Publication Cover
Applicable Analysis
An International Journal
Volume 97, 2018 - Issue 5
284
Views
23
CrossRef citations to date
0
Altmetric
Articles

A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

, &
Pages 677-704 | Received 04 Nov 2016, Accepted 11 Jan 2017, Published online: 01 Feb 2017

References

  • Boumenir A , Chanane B . Computing eigenvalues of Sturm–Liouville systems of Bessel type. P Edinburgh Math Soc. 1999;42:257–265.
  • Castillo-Pérez R , Kravchenko VV , Torba SM . Spectral parameter power series for perturbed Bessel equations. Appl Math Comput. 2013;220:676–694.
  • Castillo-Pérez R , Kravchenko VV , Torba SM . Analysis of graded-index optical fibers by the spectral parameter power series method. J Opt. 2015;17:025607 (9p).
  • Chébli H , Fitouhi A , Hamza MM . Expansion in series of Bessel functions and transmutations for perturbed Bessel operators. J Math Anal Appl. 1994;181(3):789–802.
  • Guillot J-C , Ralston JV . Inverse spectral theory for a singular Sturm-Liouville operator on [0,1]. J Differ Equ. 1988;76(2):353–373.
  • Kostenko A , Teschl G . On the singular Weyl–Titchmarsh function of perturbed spherical Schrödinger operators. J Differ Equ. 2011;250:3701–3739.
  • Okamoto K . Fundamentals of optical waveguides. San Diego (CL): Academic Press; 2000.
  • Weidmann J . Spectral theory of ordinary differential operators. Vol. 1258. Lecture notes in mathematics, Berlin: Springer. 1987.
  • Watson GN . A Treatise on the theory of Bessel functions. 2nd ed. Cambridge: Cambridge University Press; 1996. vi+804p.
  • Wilkins JE . Neumann series of Bessel functions. Trans Amer Math Soc. 1948;64:359–385.
  • Baricz A , Jankov D , Pogány TK . Neumann series of Bessel functions. Integral Trans Spec Funct. 2012;23(7):529–538.
  • Kravchenko VV , Navarro LJ , Torba SM . Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. Submitted for publication. Available from: arXiv:1508.02738.
  • Sitnik SM . On solution to the problem of unitary generalization to the Sonine–Poisson transmutations, Belgorod State University Scientific Bulletin. Math Phys. 2010;5(76):N° 18, 135–153. Russian.
  • Santana-Bejarano JYu . Operadores de transmutación para la ecuación de Bessel perturbada y aproximación analítica de sus soluciones [PhD Thesis]. CINVESTAV del IPN. 2016.
  • Kravchenko VV , Torba SM , Santana-Bejarano JYu . Generalized wave polynomials and transmutations related to perturbed Bessel equations. Submitted for publication. Available from: arXiv:1606.07850.
  • Ya V . Volk, On inversion formulas for a differential equation with a singularity at x = 0. Uspehi Matem Nauk (NS). 1953;8(4):141–151.
  • Coz M , Coudray Ch . The Riemann solution and the inverse quantum mechanical problem. J Math Phys. 1976;17(6):888–893.
  • Kostenko A , Sakhnovich A , Teschl G . Inverse eigenvalue problems for perturbed spherical Schrödinger operators. Inverse Prob. 2010;26:105013 (14p).
  • Fitouhi A , Hamza MM . A uniform expansion for the eigenfunction of a singular second-order differential operator. SIAM J. Math. Anal. 1990;21:1619–1632.
  • DeVore RA , Lorentz GG . Constructive approximation. Berlin: Springer-Verlag; 1993. x+449p.
  • Kravchenko VV , Porter RM . Spectral parameter power series for Sturm-Liouville problems. Math Methods Appl Sci. 2010;33:459–468.
  • Khmelnytskaya KV , Kravchenko VV , Rosu HC . Eigenvalue problems, spectral parameter power series, and modern applications. Math Methods Appl Sci. 2015;38:1945–1969.
  • Kravchenko VV , Torba SM . Transmutations and spectral parameter power series in eigenvalue problems. Oper Theory Adv Appl. 2013;228:209–238.
  • Adams RA . Sobolev spaces. Vol. 65. Pure and applied mathematics, New York (NY): Academic Press; 1975.
  • Katznelson Y . An introduction to harmonic analysis. 3rd ed. Cambridge: Cambridge University Press; 2004. xviii+314p.
  • Triebel H . Interpolation theory, function spaces, differential operators. 2nd ed. Amsterdam: North-Holland; 1978. p. 528.
  • Titchmarsh EC . Introduction to the theory of Fourier integrals. 3rd ed. New York (NY): Chelsea; 1986. x+394p.
  • Blancarte H , Campos H , Khmelnytskaya K . Spectral parameter power series method for discontinuous coefficients. Math Methods Appl Sci. 2015;38(10):2000–2011.
  • Suetin PK . Classical orthogonal polynomials. 3rd ed. Moscow: Fizmatlit; 2005. p. 480. Russian.
  • Jackson D . The theory of approximation. Reprint of the 1930 original.Providence (RI): American Mathematical Society; 1994.
  • Prudnikov AP , Brychkov YuA , Marichev OI . Integrals and series. Vol. 2. Special functions, New York (NY): Gordon & Breach Science Publishers; 1986. p. 750.
  • Abramovitz M , Stegun IA . Handbook of mathematical functions. New York (NY): Dover; 1972.
  • Lebedev NM . Special functions and their applications. New York (NY): Dover; 1972.
  • Olver F . Asymptotics and special functions, Wellesley. Massachusets: A K Peters; 1997.
  • Kravchenko VV , Torba SM . Analytic approximation of transmutation operators and applications to highly accurate solution of spectral problems. J Comput Appl Math. 2015;275:1–26.
  • Ledoux V , Van Daele M . Matslise 2.0.: a Matlab toolbox for Sturm-Liouville computations. ACM Trans Math Softw. 2016;42:29: 1–18.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.