### Abstract

A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature. A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.

Original language | English (US) |
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Pages (from-to) | 345-362 |

Number of pages | 18 |

Journal | Computer Methods in Applied Mechanics and Engineering |

Volume | 27 |

Issue number | 3 |

DOIs | |

State | Published - Jul 1981 |

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### ASJC Scopus subject areas

- Computational Mechanics
- Mechanics of Materials
- Mechanical Engineering
- Physics and Astronomy(all)
- Computer Science Applications

### Cite this

*Computer Methods in Applied Mechanics and Engineering*,

*27*(3), 345-362. https://doi.org/10.1016/0045-7825(81)90137-7

**Petrov-Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle.** / Bergman, Lawrence; Heinrich, J. C.

Research output: Contribution to journal › Article

*Computer Methods in Applied Mechanics and Engineering*, vol. 27, no. 3, pp. 345-362. https://doi.org/10.1016/0045-7825(81)90137-7

}

TY - JOUR

T1 - Petrov-Galerkin finite element solution for the first passage probability and moments of first passage time of the randomly accelerated free particle

AU - Bergman, Lawrence

AU - Heinrich, J. C.

PY - 1981/7

Y1 - 1981/7

N2 - A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature. A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.

AB - A numerical solution to an initial boundary value problem governing the probability of failure of a randomly accelerated free particle is obtained using a Petrov-Galerkin finite element method. This direct solution is the first successful one, and no others have been reported in the literature. A solution of the Pontriagin-Vitt equation for the time to first passage of the particle is obtained first: in this case an analytical solution is available and used to evaluate the numerical algorithm. Extensions to the solution of other stochastic differential equations, in particular those governing the probability of failure of the linear oscillator, and applications to structural dynamics are discussed.

UR - http://www.scopus.com/inward/record.url?scp=0019586844&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0019586844&partnerID=8YFLogxK

U2 - 10.1016/0045-7825(81)90137-7

DO - 10.1016/0045-7825(81)90137-7

M3 - Article

AN - SCOPUS:0019586844

VL - 27

SP - 345

EP - 362

JO - Computer Methods in Applied Mechanics and Engineering

JF - Computer Methods in Applied Mechanics and Engineering

SN - 0374-2830

IS - 3

ER -