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, 55:11 | Cite as

A Neumann series of Bessel functions representation for solutions of Sturm–Liouville equations

  • Vladislav V. Kravchenko
  • Sergii M. Torba
Article

Abstract

A Neumann series of Bessel functions (NSBF) representation for solutions of Sturm–Liouville equations and for their derivatives is obtained. The representation possesses an attractive feature for applications: for all real values of the spectral parameter \(\omega \) the estimate of the difference between the exact solution and the approximate one (the truncated NSBF) depends on N (the truncation parameter) and the coefficients of the equation and does not depend on \(\omega \). A similar result is valid when \(\omega \in {\mathbb {C}}\) belongs to a strip \(\left| \hbox {Im }\omega \right| <C\). This feature makes the NSBF representation especially useful for applications requiring computation of solutions for large intervals of \(\omega \). Error and decay rate estimates are obtained. An algorithm for solving initial value, boundary value or spectral problems for the Sturm–Liouville equation is developed and illustrated on a test problem.

Keywords

Sturm-Liouville equation Liouville transform Neumann series of Bessel functions Transmutation operator Approximate solution 

Mathematics Subject Classification

34A25 34A45 34B05 34B24 41A10 41A25 42C10 65L05 65L15 

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Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Unidad QuerétaroCINVESTAV del IPNQuerétaroMexico

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