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Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems

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  • Kravchenko, Vladislav V.
  • Morelos, Samy
  • Torba, Sergii M.

Abstract

A method for solving spectral problems for the Sturm–Liouville equation (pv′)′−qv+λrv=0 based on the approximation of the Delsarte transmutation operators combined with the Liouville transformation is presented. The problem of numerical approximation of solutions and of eigendata is reduced to approximation of a pair of functions depending on the coefficients p, q and r by a finite linear combination of certain specially constructed functions related to generalized wave polynomials introduced by Khmelnytskaya et al. (2013) and Kravchenko and Torba (2015). The method allows one to compute both lower and higher eigendata with an extreme accuracy. Several necessary results concerning the action of the Liouville transformation on formal powers arising in the method of spectral parameter power series are obtained as well as the transmutation operator for the Sturm–Liouville operator 1r(ddxpddx−q).

Suggested Citation

  • Kravchenko, Vladislav V. & Morelos, Samy & Torba, Sergii M., 2016. "Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems," Applied Mathematics and Computation, Elsevier, vol. 273(C), pages 321-336.
    • Handle: RePEc:eee:apmaco:v:273:y:2016:i:c:p:321-336
    DOI: 10.1016/j.amc.2015.10.011
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    Cited by:

    1. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & José Carlos Dias, 2019. "Pricing Double Barrier Options On Homogeneous Diffusions: A Neumann Series Of Bessel Functions Representation," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(06), pages 1-24, September.
    2. Igor V. Kravchenko & Vladislav V. Kravchenko & Sergii M. Torba & Jos'e Carlos Dias, 2017. "Pricing double barrier options on homogeneous diffusions: a Neumann series of Bessel functions representation," Papers 1712.08247, arXiv.org.
    3. Kravchenko, Vladislav V. & Navarro, Luis J. & Torba, Sergii M., 2017. "Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions," Applied Mathematics and Computation, Elsevier, vol. 314(C), pages 173-192.

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