Taiwanese Journal of Mathematics

Exact Bounds and Approximating Solutions to the Fredholm Integral Equations of Chandrasekhar Type

Sheng-Ya Feng and Der-Chen Chang

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In this paper, we study the $L^p$ solutions of the Fredholm integral equations with Chandrasekhar kernels. The Hilbert type inequality is resorted to establish an existence and uniqueness result for the Fredholm integral equation associated with Chandrasekhar kernel. A couple of examples well support the condition and extend the classical results in the literature with one generalizing the classical Chandrasekhar kernel. In order to approximate the original solution, a truncated operator is introduced to overcome the non-compactness of the integral operator. An error estimate of the convergence is made in terms of the truncated parameter, the upper bounds of the symbolic function constituting the integral kernel and initial data to the equation.

Article information

Taiwanese J. Math., Volume 23, Number 2 (2019), 409-425.

Received: 27 June 2018
Accepted: 14 November 2018
First available in Project Euclid: 22 November 2018

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Zentralblatt MATH identifier

Primary: 45B05: Fredholm integral equations
Secondary: 26D15: Inequalities for sums, series and integrals 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30]

Chandrasekhar kernel Hilbert-type inequality Fredholm integral equation $L^p$ norm approximating solution


Feng, Sheng-Ya; Chang, Der-Chen. Exact Bounds and Approximating Solutions to the Fredholm Integral Equations of Chandrasekhar Type. Taiwanese J. Math. 23 (2019), no. 2, 409--425. doi:10.11650/tjm/181108. https://projecteuclid.org/euclid.twjm/1542855640

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  • P. M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice-Hall, Englewood Cliffs, N.J., 1971.
  • I. K. Argyros, Quadratic equations and applications to Chandrasekhar's and related equations, Bull. Austral. Math. Soc. 32 (1985), no. 2, 275–292.
  • ––––, On a class of quadratic integral equations with perturbation, Funct. Approx. Comment. Math. 20 (1992), 51–63.
  • J. Banaś, M. Lecko and W. G. El-Sayed, Existence theorems for some quadratic integral equations, J. Math. Anal. Appl. 222 (1998), no. 1, 276–285.
  • J. Banaś and B. Rzepka, Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl. 332 (2007), no. 2, 1371–1379.
  • P. B. Bosma and W. A. de Rooij, Efficient methods to calculate Chandrasekhar's $H$-functions, Astron. Astrophys. 126 (1983), 283–292.
  • I. W. Busbridge, The Mathematics of Radiative Transfer, Cambridge University Press, Cambridge, 1960.
  • ––––, On solutions of Chandrasekhar's integral equation, Trans. Amer. Math. Soc. 105 (1962), 112–117.
  • J. Caballero, A. B. Mingarelli and K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differential Equations 2006 (2006), no. 57, 11 pp.
  • S. Chandrasekhar, Radiative Transfer, Oxford University Press, London, 1950.
  • D.-C. Chang and S.-Y. Feng, On integral equations of Chandrasekhar type, J. Nonlinear Convex Anal. 19 (2018), no. 3, 525–541.
  • Q. Chen and B. Yang, A survey on the study of Hilbert-type inequalities, J. Inequal Appl. 2015, 2015:302, 29 pp.
  • J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Prentice Hall Series in Computational Mathematics, Prentice Hall, Englewood Cliffs, NJ, 1983.
  • J. Di and Z. Shi, Approximate solution of the integral equation on the half line, J. Zhe Jiang Univ. Tech. 37 (2009), 326–331.
  • A. M. A. El-Sayed and H. H. G. Hashem, Integrable solution for quadratic Hammerstein and quadratic Urysohn functional integral equations, Comment. Math. 48 (2008), no. 2, 199–207.
  • ––––, A coupled system of fractional order integral equations in reflexive Banach spaces, Comment. Math. 52 (2012), no. 1, 21–28.
  • A. M. A. El-Sayed, H. H. G. Hashem and E. A. A. Ziada, Picard and Adomian decomposition methods for a coupled system of quadratic integral equations of fractional order, J. Nonlinear Anal. Optim. 3 (2012), no. 2, 171–183.
  • J. A. Ezquerro and M. A. Hernández, On the application of a fourth-order two-point method to Chandrasekhar's integral equation, Aequationes Math. 62 (2001), no. 1-2, 39–47.
  • H. H. G. Hashem and A. M. A. El-Sayed, Stabilization of coupled systems of quadratic integral equations of Chandrasekhar type, Math. Nachr. 290 (2017), no. 2-3, 341–348.
  • J. Juang, K.-Y. Lin and W.-W. Lin, Spectral analysis of some iterations in the Chandrasekhar's $H$-function, Numer. Funct. Anal. Optim. 24 (2003), no. 5-6, 575–586.
  • P. K. Kythe and P. Puri, Computational Methods for Linear Integral Equations, Birkhäuser Boston, Boston, MA, 2002.
  • R. W. Leggett, A new approach to the $H$-equation of Chandrasekhar, SIAM J. Math. Anal. 7 (1976), no. 4, 542–550.
  • J. M. Ortega and W. C. Rheinboldt, On discretization and differentiation of operators with application to Newton's method, SIAM J. Numer. Anal. 3 (1966), no. 1, 143–156.
  • H. A. H. Salem, On the quadratic integral equations and their applications, Comput. Math. Appl. 62 (2011), no. 8, 2931–2943.
  • I. Schur, Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. Reine Angew. Math. 140 (1911), 1–28.
  • T. Tanaka, Integration of Chandrasekhar's integral equation, Journal of Quantitative Spectroscopy and Radiative Transfer 76 (2003), no. 2, 121–144.
  • F. G. Tricomi, Integral Equations, Pure and Applied Mathematics V, Interscience Publishers, New York, 1957.
  • M. Y. Waziri, W. J. Leong, M. A. Hassan and M. Monsi, A low memory solver for integral equations of Chandrasekhar type in the radiative transfer problems, Math. Probl. Eng. 2011 (2011), Art. ID 467017, 12 pp.
  • H. Weyl, Singulare integral Gleichungen mit besonderer Berücksichtigung des Fourierschen integral theorems, Inaugeral-Dissertation, Gottingen, 1908.
  • B. Yang, On the norm of an integral operator and applications, J. Math. Anal. Appl. 321 (2006), no. 1, 182–192.
  • ––––, On the Norm of Operator and Hilbert-type Inequalities, Science Press, Beijing, 2009.
  • V. A. Zorich, Mathematical Analysis II, Universitext, Springer-Verlag, Berlin, 2004.