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Application of gPCRK Methods to Nonlinear Random Differential Equations with Piecewise Constant Argument

Part of: Numerical analysis: Ordinary differential equations Probabilistic methods, simulation and stochastic differential equations

Published online by Cambridge University Press:  02 May 2017

Chengjian Zhang  and
Wenjie Shi
Chengjian Zhang*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
Wenjie Shi*
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China School of Mathematics and Computer Science, Wuhan Textile University, Wuhan 430073, China
*
*Corresponding author. Email addresses:[email protected] (C. Zhang), [email protected] (W. Shi)
*Corresponding author. Email addresses:[email protected] (C. Zhang), [email protected] (W. Shi)
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Abstract

We propose a class of numerical methods for solving nonlinear random differential equations with piecewise constant argument, called gPCRK methods as they combine generalised polynomial chaos with Runge-Kutta methods. An error analysis is presented involving the error arising from a finite-dimensional noise assumption, the projection error, the aliasing error and the discretisation error. A numerical example is given to illustrate the effectiveness of this approach.

Keywords

Random differential equationspiecewise constant argumentgeneralised polynomial chaosfinite-dimensional noiseerror analysis

MSC classification

Secondary: 65L70: Error bounds 65C30: Stochastic differential and integral equations
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 306 - 324
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Babuska, I., Nobile, F. and Tempone, R., A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 1005–1034 (2007).Google Scholar
[2] Beck, J., Nobile, F., Tamellini, L. and Tempone, R., Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients, ESIAM Proc. 33, 10–21 (2011).CrossRefGoogle Scholar
[3] Ernst, O.G., Mugler, A., Starkloff, H.J. and Ullmann, E., On the convergence of generalized polynomial chaos expansions, ESIAM Math. Model. Numer. 46, 317–339 (2012).Google Scholar
[4] Ghanem, R. and Spanos, P., Stochastic Finite Elements: A Spectral Approach, Springer-Verlag, New York (1991).Google Scholar
[5] Hairer, E., Nørsett, S. P. and Wanner, G., Solving Ordinary Differential Equations I: Nonstiff Problems, Springer, Berlin (1993).Google Scholar
[6] Liu, M., Ma, S. and Yang, Z., Stability analysis of Runge-Kutta methods for unbounded retarded differential equations with piecewise continuous arguments, Appl. Math. Comput. 191, 57–66 (2007).Google Scholar
[7] Liu, X. and Liu, M., Asymptotic stability of Runge-Kutta methods for nonlinear differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 280, 265–274 (2015).CrossRefGoogle Scholar
[8] Narayan, A. and Zhou, T., Stochastic collocation on unstructured multivariate meshes, Commun. Comput. Phys. 18, 1–36 (2015).Google Scholar
[9] Shi, W. and Zhang, C., Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations, Appl. Numer. Math. 62, 1954–1964 (2012).Google Scholar
[10] Shi, W. and Zhang, C., Generalized polynomial chaos for nonlinear random Pantograph equations, Acta Math. Appl. Sin. English Ser. 32, 685–700 (2016).Google Scholar
[11] Tang, T. and Zhou, T., Recent developments in high order numerical methods for uncertainty quantification, Sci. Sin. Math. 45, 891–928 (2015).Google Scholar
[12] Wang, W. and Li, S., Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay, Appl. Math. Lett. 21, 983–991 (2008).CrossRefGoogle Scholar
[13] Wang, W., Stability of solutions of nonlinear neutral differential equations with piecewise constant delay and their discretisations, Appl. Math. Comput. 219, 4590–4600 (2013).Google Scholar
[14] Wiener, J., Generalized Solutions of Functional Differential Equations, World Scientific, Singapore (1993).Google Scholar
[15] Xiu, D., Fast numerical methods for stochastic computations: A review, Commun. Comput. Phys. 5, 242–272 (2009).Google Scholar
[16] Xiu, D., Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton, New Jersey (2010).Google Scholar
[17] Xiu, D. and Karniadakis, G.E., The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput. 24, 619–644 (2002).Google Scholar
[18] Zhou, T., A stochastic collocation method for delay differential equations with random input, Adv. Appl. Math. Mech. 6, 403–418 (2014).Google Scholar