An Analog Equation Method-Based Numerical Scheme for Initial Value Problems of Stochastic Fractional Differential Equations
摘要: 基于模拟方程法，提出了一种求解随机分数阶微分方程初值问题的数值方法.考虑含两个分数阶导数项的微分方程，引入两个线性的、非耦合的随机模拟方程，利用它们解构原方程，借助Laplace变换及逆变换，得到方程解的积分表达式，同时建立起两个模拟方程之间的联系，结合初始状态，得到求解随机微分方程初值问题的数值迭代算法.作为特例，对于含两个分数阶导数项线性常微分方程的初值问题，给出了基于模拟方程法的数值解法的显式结果.该方法是稳定的，它的误差仅存在于积分近似时的截断误差和计算软件的舍入误差.应用实例说明了数值方法在确定和随机情形的有效性和准确性.Abstract: An analog-equation-method (AEM)-based numerical scheme was proposed for initial value problems of stochastic fractional differential equations with 2 fractional derivative terms. 2 stochastic analog equations comprising respective undetermined functions were introduced, to convert the problem to a fractional differential equation with only 1 fractional derivative term. The Laplace transform and its inverse were employed to get the integration representations for the solution to the fractional differential equation and establish the relation between the 2 analog equations. In view of the initial conditions, an iterative algorithm to solve the initial value problem of the stochastic fractional differential equation was obtained. In a typical case, the numerical solution to a linear stochastic ordinary differential equation with 2 fractional derivative terms was derived based on the AEM. The numerical results of both the definite and stochastic systems demonstrate the effectiveness, stability and accuracy of the presented AEM scheme, of which the error only lies in the truncation error of the integration approximation and the rounding error of the computation software.
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