Shooting Methods for Numerical Solution of Stochastic Boundary-Value Problems

Abstract

In the present investigation, numerical methods are developed for approximate solution of stochastic boundary-value problems. In particular, shooting methods are examined for numerically solving systems of Stratonovich boundary-value problems. It is proved that these methods accurately approximate the solutions of stochastic boundary-value problems. An error analysis of these methods is performed. Computational simulations are given.     

Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.     

Numerical Solution to Hybrid Stochastic Differential Systems

Abstract

In this article numerical methods for solving hybrid stochastic differential systems of Itô-type are developed by piecewise application of numerical methods for SDEs. We prove a convergence result if the corresponding method for SDEs is numerically stable with uniform convergence in the mean square sense. The Euler and Runge–Kutta methods for hybrid stochastic differential equations are specifically described and the order of the error is given for the Euler method. A numerical example is given to illustrate the theory.     

Solution of Two-Point Boundary-Value Problems Using the Differential Transformation Method

This paper considers two-point boundary-value problems using the differential transformation method. An iterative procedure is proposed for both the linear and nonlinear cases. Using the proposed approach, an analytic solution of the two-point boundary-value problem, represented by an mth-order Taylor series expansion, can be obtained throughout the prescribed range.     

Quasilinearization Method for Nonlinear Elliptic Boundary-Value Problems

The quasilinearization method is developed for strong solutions of semilinear and nonlinear elliptic boundary-value problems. We obtain two monotone, L^p-convergent sequences of approximate solutions. The order of convergence is two. The tools are some results on the abstract quasilinearization method and from weakly–near operators theory.     

Parametric Cubic Spline Approach to the Solution of a System of Second-Order Boundary-Value Problems

We use parametric cubic spline functions to develop a numerical method for computing approximations to the solution of a system of second-order boundary-value problems associated with obstacle, unilateral, and contact problems. We show that the present method gives approximations which are better than those produced by other collocation, finite-difference, and spline methods. A numerical example is given to illustrate the applicability and efficiency of the new method.     

Lagrangian Globalization Methods for Nonlinear Complementarity Problems

This paper extends the Lagrangian globalization (LG) method to the nonsmooth equation arising from a nonlinear complementarity problem (NCP) and presents a descent algorithm for the LG phase. The aim of this paper is not to present a new method for solving the NCP, but to find such that when the NCP has a solution and is a stationary point but not a solution.     

Parametric Quintic-Spline Approach to the Solution of a System of Fourth-Order Boundary-Value Problems

In this paper, we use parametric quintic splines to derive some consistency relations which are then used to develop a numerical method for computing the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. It is known that a class of variational inequalities related to contact problems in elastostatics can be characterized by a sequence of variational inequations, which are solved using some numerical method. Numerical evidence is presented to show the applicability and superiority of the new method over other collocation, finite difference, and spline methods.     

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.     

Spline Techniques for the Numerical Solution of Singular Perturbation Problems

One-dimensional singularly-perturbed two-point boundary-value problems arising in various fields of science and engineering (for instance, fluid mechanics, quantum mechanics, optimal control, chemical reactor theory, aerodynamics, reaction-diffusion processes, geophysics, etc.) are treated. Either these problems exhibits boundary layer(s) at one or both ends of the underlying interval or they possess oscillatory behavior depending on the nature of the coefficient of the first derivative term. Some spline difference schemes are derived for these problems using splines in compression and splines in tension. Second-order uniform convergence is achieved for both kind of schemes. By making use of the continuity of the first-order derivative of the spline function, a tridiagonal system is obtained which can be solved efficiently by well-known algorithms. Numerical examples are given to illustrate the theory.     

Stochastic Method for the Solution of Unconstrained Vector Optimization Problems

We propose a new stochastic algorithm for the solution of unconstrained vector optimization problems, which is based on a special class of stochastic differential equations. An efficient algorithm for the numerical solution of the stochastic differential equation is developed. Interesting properties of the algorithm enable the treatment of problems with a large number of variables. Numerical results are given.     

Weak Convergence of a Numerical Method for a Stochastic Heat Equation

Weak convergence with respect to a space of twice continuously differentiable test functions is established for a discretisation of a heat equation with homogeneous Dirichlet boundary conditions in one dimension, forced by a space-time Brownian motion. The discretisation is based on finite differences in space and time, incorporating a spectral approximation in space to the Brownian motion.     

Kinetic energy of the Langevin particle

We compute the kinetic energy of the Langevin particle using different approaches. We build stochastic differential equations that describe this physical quantity based on both the Itô and Stratonovich stochastic integrals. It is shown that the Itô equation possesses a unique solution, whereas the Stratonovich one possesses infinitely many, all but one absent of physical meaning. We discuss how this fact matches with the existent discussion on the Itô vs Stratonovich dilemma and the apparent preference toward the Stratonovich interpretation in the physical literature.     

Khasminskii-Type Theorems for Stochastic Differential Delay Equations

Abstract

The classical Khasminskii theorem (see [6 Khasminskii , R. Z. 1980 . Stochastic Stability of Differential Equations . Alphen : Sijtjoff and Noordhoff (translation of the Russian edition, Moscow: Nauka 1969) .[Crossref] , [Google Scholar]]) on the nonexplosion solutions of stochastic differential equations (SDEs) is very important since it gives a powerful test for SDEs to have nonexplosion solutions without the linear growth condition. Recently, Mao [13 Mao , X. 2002 . A note on the LaSalle-type theorems for stochastic differential delay equations . J. Math. Anal. Appl. 268 : 125 – 142 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]] established a Khasminskii-type test for stochastic differential delay equations (SDDEs). However, the Mao test can not still be applied to many important SDDEs, e.g., the stochastic delay power logistic model in population dynamics. The main aim of this paper is to establish an even more general Khasminskii-type test for SDDEs that covers a wide class of highly nonlinear SDDEs. As an application, we discuss a stochastic delay Lotka-Volterra model of the food chain to which none of the existing results but our new Khasminskii-type test can be applied.     

A Peano-Like Theorem for Stochastic Differential Equations with Nonlocal Sample Dependence

The existence of a not-necessarily-unique strong solution for a stochastic differential equations with nonlocal sample dependence is established under the assumption that the coefficients satisfy an asymptotically local boundedness condition in addition to continuity. The proof is by an Euler-like construction of approximations. These equations include mean-field stochastic differential equations, but the nonlocal sample dependence can be more general than just the dependence on moments of the solution.     

A Combination of Finite Difference and Wong-Zakai Methods for Hyperbolic Stochastic Partial Differential Equations

Abstract

The present article focuses on the use of difference methods together with Wong-Zakai methods in order to approximate the solutions of stochastic hyperbolic differential equations of Itô type. We prove convergence, consistency, and stability for the schemes we use. Whereby the consistency and stability imply convergence.     

Stochastic differential delay equations with jumps,under nonlinear growth condition

In this paper, we investigate the existence and uniqueness of solutions to stochastic differential delay equations under a local Lipschitz condition but without linear growth condition on its coefficients. Moreover, we prove convergence in probability of the Euler–Maruyama approximation as well as of the stochastic theta method approximation to the exact solution.     

Spline Techniques for Solving Singularly-Perturbed Nonlinear Problems on Nonuniform Grids

A numerical method based on cubic splines with nonuniform grid is given for singularly-perturbed nonlinear two-point boundary-value problems. The original nonlinear equation is linearized using quasilinearization. Difference schemes are derived for the linear case using a variable-mesh cubic spline and are used to solve each linear equation obtained via quasilinearization. Second-order uniform convergence is achieved. Numerical examples are given in support of the theoretical results.     

非线性中立型随机延迟微分方程随机θ方法的稳定性

本文研究非线性中立型随机延迟微分方程随机θ方法的均方稳定性.在方程解析解均方稳定的条件下,证明了如下结论:当θ∈[0,1/2)时,随机θ方法对于适当小的时间步长是均方稳定的;当θ∈[1/2,1]时,随机θ方法对于任意步长都是均方稳定的.数值结果验证了所获结论的正确性.     

Asymptotic Solution of a Boundary-Value Problem for Linear Singularly-Perturbed Functional Differential Equations Arising in Optimal Control Theory

The Hamiltonian boundary-value problem, associated with a singularly-perturbed linear-quadratic optimal control problem with delay in the state variables, is considered. A formal asymptotic solution of this boundary-value problem is constructed by application of the boundary function method. The justification of this asymptotic solution is done. The asymptotic solution of the Hamiltonian boundary-value problem is constructed and justified assuming boundary-layer stabilizability and detectability.