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


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).


Integral Equation Rational Function Eigenvalue Problem Numerical Evaluation Kernel Approximation 
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  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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