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Mathematical Notes

, Volume 81, Issue 5–6, pp 620–627 | Cite as

On the convolution equation with positive kernel expressed via an alternating measure

  • B. N. Engibaryan
Article
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Abstract

We consider the integral convolution equation on the half-line or on a finite interval with kernel
$$K(x - t) = \int_a^b {e^{ - \left| {x - t} \right|s} d\sigma (s)} $$
with an alternating measure under the conditions
$$K(x) > 0, \int_a^b {\frac{1}{s}\left| {d\sigma (s)} \right| < + \infty } , \int_{ - \infty }^\infty {K(x)dx = 2} \int_a^b {\frac{1}{s}d\sigma (s) \leqslant 1} .$$
The solution of the nonlinear Ambartsumyan equation
$$\varphi (s) = 1 + \varphi (s) \int_a^b {\frac{{\varphi (p)}}{{s + p}}d\sigma (p)} ,$$
is constructed; it can be effectively used for solving the original convolution equation.

Key words

integral convolution equation nonlinear Ambartsumyan equation alternating measure Wiener-Hopf operator nonlinear factorization equation Volterra equation 

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References

  1. 1.
    V. A. Ambartsumyan, Selected Works (Izd. AN Armyan. SSR, Yerevan, 1960), Vol. 1 [in Russian].Google Scholar
  2. 2.
    S. Chandrasekhar, Radiative Transfer (Oxford University Press, 1950; Inostr. Lit., Moscow, 1953).Google Scholar
  3. 3.
    V. V. Sobolev, Scattering of Light in the Atmospheres of Planets (Nauka, Moscow, 1972) [in Russian].Google Scholar
  4. 4.
    C. Cercignani, Theory and Application of the Boltzmann Equation (Scott. Acad. Press, Edinburgh-London, 1975).MATHGoogle Scholar
  5. 5.
    N. B. Engibaryan and A. A. Arutyunyan, “Integral equations on the half-line with difference kernels and nonlinear functional equations,” Mat. Sb. 97(139) (1 (5)), 35–58 (1975).Google Scholar
  6. 6.
    L. G. Arabadzhyan and N. B. Engibaryan, “Convolution equations and nonlinear functional equations,” in Mathematical Analysis (VINITI, Itogi Naukii Tekhniki, Moscow, 1984), Vol. 22, pp. 175–244 [in Russian].Google Scholar
  7. 7.
    I. Ts. Gokhberg and I. A. Fel’dman, Convolution Equations and Projective Methods of Their Solution (Nauka, Moscow, 1971) [in Russian].Google Scholar
  8. 8.
    L. A. Sakhnovich, “Equations with difference kernel on a finite interval,” Uspekhi Mat. Nauk 35(4), 69–129 (1980).MATHGoogle Scholar
  9. 9.
    J. Casti and R. Kalaba, Imbedding Methods in Applied Mathematics (Addison-Wesley Publ., Mass.-London-Amsterdam, 1973).MATHGoogle Scholar
  10. 10.
    N. B. Engibaryan and M. A. Mnatsakyan, “On an integral equation with difference kernel,” Mat. Zametki 19(6), 927–932 (1976).MATHGoogle Scholar
  11. 11.
    N. B. Engibaryan and B. N. Engibaryan, “Integral convolution equation on the half-line with a totally monotone kernel,” Mat. Sb. 187(10), 53–72 (1996) [Russian Acad. Sci. Sb. Math. 187 (10), 1465–1485 (1996)].Google Scholar
  12. 12.
    B. N. Engibaryan, “On multiple factorization of integral operators,” Zh. Vychisl. Mat. i Mat. Fiz. 37(4), 447–458 (1997) [Comput. Math. Math. Phys. 37 (4), 435–446 (1997)].MATHGoogle Scholar
  13. 13.
    V. V. Ivanov, G. B. Rybicky, and A. M. Kasaurov, Albedo Shifting, Preprint, Ser. no. 3478 (Harvard-Smithsonian Center for Astrophysics, 1992).Google Scholar
  14. 14.
    N. B. Engibaryan and B. N. Engibaryan, “The albedo shifting method,” Astrophizika 38(3), 417–431 (1995) [Astrophysics 38 (3), 232–240 (1995)].Google Scholar
  15. 15.
    A. Kh. Khachatryan and A. N. Afyan, “On the analytic and numerical solution of a radiative transfer problem in the presence of a reflecting surface,” Zh. Vychisl. Mat. i Mat. Fiz. 41(8), 1217–1228 (2001) [Comput. Math. Math. Phys. 41 (8), 1158–1168 (2001)].Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2007

Authors and Affiliations

  • B. N. Engibaryan
    • 1
  1. 1.Mathematics InstituteNational Academy of Sciences of ArmeniaYerevanRussia

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