solutions for them are obtained. 相似文献
13.
构造一类求解三种类型偏微分方程的间断Petrov-Galerkin方法.求解的方程分别含有二阶、三阶和四阶偏导数,包括Burgers型方程、KdV型方程和双调和型方程.首先将高阶微分方程转化成为与之等价的一阶微分方程组,再将求解双曲守恒律的间断Petrov-Galerkin方法用于求解微分方程组.该方法具有四阶精度且具有间断Petrov-Galerkin方法的优点.数值实验表明该方法可以达到最优收敛阶而且可以模拟复杂波形相互作用,如孤立子的传播及相互碰撞等. 相似文献
14.
In this work, by means of a new
more general ansatz and the symbolic computation system Maple, we extend the Riccati equation rational expansion method [Chaos, Solitons & Fractals
25 (2005) 1019] to
uniformly construct a series of stochastic nontravelling wave
solutions for nonlinear stochastic evolution equation. To illustrate
the effectiveness of our method, we take the stochastic mKdV
equation as an example, and successfully construct some new and more
general solutions including a series of rational formal nontraveling
wave and coefficient functions' soliton-like solutions and
trigonometric-like function solutions. The method can also be
applied to solve other nonlinear stochastic evolution equation or equations. 相似文献
15.
F. Bagarello 《International Journal of Theoretical Physics》2005,44(8):1193-1216
We adapt ideas coming from Quantum Mechanics to develop a non-commutative strategy for the analysis of some systems of ordinary
differential equations. We show that the solution of such a system can be described by an unbounded, self-adjoint and densely
defined operator H which we call, in analogy with Quantum Mechanics, the Hamiltonian of the system.
We discuss the role of H in the analysis of the integrals of motion of the system. Finally, we apply this approach to several examples.
Pacs Numbers: 02.30.Hq, 03.65.-w, 03.65.Db 相似文献
16.
We adapt ideas coming from Quantum Mechanics to develop a non-commutative strategy for the analysis of some systems of ordinary differential equations. We show that the solution of such a system can be described by an unbounded, self-adjoint and densely defined operator H which we call, in analogy with Quantum Mechanics, the Hamiltonian of the system.We discuss the role of H in the analysis of the integrals of motion of the system. Finally, we apply this approach to several examples. 相似文献
17.
In this paper, the (G'/G)-expansion method is extended to solve fractional partial differential equations in the sense of modified Riemann-Liouville derivative. Based on a nonlinear fractional complex transformation, a certain fractional partial differential equation can be turned into another ordinary differential equation of integer order. For illustrating the validity of this method, we apply it to the space-time fractional generalized Hirota-Satsuma coupled KdV equations and the time-fractional fifth-order Sawada-Kotera equation. As a result, some new exact solutions for them are successfully established. 相似文献
18.
19.
Homogeneous balance method for solving nonlinear partial differential equation(s) is extended to solving initial-value problem and getting new solution(s) from a known solution of the equation(s) under consideration. The approximate equations for long water waves are chosen to illustrate the method, infinitely many simple-solitary-wave solutions and infinitely many rational function solutions, especially the closed form of the solution for initial-value problem, are obtained by using the extended homogeneous balance method given here. 相似文献
20.
A. H. Khater D. K. Callebaut S. M. Sayed 《International Journal of Theoretical Physics》2006,45(3):589-616
An effective algorithmic method (Anco, S. C. and Bluman, G. (1996). Journal of Mathematical Physics 37, 2361; Anco, S. C. and Bluman, G. (1997). Physical Review Letters 78, 2869; Anco, S. C. and Bluman, G. (1998). European Journal of Applied Mathematics 9, 254; Anco, S. C. and Bluman, G. (2001). European Journal of Applied Mathematics 13, 547; Anco, S. C. and Bluman, G. (2002). European Journal of Applied Mathematics 13, 567 is used for finding the local conservation laws for some nonlinear partial differential equations. The method does not require the use or existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that of finding symmetries. An explicit construction formula is derived which yields a conservation law for each solution of the determining system. Different methods to construct new exact solution classes for the same nonlinear partial differential equations are also presented, which are named hyperbolic function method and the Bäcklund transformations. On the other hand, other methods and transformations are developed to obtain exact solutions for some nonlinear partial differential equations. 相似文献