### Abstract

In a recent paper [Phys. Lett. A 205, 130 (1995)], we investigated the inverse problem of solving g(x_{1},...,x_{q}) from the integral equation n(y_{1},...,y_{q}) = ∫ K(y_{1},...,y_{q}|x_{1},...,x_{q})g(x _{1},...,x_{q})dx_{1}⋯-dx_{q}, with the given integral n and kernel K by analytically dilating variable y to the complex plane. We showed, by studying the singularities and discontinuities of the dilated kernel and integral, that the unknown function g can be obtained from an algebraic relation in the case where the dilated kernel contains a simple and single-valued pole. The present paper intends to generalize this result to the case where the kernel contains higher-order and/or multivalued poles. We show that the integral equation in these more general cases can be transformed to algebraic, ordinary, or partial differential equations, depending on the type of the singularities of the kernel and the dimension of the inverse problem. Moreover, some conditions constraining the integral n, which are independent of the integrand g, are revealed when K has multivalued or high-order singularities.

Original language | English (US) |
---|---|

Pages (from-to) | 2384-2391 |

Number of pages | 8 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 54 |

Issue number | 3 |

State | Published - 1996 |

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### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*54*(3), 2384-2391.

**Inverse problem with a dilated kernel containing different singularities.** / Gang, Hu; Ning, Cun-Zheng; Haken, H.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 54, no. 3, pp. 2384-2391.

}

TY - JOUR

T1 - Inverse problem with a dilated kernel containing different singularities

AU - Gang, Hu

AU - Ning, Cun-Zheng

AU - Haken, H.

PY - 1996

Y1 - 1996

N2 - In a recent paper [Phys. Lett. A 205, 130 (1995)], we investigated the inverse problem of solving g(x1,...,xq) from the integral equation n(y1,...,yq) = ∫ K(y1,...,yq|x1,...,xq)g(x 1,...,xq)dx1⋯-dxq, with the given integral n and kernel K by analytically dilating variable y to the complex plane. We showed, by studying the singularities and discontinuities of the dilated kernel and integral, that the unknown function g can be obtained from an algebraic relation in the case where the dilated kernel contains a simple and single-valued pole. The present paper intends to generalize this result to the case where the kernel contains higher-order and/or multivalued poles. We show that the integral equation in these more general cases can be transformed to algebraic, ordinary, or partial differential equations, depending on the type of the singularities of the kernel and the dimension of the inverse problem. Moreover, some conditions constraining the integral n, which are independent of the integrand g, are revealed when K has multivalued or high-order singularities.

AB - In a recent paper [Phys. Lett. A 205, 130 (1995)], we investigated the inverse problem of solving g(x1,...,xq) from the integral equation n(y1,...,yq) = ∫ K(y1,...,yq|x1,...,xq)g(x 1,...,xq)dx1⋯-dxq, with the given integral n and kernel K by analytically dilating variable y to the complex plane. We showed, by studying the singularities and discontinuities of the dilated kernel and integral, that the unknown function g can be obtained from an algebraic relation in the case where the dilated kernel contains a simple and single-valued pole. The present paper intends to generalize this result to the case where the kernel contains higher-order and/or multivalued poles. We show that the integral equation in these more general cases can be transformed to algebraic, ordinary, or partial differential equations, depending on the type of the singularities of the kernel and the dimension of the inverse problem. Moreover, some conditions constraining the integral n, which are independent of the integrand g, are revealed when K has multivalued or high-order singularities.

UR - http://www.scopus.com/inward/record.url?scp=5544228313&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=5544228313&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:5544228313

VL - 54

SP - 2384

EP - 2391

JO - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

JF - Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

SN - 1539-3755

IS - 3

ER -