Journal of engineering physics

, Volume 39, Issue 2, pp 869–872 | Cite as

Regularizing algorithm for inverting the Abel equation

  • Yu. E. Voskoboinikov
Article
  • 16 Downloads

Abstract

The article presents a regularizing algorithm for solving the Abel equation using information on the statistics of the error of measurement of the right-hand side of the equation.

Keywords

Statistical Physic Abel Equation 

Notation

ϕ(r), f(x)

solution and right-hand side of the Abel equation, respectively

fi

value of the right-hand side measured at point xi

ξi

uncertainty of the i-th measurement

n

number of measurements of the right-hand side

Vξ

correlation matrix of the uncertainty of measurement

α

smoothing parameter

Sn(x)

interpolating spline

Sn,α(x)

smoothing spline

ai, bi, ci, di

coefficients of the smoothing spline

ϕα(r)

regularized solution of the Abel equation

e(α)

discrepancy vector

Sp[Vξ]

trace of the matrix Vξ

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Literature cited

  1. 1.
    S. G. Mikhlin, Lectures on Linear Integral Equations [in Russian], Fizmatgiz, Moscow (1959).Google Scholar
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    A. N. Tikhonov and V. Ya. Arsenin, Methods of Solving Incorrect Problems [in Russian], Nauka, Moscow (1974).Google Scholar
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    V. V. Pikalov and N. G. Preobrazhenskii, “Some problems of diagnostics of low-temperature plasma solved with computers,” in: Properties of Low-Temperature Plasma and Methods of Its Diagnostics [in Russian], Nauka, Novosibirsk, Izv. Sib. Otd. Akad. Nauk SSSR (1977), pp. 138–176.Google Scholar
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    S. B. Stechkin and Yu. N. Subbotin, Splines in Computer Mathematics [in Russian], Nauka, Moscow (1976).Google Scholar
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    Yu. E. Voskoboinikov, “Criterion and algorithms for selecting a parameter in smoothing with spline functions,” in: Algorithms for Processing and Means of Automating Thermophysical Experiments [in Russian], Izd. Inst. Teplofiziki Sib. Otd. Akad. Nauk SSSR, Novosibirsk (1978), pp. 30–45.Google Scholar
  6. 6.
    Yu. E. Voskoboinikov and Ya. Ya. Tomsons, “Construction of a regularized solution of one inverse problem of heat conductivity with random errors in the initial data,” Inzh.-Fiz. Zh.,33, No. 6, 1097–1102 (1977).Google Scholar

Copyright information

© Plenum Publishing Corporation 1981

Authors and Affiliations

  • Yu. E. Voskoboinikov
    • 1
  1. 1.Institute of Thermophysics, Siberian BranchAcademy of Sciences of the USSRNovosibirsk

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