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A GENERAL CLASS OF ONE-STEP APPROXIMATION FOR INDEX-1 STOCHASTIC DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS

Tingting Qin1,2, Chengjian Zhang1,2   

  1. 1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
    2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan 430074, China
  • Received:2016-12-12 Revised:2017-04-28 Online:2019-03-15 Published:2019-03-15
  • Supported by:

    The authors would like to thank the anonymous referees for their valuable comments and helpful suggestions. The first author Tingting Qin is supported by Fundamental Research Funds for the Central Universities (Grant No. HUST:2016YXMS009) and the second author Chengjian Zhang (corresponding author) is supported by NSFC (Grant No. 11571128).

Tingting Qin, Chengjian Zhang. A GENERAL CLASS OF ONE-STEP APPROXIMATION FOR INDEX-1 STOCHASTIC DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS[J]. Journal of Computational Mathematics, 2019, 37(2): 151-169.

This paper develops a class of general one-step discretization methods for solving the index-1 stochastic delay differential-algebraic equations. The existence and uniqueness theorem of strong solutions of index-1 equations is given. A strong convergence criterion of the methods is derived, which is applicable to a series of one-step stochastic numerical methods. Some specific numerical methods, such as the Euler-Maruyama method, stochastic θ-methods, split-step θ-methods are proposed, and their strong convergence results are given. Numerical experiments further illustrate the theoretical results.

CLC Number: 

[1] A. Alabert and M. Ferrantey, Linear stochatic differential-algebraic equations with constant coefficients, Elect. Comm. in Probab., 11(2006), 316-335.

[2] U. Ascher and L.R. Petzold, The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. Numer. Anal., 32(1995), 1635-1657.

[3] U. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1998.

[4] E. Buckwar, One-step approximations for stochastic functional differential equations, Appl. Numer. Math., 56(2006), 667-681.

[5] S. Gan, H. Schurz and H. Zhang, Mean square convergence of stochastic θ-methods for nonlinear neutral stochastic differential delay equations, Int. J. Numer. Anal. Model., 8(2011), 201-213

[6] E. Hairer and G. Wanner, Solving Ordinary Differential Equations Ⅱ:Stiff and DifferentialAlgebraic Problems, Springer-Verlag, Berlin, 1996.

[7] R. Hauber, Numerical treatment of retarded differential-algebraic equations by collocation methods, Adv. Comput. Math., 7(1997), 573-592.

[8] D. Küpper, A. Kvænø and A. Rößler, A Runge-Kutta method for index 1 stochastic differentialalgebraic equations with scalar noise, BIT Numer. Math., 52(2012), 437-455.

[9] D. Küpper, A. Kvænø and A. Rößler, Stability analysis and classification of Runge-Kutta methods for index 1 stochastic differential-algebraic equations with scalar noise, Appl. Numer. Math., 96(2015), 24-44.

[10] T. Luzyanina and D. Roose, Periodic solutions of differential algebraic equations with time-delays:computation and stability analysis, Int. J. Bifurcat. Chaos, 16(2006), 67-84.

[11] X. Mao, Stochastic Differential Equations and Applications, Horwood, England, 1997.

[12] X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151(2003), 215-227.

[13] G.N. Milstien, Numerical Integration of Stochastic Differential Equations, Kluwer Academic, Dordrecht, 1995.

[14] Y. Niu, C. Zhang and K. Burrage, Strong predictor-corrector approximation for stochastic delay differential equations, J. Comput. Math., 33(2015), 587-605.

[15] C. Penski, A new numerical method for SDEs and its application in circuit simulation, J. Comput. App. Math., 115(2000), 461-470.

[16] O. Schein and G. Denk, Numerical solution of stochastic differential-algebraic equations with applications to transient noise simulation of microelectronic circuits, J. Comput. Appl. Math., 100(1998), 77-92.

[17] T. Sickenberger, E. Weinmüller and R. Winkler, Local error estimates for moderately smooth problems:Part Ⅱ-SDEs and SDAEs, BIT Numer. Math., 49(2009), 217-245.

[18] X. Wang, S. Gan and D. Wang, θ-Maruyama methods for nonlinear stochastic differential delay equations, Appl. Numer. Math., 98(2015), 38-58.

[19] W. Wang, C. Zhang, Preserving stability implicit Euler method for nonlinear Volterra and neutral functional differential equations in Banach space, Numer. Math., 115(2010), 451-474.

[20] R. Winkler, Stochastic differential algebraic equations of index 1 and applications in circuit simulation, J. Comput. Appl. Math., 163(2004), 435-463.

[21] F. Xiao and C. Zhang, Existence and uniqueness of the solution of stochastic differential algebraic equations with delay, Adv. Syst. Sci. Appl., 9(2009), 121-127.

[22] F. Xiao and C. Zhang, Euler-Maruyama methods for a class of stochastic differential algebraic system with time delay, Acta Math. Appl. Sinica, 33(2010), 590-600.

[23] C. Zhang and G. Sun, The discrete dynamics of nonlinear infinte-delay-differential equations, Appl. Math. Lett., 15(2002), 521-526.

[24] C. Zhang and S. Vandewalle, Stability criteria for exact and discrete solutions of neutral multidelay-integro-differential equations, Adv. Comput. Math., 28(2008), 383-399.

[25] Y. Zhang, Y. Zheng, X. Liu, Q. Zhang and A. Li, Dynamical analysis in a differential algebraic bio-economic model with stage-structured and stochastic fluctuations, Phys. A, 462(2016), 222- 229.

[26] W. Zhu and L.R. Petzold, Asymptotic stability of linear delay differential-algebraic equations and numerical methods, Appl. Numer. Math., 24(1997), 247-264.
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