Analysis of solutions of some multi-term fractional Bessel equations

Pavel B. Dubovski, Jeffrey Slepoi

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We construct the existence theory for generalized fractional Bessel differential equations and find the solutions in the form of fractional or logarithmic fractional power series. We figure out the cases when the series solution is unique, non-unique, or does not exist. The uniqueness theorem in space Cp is proved for the corresponding initial value problem. We are concerned with the following homogeneous generalized fractional Bessel equation i=1mdix α iD α iu(x)+(x β-ν 2)u(x)=0,αi> - >0,β> - >0, which includes the standard fractional and classical Bessel equations as particular cases. Mostly, we consider fractional derivatives in Caputo sense and construct the theory for positive coefficients di. Our theory leads to a threshold admissible value for ν2, which perfectly fits to the known results. Our findings are supported by several numerical examples and counterexamples that justify the necessity of the imposed conditions. The key point in the investigation is forming proper fractional power series leading to an algebraic characteristic equation. Depending on its roots and their multiplicity/complexity, we find the system of linearly independent solutions.

Original languageEnglish
Pages (from-to)1380-1408
Number of pages29
JournalFractional Calculus and Applied Analysis
Volume24
Issue number5
DOIs
StatePublished - 1 Oct 2021

Keywords

  • characteristic equation
  • existence
  • fractional power series
  • generalized fractional Bessel equation
  • logarithmic fractional series
  • threshold value
  • uniqueness

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