Exponential mean-square stability properties of stochastic linear multistep methods

The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand whether a stochastic linear multistep method can provide an analogous behaviour and which restrictions on the employed stepsize should be imposed in order to reproduce the contractive behaviour. Numerical experiments confirming the theoretical analysis are also given.

Four-Step One Hybrid Block Methods for Solution of Fourth Derivative Ordinary Differential Equations

We consider developing a four-step one offgrid block hybrid method for the solution of fourth derivative Ordinary Differential Equations. Method of interpolation and collocation of power series approximate solution was used as the basis function to generate the continuous hybrid linear multistep method, which was then evaluated at non-interpolating points to give a continuous block method. The discrete block method was recovered when the continuous block was evaluated at all step points. The basic properties of the methods were investigated and said to be converge. The developed four-step method is applied to solve fourth derivative problems of ordinary differential equations from the numerical results obtained; it is observed that the developed method gives better approximation than the existing method compared with.

FORMULATION OF BLOCK SCHEMES WITH LINEAR MULTISTEP METHOD FOR THE APPROXIMATION OF FIRST-ORDER IVPS

ABSTRACT In this paper, formulation of an efficient numerical schemes for the approximation first-order initial value problems (IVPs) of ordinary differential equations (ODE) is presented. The method is a block scheme for some k-step linear multi-step methods (and) using the Hermite Polynomials a basis function. The continuous and discrete linear multi-step methods (LMM) are formulated through the technique of collocation and interpolation. Numerical examples of ODE have been examined and results obtained show that the proposed scheme can be efficient in solving initial value problems of first order ODE.

The Double Step Hybrid Linear Multistep Method for Solving Second Order Initial Value Problems

In this paper, we develop the double step hybrid linear multistep method for solving second order initial value problems via interpolation and collocation method of power series approximate solution to give a system of nonlinear equations which is solved to give a continuous hybrid linear multistep method. The continuous hybrid linear multistep method is solved for the independent solutions to give a continuous hybrid block method which is then evaluated at some selected grid points to give a discrete block method. The basic numerical properties of the hybrid block method was established and found to be zero-stable, consistent and convergent. The efficiency of the new method was conformed on some initial value problems and found to give better approximation than the existing methods.

Explicit Fourth-Derivative Two-Step Linear Multistep Method for Ordinary Differential Equations

Many problems from science and engineering are modeled by Ordinary Differential Equations (ODEs) whose solutions describe the temporal evolution of the modeled processes. In most cases however, the arising equations are too complex to be solved analytically. Consequently, their solutions have to be approximated by numerical methods. In this article, we propose an explicit fourth-derivative two-step linear multistep method (FD2LMM) for ordinary differential equations. The proposed method is constructed by using the maximal order criteria which is obtained through the associated linear difference operator. The starting values used by the proposed method are obtained by suitable single-step method. The order, consistency, linear stability, and the convergence properties of the method are discussed. Numerical experiments are performed and the results are compared with those of existing methods in the literature.

Construction and Implementation of Optimal 8--Step Linear Multistep method

In this paper, the optimal 8--step linear multistep method for solving $y^{\prime}=f(x,y)$ is constructed and implemented. The construction was carried out using the technique based on the Taylor expansion of $y(x + jh)$ and $y^{\prime}(x + jh)$ about $x + t h$, where \emph{t} need not necessarily be an integer. The consistency, stability and convergence of the proposed method are investigated. To investigate the accuracy of the method, a comparison with the classical 8-stage Runge--Kutta method is carried out on two numerical examples. The results obtained by the constructed method are accurate up to certain degrees and compete favourably with those produced by the classical 8-stage Runge--Kutta method