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Numerical computation and properties of the two dimensional exponential integrals

Year 2016, Volume: 45 Issue: 2, 483 - 495, 01.04.2016

Abstract

In this paper, we investigate problem of convergence of the two dimensional exponential integral (TDEI) functions arising in the study of the radiative transfer in a multi-dimensional medium. In our study, generalized exponential integral function’s ( GEIF ) are expressed with double improper integrals as given in the original expression. Then we study the properties and asymptotic behaviour of the TDEI functions. We also give numerical computations of the values of TDEI functions.

References

  • Z. Altac, Integrals involving Bickley and Bessel functions in radiative transfer amd generalized exponential integral functions, ASME Journal of Heat Transfer, Volume 118, 1996, Pages 789-792.
  • Z. Altac, Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 104, Issue 3, April 2007, Pages 310-325.
  • Y. Aygar, E. Bairamov, Properties of the two-dimensional exponential integral functions, Journal of Mathematical Chemistry, May 2011, Volume 49, Issue 5, pp 1014-1025.
  • E. Bairamov, N. Ozalp, On convergence, asymptotic behaviour and computational algorithm of generalized exponential integral functions arising in a radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 113, Issue 14, September 2012, Pages 1818-1825.
  • E. Bairamov, S. Yardimci, Uniform convergence and the properties of the exponential and generalized exponential integral functions, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 111, Issue 16, November 2010, Pages 2471-2473. [6] W.F. Breig, A.L. Crosbie, Two-dimensional radiative equilibrium semi-infinite medium subjected to cosine varying radiation. J. Quantitative Spectroscopy & Radiative Transfer 13 (1973) 1395–1419.
  • W.F. Breig, A.L. Crosbie, Numerical computation of a generalized exponential integral function. Math. Comput. 28 (1974) 575–579. I.M. Busbridge, The mathematics of radiative transfer. Cambridge 1960.
  • S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.
  • C. Chiccoli, S. Lorenzutta, G. Maino; Concerning some integrals of the generalized exponential-integral function. Computers & Mathematics with Applications, Volume 23, Issue 11, June 1992, Pages 13-21.
  • A.L. Crosbie, R.L. Dougherty, Two-dimensional in a cylindrical geometry with anisotropic scattering. J. Quantitative Spectroscopy & Radiative Transfer, 25 (1981) 551–569.
  • A.L. Crosbie, L.C. Lee, Relation between multidimensional radiative transfer in cylindrical and rectangular coordinates with anisotropic scattering. J. Quantitative Spectroscopy & Radiative Transfer, 38 (1987) 231–241.
  • A.L. Crosbie, L.C. Lee, Multidimensional radiative transfer: a single integral represantation of anisotropic scattering kernels J. Quantitative Spectroscopy & Radiative Transfer, 42 (1989) 239–246.
  • I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products Academic Press, NewYork, 2007.
  • I.I. Guseinov, B.A. Mamedov, On the evalution of generalized exponential integral functions, J. Quantitative Spectroscopy & Radiative Transfer, 102 (2006) 251-256.
  • V. Kourganoff, Basic Methods in Transfer Problems, Dover Publications, New York, 1963.
  • B.A. Mamedov; Evaluation of generalized exponential integral function using binomial expansion theorems, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 94, Issues 3-4, 1 September 2005, Pages 507-514.
  • S.M. Nikolsky,. Course of Mathematical Analysis. Vol. 2, Mir Publishers, Moscow 1977.
  • R.H. Prabha, R.D.S. Yadav, Polynomial expressions for Bickley and exponential integral functions. Ann. Nucl. Energy, 23 (1996) 1021–1025.
  • V.V. Sobelev, A Treatise on Radiative Transfer Princeton, New York, 1963
  • H.C. Van de Hulst, Multiple Light Scattering, Vol. 1, Academic Press, New York, 1980.
  • W.W Yuen, C.F. Ho, Analysis of two-dimensional radiative heat transfer in a gray medium with internal heat generation, Int. J. Heat Mass Transfer, 28 (1985) 17-23.
  • V.A. Zorich, Mathematical Analysis. Vol. 2, Springer, New York 2004.
Year 2016, Volume: 45 Issue: 2, 483 - 495, 01.04.2016

Abstract

References

  • Z. Altac, Integrals involving Bickley and Bessel functions in radiative transfer amd generalized exponential integral functions, ASME Journal of Heat Transfer, Volume 118, 1996, Pages 789-792.
  • Z. Altac, Exact series expansions, recurrence relations, properties and integrals of the generalized exponential integral functions, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 104, Issue 3, April 2007, Pages 310-325.
  • Y. Aygar, E. Bairamov, Properties of the two-dimensional exponential integral functions, Journal of Mathematical Chemistry, May 2011, Volume 49, Issue 5, pp 1014-1025.
  • E. Bairamov, N. Ozalp, On convergence, asymptotic behaviour and computational algorithm of generalized exponential integral functions arising in a radiative transfer, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 113, Issue 14, September 2012, Pages 1818-1825.
  • E. Bairamov, S. Yardimci, Uniform convergence and the properties of the exponential and generalized exponential integral functions, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 111, Issue 16, November 2010, Pages 2471-2473. [6] W.F. Breig, A.L. Crosbie, Two-dimensional radiative equilibrium semi-infinite medium subjected to cosine varying radiation. J. Quantitative Spectroscopy & Radiative Transfer 13 (1973) 1395–1419.
  • W.F. Breig, A.L. Crosbie, Numerical computation of a generalized exponential integral function. Math. Comput. 28 (1974) 575–579. I.M. Busbridge, The mathematics of radiative transfer. Cambridge 1960.
  • S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960.
  • C. Chiccoli, S. Lorenzutta, G. Maino; Concerning some integrals of the generalized exponential-integral function. Computers & Mathematics with Applications, Volume 23, Issue 11, June 1992, Pages 13-21.
  • A.L. Crosbie, R.L. Dougherty, Two-dimensional in a cylindrical geometry with anisotropic scattering. J. Quantitative Spectroscopy & Radiative Transfer, 25 (1981) 551–569.
  • A.L. Crosbie, L.C. Lee, Relation between multidimensional radiative transfer in cylindrical and rectangular coordinates with anisotropic scattering. J. Quantitative Spectroscopy & Radiative Transfer, 38 (1987) 231–241.
  • A.L. Crosbie, L.C. Lee, Multidimensional radiative transfer: a single integral represantation of anisotropic scattering kernels J. Quantitative Spectroscopy & Radiative Transfer, 42 (1989) 239–246.
  • I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series and products Academic Press, NewYork, 2007.
  • I.I. Guseinov, B.A. Mamedov, On the evalution of generalized exponential integral functions, J. Quantitative Spectroscopy & Radiative Transfer, 102 (2006) 251-256.
  • V. Kourganoff, Basic Methods in Transfer Problems, Dover Publications, New York, 1963.
  • B.A. Mamedov; Evaluation of generalized exponential integral function using binomial expansion theorems, Journal of Quantitative Spectroscopy and Radiative Transfer, Volume 94, Issues 3-4, 1 September 2005, Pages 507-514.
  • S.M. Nikolsky,. Course of Mathematical Analysis. Vol. 2, Mir Publishers, Moscow 1977.
  • R.H. Prabha, R.D.S. Yadav, Polynomial expressions for Bickley and exponential integral functions. Ann. Nucl. Energy, 23 (1996) 1021–1025.
  • V.V. Sobelev, A Treatise on Radiative Transfer Princeton, New York, 1963
  • H.C. Van de Hulst, Multiple Light Scattering, Vol. 1, Academic Press, New York, 1980.
  • W.W Yuen, C.F. Ho, Analysis of two-dimensional radiative heat transfer in a gray medium with internal heat generation, Int. J. Heat Mass Transfer, 28 (1985) 17-23.
  • V.A. Zorich, Mathematical Analysis. Vol. 2, Springer, New York 2004.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Şeyhmus Yardımcı

Murat Olgun

Çağla Can This is me

Publication Date April 1, 2016
Published in Issue Year 2016 Volume: 45 Issue: 2

Cite

APA Yardımcı, Ş., Olgun, M., & Can, Ç. (2016). Numerical computation and properties of the two dimensional exponential integrals. Hacettepe Journal of Mathematics and Statistics, 45(2), 483-495.
AMA Yardımcı Ş, Olgun M, Can Ç. Numerical computation and properties of the two dimensional exponential integrals. Hacettepe Journal of Mathematics and Statistics. April 2016;45(2):483-495.
Chicago Yardımcı, Şeyhmus, Murat Olgun, and Çağla Can. “Numerical Computation and Properties of the Two Dimensional Exponential Integrals”. Hacettepe Journal of Mathematics and Statistics 45, no. 2 (April 2016): 483-95.
EndNote Yardımcı Ş, Olgun M, Can Ç (April 1, 2016) Numerical computation and properties of the two dimensional exponential integrals. Hacettepe Journal of Mathematics and Statistics 45 2 483–495.
IEEE Ş. Yardımcı, M. Olgun, and Ç. Can, “Numerical computation and properties of the two dimensional exponential integrals”, Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, pp. 483–495, 2016.
ISNAD Yardımcı, Şeyhmus et al. “Numerical Computation and Properties of the Two Dimensional Exponential Integrals”. Hacettepe Journal of Mathematics and Statistics 45/2 (April 2016), 483-495.
JAMA Yardımcı Ş, Olgun M, Can Ç. Numerical computation and properties of the two dimensional exponential integrals. Hacettepe Journal of Mathematics and Statistics. 2016;45:483–495.
MLA Yardımcı, Şeyhmus et al. “Numerical Computation and Properties of the Two Dimensional Exponential Integrals”. Hacettepe Journal of Mathematics and Statistics, vol. 45, no. 2, 2016, pp. 483-95.
Vancouver Yardımcı Ş, Olgun M, Can Ç. Numerical computation and properties of the two dimensional exponential integrals. Hacettepe Journal of Mathematics and Statistics. 2016;45(2):483-95.