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Solution of generalized density evolution equation via a family of δ sequences

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Abstract

The traditional probability density evolution equations of stochastic systems are usually in high dimensions. It is very hard to obtain the solutions. Recently the development of a family of generalized density evolution equation (GDEE) provides a new possibility of tackling nonlinear stochastic systems. In the present paper, a numerical method different from the finite difference method is developed for the solution of the GDEE. In the proposed method, the formal solution is firstly obtained through the method of characteristics. Then the solution is approximated by introducing the asymptotic sequences of the Dirac δ function combined with the smart selection of representative point sets in the random parameters space. The implementation procedure of the proposed method is elaborated. Some details of the computation including the selection of the parameters are discussed. The rationality and effectiveness of the proposed method is verified by some examples. Some features of the numerical results are observed.

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Authors and Affiliations

  1. School of Civil Engineering, Tongji University, 1239 Siping Road, 200092, Shanghai, People’s Republic of China

    Wenliang Fan

  2. State Key Laboratory for Disaster Reduction in Civil Engineering, School of Civil Engineering, Tongji University, 1239 Siping Road, 200092, Shanghai, People’s Republic of China

    Jianbing Chen & Jie Li

Authors
  1. Wenliang Fan
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  2. Jianbing Chen
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  3. Jie Li
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Correspondence to Jie Li.

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Fan, W., Chen, J. & Li, J. Solution of generalized density evolution equation via a family of δ sequences. Comput Mech 43, 781–796 (2009). https://doi.org/10.1007/s00466-008-0345-4

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  • DOI: https://doi.org/10.1007/s00466-008-0345-4

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