带跳随机比例微分方程补偿分步θ方法的收敛性和稳定性
Convergence and Stability of Compenseted Split-Step Theta Methods for Stochastic Pantograph Differential Equations with Jumps
摘要: 我们探讨了带跳的随机比例微分方程的补偿分步
θ方法。首先，证明了该数值方法收敛，并且收敛阶为12，其次证明了解析解的均方稳定性，并且证明了补偿分步
θ方法能保持解析解的均方稳定性，最后给出数值算例验证理论结果的正确性。
Abstract:
A semi-implicit compensated split-step θ method for stochastic pantograph differential equations with Poisson jumps is investigated in the paper. Firstly, the convergence of the numerical method is discussed, and it is proved that the convergence order is 1/2. Secondly, the mean-square stability of the analytical solution is proved, and it is found that the compensated split-step θ methods can maintain the mean square stability of the analytical solution under some conditions. Finally, two numerical examples are given to verify the correctness of the theoretical results.
参考文献
|
[1]
|
Ockendon, J.R. and Tayler, A.B. (1971) The Dynamics of a Current Collection System for an Electric Locomotive. Proceedings of the Royal Society A. Mathematical and Physical Sciences, 322, 447-468. [Google Scholar] [CrossRef]
|
|
[2]
|
毛伟. 一类带跳的随机比例微分方程解的存在唯一性和Euler数值方法的收敛性[J]. 江苏教育学院学报(自然科学版), 2007(4): 1-5.
|
|
[3]
|
Mao, W., Han, X.J. and Chen, B. (2011) The Semi-Implicit Euler Method for Stochastic Pantograph Equations with Jumps. Chinese Quarterly Journal of Mathematics, 3, 405-409.
|
|
[4]
|
Mao, W. and Mao, X.R. (2013) Approximate Solutions of Hybrid Stochastic Pantograph Equations with Levy Jumps. Abstract and Applied Analysis, 2013, Article ID: 718627. [Google Scholar] [CrossRef]
|
|
[5]
|
Hu, L. and Gan, S.Q. (2013) Numerical Analysis of the Balanced Implicit Methods for Stochastic Pantograph Equations with Jumps. Applied Mathematics and Computation, 223, 281-297. [Google Scholar] [CrossRef]
|
|
[6]
|
Yang, H. and Jiang, F. (2014) Stochastic-Methods for a Class of Jump-Diffusion Stochastic Pantograph Equations with Random Magnitude. The Scientific World Journal, 2014, Article ID: 589167. [Google Scholar] [CrossRef] [PubMed]
|
|
[7]
|
Lu, Y.L., Song, M.H. and Liu, M.Z. (2018) Convergence and Stability of the Compensated Split-Step Theta Method for Stochastic Differential Equations with Piecewise Continuous Arguments Driven by Poisson Random Measure. Journal of Computational and Applied Mathematics, 340, 296-317. [Google Scholar] [CrossRef]
|
|
[8]
|
Xiao, Y., Eissa, M.A. and Tian, B.P. (2017) Convergence and Stability of Split-Step Theta Methods with Variable Step-Size for Stochastic Pantograph Differential Equations. International Journal of Computer Mathematics, 95, 939-960. [Google Scholar] [CrossRef]
|