Abstract

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.

Original languageEnglish (US)
Pages (from-to)2384-2391
Number of pages8
JournalPhysical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
Volume54
Issue number3
StatePublished - 1996

Fingerprint

Inverse Problem
Singularity
kernel
integral equations
poles
Multivalued
Pole
Integral Equations
partial differential equations
Higher Order
discontinuity
Single valued
G-function
Integrand
Argand diagram
Discontinuity
Ordinary differential equation
Partial differential equation
Unknown
Generalise

ASJC Scopus subject areas

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

Cite this

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title = "Inverse problem with a dilated kernel containing different singularities",
abstract = "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.",
author = "Hu Gang and Cun-Zheng Ning and H. Haken",
year = "1996",
language = "English (US)",
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publisher = "American Physical Society",
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T1 - Inverse problem with a dilated kernel containing different singularities

AU - Gang, Hu

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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.

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