泊松白噪声激励下的随机时滞系统的数值解及应用研究
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摘要
随机时滞系统主要是用随机延迟微分方程来描述的,目前随机延迟微分方程已经广泛地应用在经济学、物理学、生物学等各个领域中。然而一般情况下,很难求出它的解析解,因此研究随机延迟微分方程的数值解法既有理论意义又有广泛的应用价值。
近几年来,虽然也有不少学者研究了随机常延迟微分方程及其数值解法,也取得了一些成果,但对于带跳跃的随机延迟微分方程、随机延迟积分微分方程和跳跃的中立型随机延迟微分方程的研究却刚刚开始,还未见成果报导。因此,本文分别对这三种不同类型的随机延迟微分方程的数值方法及收敛性与均方稳定性进行了深入地研究,并具体应用到生物学领域中,取得的成果如下:
1.将泊松白噪声加入到随机延迟微分方程中,得到了一种带跳跃的随机延迟微分方程,首先给出该类方程数值解均方稳定性的定义,然后研究了该类方程的Euler-Maruyama数值解的均方稳定性,并研究了该方法得到的数值解均方稳定性的充分条件,最后通过数值实验验证该理论结果的正确性。
2.考虑带有分布式延迟项的线性随机延迟积分微分方程,将分步向后欧拉法用于具有一般性的线性随机延迟积分微分方程中,研究了该方法得到的数值解的收敛性和均方稳定性,并证明了该数值解收敛于解析解的阶为1/2,同时还得到了当方程的解析解渐近均方稳定时,该数值法得到的数值解保持均方稳定性的充分条件,最后,通过数值试验说明了该方法的有效性和正确性。
3.将泊松白噪声加入中立型随机泛函微分方程中,得到了一类带跳跃的中立型随机泛函微分方程,首先研究了其解的存在唯一性定理,然后研究了一类带跳跃的中立型随机延迟微分方程的Euler-Maruyama数值方法的收敛性,最后证明了Euler-Maruyama法得到的数值解在局部Lipchitz条件下收敛于解析解,并且收敛阶为1/2。
4.将带跳跃的随机延迟微分方程和随机延迟积分微分方程应用到生物种群动力学中,利用本文提出的数值方法对生物种群系统的响应进行了数值模拟,结果发现了生物种群的稳定性和Hopf分岔点。
Stochastic systems with time delays are mainly described by the sotchasic delay differential equations (SDDEs). Stochastic systems with delays are becoming increasingly used nowdays in dfferent fields, such as, economics, physics, medicine, ecology and so on. However few analytical solutions can be obtained for SDDEs, thus, developing appropriate numerical methods for SDDEs is interesting topic both in theory and in applications.
In recent years, many scholars studied the approximate schemes for SDDEs and have achieved fruitful results. However, the study on the numerical schemes for stochastic delay differential equations with jumps (SDDEwJs), stochastic delay integro-differential equations (SDIDEs) and neutral stochastic delay differential equations with jumps (NSDDEwJs) has just begun. In this paper, we investigate the convergence and mean square stability of the numerical methods for these three types stochastic delay differential equations,at the same time we apply these stochachtic systems to biology and obtained the following results:
1. For the SDDEs, Poisson white noise is added to this class equation to obtain a class of SDDEwJs. First, the definition of mean-square stability of numerical methods for SDDEwJs is established, and then the sufficient condition of mean square stability of the Euler-Maruyama method for SDDEwJs is derived, finally a class scalar test equation is simulated, the numerical experiments verify the results obtained from theory.
2. The convergence and mean square stability of the numerical approximation of solutions for linear stochastic delay integro-differential equations (SDIDEs) was studied. Split-step backward Euler (SSBE) method for solving linear stochastic delay integro-differential equations is derived. It is proved that the SSBE method is convergent with strong order 12 in the mean-square sense. The condition under which the SSBE method is mean-square stable (MS-stable) is obtained. At last, the numerical experiments illustrate the validity and correctness the method.
3. For the neutral stochastic functional differential equations (NSFDEs), Poisson white noise is added to this class equation to obtain a class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwJs). First, the existence and uniqueness of solutions for NSFDEwJs is studied, then the Euler-Maruyama method for neutral stochastic delay differential equations with Poisson jumps (NSDDEwJs) is developed, and last we prove that the numerical solutions will converge to the true solutions for NSDDEwJs with strong order 1/2 under the local Lipschitz condition.
4. The stochastic delay differential equations with jumps and stochastic delay integro-differential equations are applied to biological population dynamics. Using the numerical methods in this paper, we simulate the response of the systems and found that the stability and Hopf bifurcation point of the biological populations.
近几年来,虽然也有不少学者研究了随机常延迟微分方程及其数值解法,也取得了一些成果,但对于带跳跃的随机延迟微分方程、随机延迟积分微分方程和跳跃的中立型随机延迟微分方程的研究却刚刚开始,还未见成果报导。因此,本文分别对这三种不同类型的随机延迟微分方程的数值方法及收敛性与均方稳定性进行了深入地研究,并具体应用到生物学领域中,取得的成果如下:
1.将泊松白噪声加入到随机延迟微分方程中,得到了一种带跳跃的随机延迟微分方程,首先给出该类方程数值解均方稳定性的定义,然后研究了该类方程的Euler-Maruyama数值解的均方稳定性,并研究了该方法得到的数值解均方稳定性的充分条件,最后通过数值实验验证该理论结果的正确性。
2.考虑带有分布式延迟项的线性随机延迟积分微分方程,将分步向后欧拉法用于具有一般性的线性随机延迟积分微分方程中,研究了该方法得到的数值解的收敛性和均方稳定性,并证明了该数值解收敛于解析解的阶为1/2,同时还得到了当方程的解析解渐近均方稳定时,该数值法得到的数值解保持均方稳定性的充分条件,最后,通过数值试验说明了该方法的有效性和正确性。
3.将泊松白噪声加入中立型随机泛函微分方程中,得到了一类带跳跃的中立型随机泛函微分方程,首先研究了其解的存在唯一性定理,然后研究了一类带跳跃的中立型随机延迟微分方程的Euler-Maruyama数值方法的收敛性,最后证明了Euler-Maruyama法得到的数值解在局部Lipchitz条件下收敛于解析解,并且收敛阶为1/2。
4.将带跳跃的随机延迟微分方程和随机延迟积分微分方程应用到生物种群动力学中,利用本文提出的数值方法对生物种群系统的响应进行了数值模拟,结果发现了生物种群的稳定性和Hopf分岔点。
Stochastic systems with time delays are mainly described by the sotchasic delay differential equations (SDDEs). Stochastic systems with delays are becoming increasingly used nowdays in dfferent fields, such as, economics, physics, medicine, ecology and so on. However few analytical solutions can be obtained for SDDEs, thus, developing appropriate numerical methods for SDDEs is interesting topic both in theory and in applications.
In recent years, many scholars studied the approximate schemes for SDDEs and have achieved fruitful results. However, the study on the numerical schemes for stochastic delay differential equations with jumps (SDDEwJs), stochastic delay integro-differential equations (SDIDEs) and neutral stochastic delay differential equations with jumps (NSDDEwJs) has just begun. In this paper, we investigate the convergence and mean square stability of the numerical methods for these three types stochastic delay differential equations,at the same time we apply these stochachtic systems to biology and obtained the following results:
1. For the SDDEs, Poisson white noise is added to this class equation to obtain a class of SDDEwJs. First, the definition of mean-square stability of numerical methods for SDDEwJs is established, and then the sufficient condition of mean square stability of the Euler-Maruyama method for SDDEwJs is derived, finally a class scalar test equation is simulated, the numerical experiments verify the results obtained from theory.
2. The convergence and mean square stability of the numerical approximation of solutions for linear stochastic delay integro-differential equations (SDIDEs) was studied. Split-step backward Euler (SSBE) method for solving linear stochastic delay integro-differential equations is derived. It is proved that the SSBE method is convergent with strong order 12 in the mean-square sense. The condition under which the SSBE method is mean-square stable (MS-stable) is obtained. At last, the numerical experiments illustrate the validity and correctness the method.
3. For the neutral stochastic functional differential equations (NSFDEs), Poisson white noise is added to this class equation to obtain a class of neutral stochastic functional differential equations with Poisson jumps (NSFDEwJs). First, the existence and uniqueness of solutions for NSFDEwJs is studied, then the Euler-Maruyama method for neutral stochastic delay differential equations with Poisson jumps (NSDDEwJs) is developed, and last we prove that the numerical solutions will converge to the true solutions for NSDDEwJs with strong order 1/2 under the local Lipschitz condition.
4. The stochastic delay differential equations with jumps and stochastic delay integro-differential equations are applied to biological population dynamics. Using the numerical methods in this paper, we simulate the response of the systems and found that the stability and Hopf bifurcation point of the biological populations.
引文
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