求解非线性时滞脉冲双曲微分方程的隐式差分格式

构造了一类求解非线性时滞脉冲双曲型偏微分方程的隐式差分格式.在一定条件下,获得了该差分格式的唯一可解性、收敛性和无条件稳定性,且空间和时间均二阶精度.最后,数值实验表明了所得格式的精度和有效性.     

Mean-square stability of semi-implicit Euler method for nonlinear neutral stochastic delay differential equations

There are few results on the numerical stability of nonlinear neutral stochastic delay differential equations (NSDDEs). The aim of this paper is to establish some new results on the numerical stability for nonlinear NSDDEs. It is proved that the semi-implicit Euler method is mean-square stable under suitable condition. The theoretical result is also confirmed by a numerical experiment.     

Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise

In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed.     

An implicit Euler scheme with non-uniform time discretization for heat equations with multiplicative noise

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion W. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of W. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of W. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations. AMS subject classification (2000)  60H15, 60H35, 65C30     

T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise

The paper deals with the T-stability of the semi-implicit Euler method for delay differential equations with multiplicative noise. A difference equation is obtained by applying the numerical method to a linear test equation, in which the Wiener increment is approximated by a discrete random variable with two-point distribution. The conditions under which the method is T-stable are considered and the numerical experiments are given.     

Degenerate parabolic stochastic partial differential equations

We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

Higher-order stochastic partial differential equations with branching noises

In this paper, we propose a class of higher-order stochastic partial differential equations (SPDEs) with branching noises. The existence of weak (mild) solutions is established through weak convergence and tightness arguments.      

Exact solutions for nonlinear partial differential equations via Exp‐function method

In this article, the Exp‐function method is applied to nonlinear Burgers equation and special fifth‐order partial differential equation. Using this method, we obtain exact solutions for these equations. The method is straightforward and concise, and its applications are promising. This method can be used as an alternative to obtain analytical and approximate solutions of different types of nonlinear differential equations. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

延迟微分方程隐式Euler方法的收敛性

本文讨论了用隐式Euler方法求解一类延迟量满足Lipschitz条件且Lipschitz常数小于1的非线性变延迟微分方程初值问题的收敛性.获得了带线性插值的隐式Euler方法的收敛性结果.     

A link of stochastic differential equations to nonlinear parabolic equations

Using Girsanov transformation,we derive a new link from stochastic differential equations of Markovian type to nonlinear parabolic equations of Burgers-KPZ type,in such a manner that the obtained BurgersKPZ equation characterizes the path-independence property of the density process of Girsanov transformation for the stochastic differential equation.Our assertion also holds for SDEs on a connected differential manifold.     

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction–diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier–Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.     

A numerical algorithm for the solution of nonlinear fractional differential equations via beta‐derivatives

In this paper, the sinc‐collocation method (SCM) is investigated to obtain the solution of the nonlinear fractional order differential equations based on the relatively new defined fractional derivative, beta‐derivative. For this purpose, a theorem is proved for the approximate solution obtained from the SCM. Moreover, convergence analysis of the SCM is presented. To show the efficiency and the simplicity of the proposed method, some examples are solved, and the results are compared with the exact solutions of the considered equations.     

Strong convergence rate for multivalued stochastic differential equations via stochastic theta method

In this paper we study the stochastic theta method for multivalued stochastic differential equations driven by standard Brownian motions and obtain the strong convergence rate of this numerical scheme.     

Multiscale modeling and analysis for some special additive noises driven stochastic partial differential equations

This paper is concerned with some special additive noises driven stochastic partial differential equations with multiscale parameters. Then, the constraint energy minimizing generalized multiscale finite element method with a novel multiscale spectral representation of the noise is constructed to solve the multiscale models. The corresponding convergence analysis and error estimates are derived, and the effects of noises on the accuracy of the multiscale computation are demonstrated. Some numerical examples are provided to validate our theoretic analysis, and numerical results show the highly efficient computational performance of our method, which is a beneficial attempt to deal with the noises in the complex multiscale stochastic physical system.     

Efficient hybrid method for solving special type of nonlinear partial differential equations

In this article, an efficient hybrid method has been developed for solving some special type of nonlinear partial differential equations. Hybrid method is based on tanh–coth method, quasilinearization technique and Haar wavelet method. Nonlinear partial differential equations have been converted into a nonlinear ordinary differential equation by choosing some suitable variable transformations. Quasilinearization technique is used to linearize the nonlinear ordinary differential equation and then the Haar wavelet method is applied to linearized ordinary differential equation. A tanh–coth method has been used to obtain the exact solutions of nonlinear ordinary differential equations. It is easier to handle nonlinear ordinary differential equations in comparison to nonlinear partial differential equations. A distinct feature of the proposed method is their simple applicability in a variety of two‐ and three‐dimensional nonlinear partial differential equations. Numerical examples show better accuracy of the proposed method as compared with the methods described in past. Error analysis and stability of the proposed method have been discussed.     

A numerical scheme based on Bernoulli wavelets and collocation method for solving fractional partial differential equations with Dirichlet boundary conditions

In this paper, an efficient and accurate numerical method is presented for solving two types of fractional partial differential equations. The fractional derivative is described in the Caputo sense. Our approach is based on Bernoulli wavelets collocation techniques together with the fractional integral operator, described in the Riemann‐Liouville sense. The main characteristic behind this approach is to reduce such problems to those of solving systems of algebraic equations, which greatly simplifies the problem. By using Newton's iterative method, this system is solved and the solution of fractional partial differential equations is achieved. Some results concerning the error analysis are obtained. The validity and applicability of the method are demonstrated by solving four numerical examples. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions much easier.