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Analysis of solutions of some multi-term fractional Bessel equations

  • Pavel B. Dubovski EMAIL logo and Jeffrey Slepoi
Published/Copyright: October 28, 2021
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 24 Issue 5

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=1mdixαiDαiu(x)+(xβ−ν2)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.

Key Words and Phrases: generalized fractional Bessel equation; characteristic equation; fractional power series; logarithmic fractional series; existence; uniqueness; threshold value
MSC 2010: Primary 26A33; 33C10; Secondary 33D05; 34A05; 34B30

Acknowledgements

The authors would like to thank Professor L.~Boyadjiev for drawing our attention to the fractional Bessel equation and Professor V.~Kiryakova for the valuable remarks, which helped to improve the paper.

References

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Received: 2020-08-14
Revised: 2021-06-26
Published Online: 2021-10-28
Published in Print: 2021-10-26

© 2021 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books
  4. Survey Paper
  5. Towards a unified theory of fractional and nonlocal vector calculus
  6. Research Paper
  7. An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
  8. Analysis of solutions of some multi-term fractional Bessel equations
  9. Existence of solutions for the semilinear abstract Cauchy problem of fractional order
  10. Summability of formal solutions for a family of generalized moment integro-differential equations
  11. Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
  12. Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
  13. First order plus fractional diffusive delay modeling: Interconnected discrete systems
  14. On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
  15. On the decomposition of solutions: From fractional diffusion to fractional Laplacian
  16. Output error MISO system identification using fractional models
  17. Short Paper
  18. Identification of system with distributed-order derivatives
  19. Short note
  20. On the Green function of the killed fractional Laplacian on the periodic domain
https://doi.org/10.1515/fca-2021-0059
Keywords for this article
generalized fractional Bessel equation; characteristic equation; fractional power series; logarithmic fractional series; existence; uniqueness; threshold value

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books
  4. Survey Paper
  5. Towards a unified theory of fractional and nonlocal vector calculus
  6. Research Paper
  7. An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative
  8. Analysis of solutions of some multi-term fractional Bessel equations
  9. Existence of solutions for the semilinear abstract Cauchy problem of fractional order
  10. Summability of formal solutions for a family of generalized moment integro-differential equations
  11. Analysis and fast approximation of a steady-state spatially-dependent distributed-order space-fractional diffusion equation
  12. Green’s function for the fractional KdV equation on the periodic domain via Mittag–Leffler function
  13. First order plus fractional diffusive delay modeling: Interconnected discrete systems
  14. On a solution of a fractional hyper-Bessel differential equation by means of a multi-index special function
  15. On the decomposition of solutions: From fractional diffusion to fractional Laplacian
  16. Output error MISO system identification using fractional models
  17. Short Paper
  18. Identification of system with distributed-order derivatives
  19. Short note
  20. On the Green function of the killed fractional Laplacian on the periodic domain
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