TY - JOUR
T1 - A Neumann series of Bessel functions representation for solutions of perturbed Bessel equations
AU - Kravchenko, Vladislav V.
AU - Torba, Sergii M.
AU - Castillo-Pérez, Raúl
N1 - Publisher Copyright:
© 2017 Informa UK Limited, trading as Taylor & Francis Group.
PY - 2018/4/4
Y1 - 2018/4/4
N2 - A new representation for a regular solution of the perturbed Bessel equation of the form (Formula presented.) is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to (Formula presented.). For the coefficients of the series, explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier–Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley–Wiener theorem, and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.
AB - A new representation for a regular solution of the perturbed Bessel equation of the form (Formula presented.) is obtained. The solution is represented as a Neumann series of Bessel functions uniformly convergent with respect to (Formula presented.). For the coefficients of the series, explicit direct formulas are obtained in terms of the systems of recursive integrals arising in the spectral parameter power series (SPPS) method, as well as convenient for numerical computation recurrent integration formulas. The result is based on application of several ideas from the classical transmutation (transformation) operator theory, recently discovered mapping properties of the transmutation operators involved and a Fourier–Legendre series expansion of the transmutation kernel. For convergence rate estimates, asymptotic formulas, a Paley–Wiener theorem, and some results from constructive approximation theory were used. We show that the analytical representation obtained among other possible applications offers a simple and efficient numerical method able to compute large sets of eigendata with a nondeteriorating accuracy.
KW - Legendre polynomials
KW - Neumann series of Bessel functions
KW - Perturbed Bessel equation
KW - convergent rate estimates
KW - numerical methods
KW - representation of solution
KW - spectral problems
KW - transmutation operator
UR - http://www.scopus.com/inward/record.url?scp=85011299413&partnerID=8YFLogxK
U2 - 10.1080/00036811.2017.1284313
DO - 10.1080/00036811.2017.1284313
M3 - Artículo
SN - 0003-6811
VL - 97
SP - 677
EP - 704
JO - Applicable Analysis
JF - Applicable Analysis
IS - 5
ER -