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

  • Chengjian Zhang (a1) and Wenjie Shi (a1) (a2)

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.

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*Corresponding author. Email addresses: (C. Zhang), (W. Shi)
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[1] I. Babuska , F. Nobile and R. Tempone , A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal. 45, 10051034 (2007).

[2] J. Beck , F. Nobile , L. Tamellini and R. Tempone , Implementation of optimal Galerkin and collocation approximations of PDEs with random coefficients, ESIAM Proc. 33, 1021 (2011).

[3] O.G. Ernst , A. Mugler , H.J. Starkloff and E. Ullmann , On the convergence of generalized polynomial chaos expansions, ESIAM Math. Model. Numer. 46, 317339 (2012).

[7] X. Liu and M. Liu , Asymptotic stability of Runge-Kutta methods for nonlinear differential equations with piecewise continuous arguments, J. Comput. Appl. Math. 280, 265274 (2015).

[9] W. Shi and C. Zhang , Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations, Appl. Numer. Math. 62, 19541964 (2012).

[10] W. Shi and C. Zhang , Generalized polynomial chaos for nonlinear random Pantograph equations, Acta Math. Appl. Sin. English Ser. 32, 685700 (2016).

[12] W. Wang and S. Li , Dissipativity of Runge-Kutta methods for neutral delay differential equations with piecewise constant delay, Appl. Math. Lett. 21, 983991 (2008).

[14] J. Wiener , Generalized Solutions of Functional Differential Equations, World Scientific, Singapore (1993).

[18] T. Zhou , A stochastic collocation method for delay differential equations with random input, Adv. Appl. Math. Mech. 6, 403418 (2014).

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East Asian Journal on Applied Mathematics
  • ISSN: 2079-7362
  • EISSN: 2079-7370
  • URL: /core/journals/east-asian-journal-on-applied-mathematics
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