### Abstract

In this study, one-dimensional stochastic Korteweg-de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.

Original language | English |
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Pages (from-to) | 3080-3093 |

Number of pages | 14 |

Journal | Applied Mathematical Modelling |

Volume | 36 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2012 Jul 1 |

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### Keywords

- KdV equation
- Polynomial chaos
- Spectral method
- Stochastic differential equation

### ASJC Scopus subject areas

- Applied Mathematics
- Modelling and Simulation

### Cite this

*Applied Mathematical Modelling*,

*36*(7), 3080-3093. https://doi.org/10.1016/j.apm.2011.09.086

**Dependence of polynomial chaos on random types of forces of KdV equations.** / Kim, Hongjoong; Kim, Yoontae; Yoon, Daeki.

Research output: Contribution to journal › Article

*Applied Mathematical Modelling*, vol. 36, no. 7, pp. 3080-3093. https://doi.org/10.1016/j.apm.2011.09.086

}

TY - JOUR

T1 - Dependence of polynomial chaos on random types of forces of KdV equations

AU - Kim, Hongjoong

AU - Kim, Yoontae

AU - Yoon, Daeki

PY - 2012/7/1

Y1 - 2012/7/1

N2 - In this study, one-dimensional stochastic Korteweg-de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.

AB - In this study, one-dimensional stochastic Korteweg-de Vries equation with uncertainty in its forcing term is considered. Extending the Wiener chaos expansion, a numerical algorithm based on orthonormal polynomials from the Askey scheme is derived. Then dependence of polynomial chaos on the distribution type of the random forcing term is inspected. It is numerically shown that when Hermite (Laguerre or Jacobi) polynomial chaos is chosen as a basis in the Gaussian (Gamma or Beta, respectively) random space for uncertainty, the solution to the KdV equation converges exponentially. If a proper polynomial chaos is not used, however, the solution converges with slower rate.

KW - KdV equation

KW - Polynomial chaos

KW - Spectral method

KW - Stochastic differential equation

UR - http://www.scopus.com/inward/record.url?scp=84858334190&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858334190&partnerID=8YFLogxK

U2 - 10.1016/j.apm.2011.09.086

DO - 10.1016/j.apm.2011.09.086

M3 - Article

VL - 36

SP - 3080

EP - 3093

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 7

ER -