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Astrophysics and Space Science

, Volume 106, Issue 2, pp 355–369 | Cite as

On numerical evaluation of theH-functions of transport problems by kernel approximation for the albedo 0<ω≤1

  • Z. Islam
  • S. R. Das Gupta
Article
  • 28 Downloads

Abstract

Das Gupta represented theH-functions of transport problems for the albedo ω∈[0, 1] in the formH(z)=R(z)−S(z) (see Das Gupta, 1977) whereR(z) is a rational function ofz andS(z) is regular on [−1, 0] c . In this paper we have representedS(z) through a Fredholm integral equation of the second kind with a symmetric real kernelL(y, z) as\(S(z) = f(z) - \int_0^1 {L(y,{\text{ }}z)S(y){\text{ d}}y} \). The problem is then solved as an eigenvalue problem. The kernel is converted into a degenerate kernel through finite Taylor's expansion and the integral equation forS(z) takes the form:\(S(z) = f(z) - \sum\nolimits_{i - 1}^N {{\text{ }}\chi _i \int_0^1 {F_i (z)F_i (y)S(y){\text{ d}}y} } \) (which is solved by the usual procedure) where χ r 's are the discrete eigenvalues andF r 's the corresponding eigenfunctions of the real symmetric kernelL(y, z).

Keywords

Integral Equation Rational Function Eigenvalue Problem Numerical Evaluation Kernel Approximation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Chandrasekhar, S.: 1950,Radiative Transfer, Clarendon Press, Oxford.Google Scholar
  2. Das Gupta, S. R.: 1974,Astrophys. Space Sci. 30, 327.Google Scholar
  3. Das Gupta, S. R.: 1977,Astrophys. Space Sci. 50, 187.Google Scholar
  4. Whittaker, E. T. and Watson, G. N.: 1950,A Course of Modern Analysis, Cambridge University Press.Google Scholar

Copyright information

© D. Reidel Publishing Company 1984

Authors and Affiliations

  • Z. Islam
    • 1
  • S. R. Das Gupta
    • 2
  1. 1.Dept. of MathematicsMalda CollegeMaldaIndia
  2. 2.Dept. of MathematicsUniversity of North BengalIndia

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