Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations

Numerical Algorithms - The existence and uniqueness of the stationary distribution of the numerical solution generated by the stochastic theta method are studied. When the parameter theta takes...     

An approximate method via Taylor series for stochastic functional differential equations

The subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals, sufficiently smooth in the sense of Fréchet derivatives. The approximate equations are defined on equidistant partitions of the time interval, and their coefficients are general Taylor expansions of the coefficients of the initial equation. It will be shown that the approximate solutions converge in the Lp-norm and with probability one to the solution of the initial equation, and also that the rate of convergence increases when degrees in Taylor expansions increase, analogously to real analysis.     

A method for solution of nonlinear ordinary differential equations

We present a method for the solution of nonlinear second-order differential equations by using a system of Fredholm equations of the second kind.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1254–1260, September, 1995.     

Local Linearization method for the numerical solution of stochastic differential equations

The Local Linearization (LL) approach for the numerical solution of stochastic differential equations (SDEs) is extended to general scalar SDEs, as well as to non-autonomous multidimensional SDEs with additive noise. In case of autonomous SDEs, the derivation of the method introduced gives theoretical support to one of the previously proposed variants of the LL approach. Some numerical examples are given to demonstrate the practical performance of the method.     

Coupled nonlinear stochastic differential equations

Systems of n coupled linear or nonlinear differential equations which may be deterministic or stochastic are solved by methods of the first author and his co-workers. Examples include two coupled Riccati equations, coupled linear equations, stochastic coupled equations with product terms, and n coupled stochastic differential equations.     

Numerical solution of linear and nonlinear partial differential equations using the peridynamic differential operator

This study presents numerical solutions to linear and nonlinear Partial Differential Equations (PDEs) by using the peridynamic differential operator. The solution process involves neither a derivative reduction process nor a special treatment to remove a jump discontinuity or a singularity. The peridynamic discretization can be both in time and space. The accuracy and robustness of this differential operator is demonstrated by considering challenging linear, nonlinear, and coupled PDEs subjected to Dirichlet and Neumann‐type boundary conditions. Their numerical solutions are achieved using either implicit or explicit methods. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1726–1753, 2017     

An approximate solution for nonlinear backward parabolic equations

We consider the backward parabolic equation

     

线性随机时滞微分方程裂步Milstein方法的收敛性(英文)

给出一个新的求解线性随机时滞微分方程的显式分裂步长Milstein格式.运用ItoTaylor展开式证明该格式相对于已有的求解随机时滞微分方程的分裂步长方法而言具有更好的收敛性.数值实验验证了理论分析的正确性.     

Factoring linear partial differential operators and the Darboux method for integrating nonlinear partial differential equations

Using a new definition of the generalized factorization of linear partial differential operators, we discuss possible generalizations of the Darboux integrability of nonlinear partial differential equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 1, pp. 144–160, January, 1999.     

Implicit Euler method for numerical solution of nonlinear stochastic partial differential equations with multiplicative trace class noise

In this paper, we consider the numerical approximation of stochastic partial differential equations with nonlinear multiplicative trace class noise. Discretization is obtained by spectral collocation method in space, and semi‐implicit Euler method is used for the temporal approximation. Our purpose is to investigate the convergence of the proposed method. The rate of convergence is obtained, and some numerical examples are included to illustrate the estimated convergence rate.     

Inverse coefficient problem for a nonlinear differential equation and an iterative solution method

The article considers the determination of the solution-dependent coefficient of a nonautonomous ordinary differetial equation with a parameter. Reduction of the inverse problem to a nonlinear operator equation is used to prove existence and uniqueness theorems and to propose an iterative solution method. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 5–17.     

Modified Householder iterative method for nonlinear equations

In this paper, we suggest and analyze a new two-step predictor–corrector type iterative method for solving nonlinear equations of the type f(x)=0. This new method includes the two-step Newton method as a special case. We show that this new two-step method is a sixth-order convergent method. Several examples are given to illustrate the efficiency of this new method and its comparison with other sixth-order methods. This method can be considered as a significant improvement of the Newton method and its variant forms.     

An iterative method for solving nonlinear equations

An iterative method for finding a solution of the equation f(x)=0$f(x)=0$ is presented. The method is based on some specially derived quadrature rules. It is shown that the method can give better results than the Newton method.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Positivity and stabilisation for nonlinear stochastic delay differential equations

We use state dependent Gaussian perturbations to stabilise the solutions of differential equations with coefficients that take, as arguments, averaged sets of information from the history of the solution, as well as isolated past and present states. The properties that guarantee stability also guarantee positivity of solutions as long as the initial value is nonzero.

We do not require that any component of the coefficients of the equations satisfy Lipschitz conditions. Instead, we require that the functional part of each coefficient which feeds back the present state of the process admit to bounds imposed by a member of a particular class of concave functions. Lipschitz conditions are included as a special case of these bounds.

We generalise these results to the finite dimensional case, also constructing perturbations that can destabilise the otherwise stable solutions of a deterministic system of equations.     

Approximating solutions to linear and nonlinear differential equations by the method of maximum entropy

A technique to approximate the solution to linear and nonlinear boundary value problems is developed and numerical examples are presented. The technique is based on the method of maximum entropy with moments of the differential equation used as constraints. The method is very general and has the advantage that additional information can be fed into the solution, such as the function's domain or the positivity or negativity of the solution. The technique should find applications in approximating solutions to equations which may or may not contain noise and as an alternative to finite difference and Fourier series solutions and may have applications to large scale simulations.