Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-05T16:44:40.000Z Has data issue: false hasContentIssue false
  • English
  • Français

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)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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