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Improvements in a method for solving fractional integral equations with some links with fractional differential equations

  • Daniel Cao Labora EMAIL logo und Rosana Rodríguez-López
Veröffentlicht/Copyright: 13. März 2018
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 21 Heft 1

Abstract

In this work, we apply and extend our ideas presented in [4] for solving fractional integral equations with Riemann-Liouville definition. The approach made in [4] turned any linear fractional integral equation with constant coefficients and rational orders into a similar one, but with integer orders. If the right hand side was smooth enough we could differentiate at both sides to arrive to a linear ODE with constant coefficients and some initial conditions, that can be solved via an standard procedure.

In this procedure, there were two major obstacles that did not allow to obtain a full result. These were the assumptions over the smoothness of the source term and the assumption about the rationality of the orders.

So, one of the main topics of this document is to describe a modification of the procedure presented in [4], when the source term is not smooth enough to differentiate the required amount of times. Furthermore, we will also study the fractional integral equations with non-rational orders by a limit process of fractional integral equations with rational orders.

Finally, we will connect the previous material with some fractional differential equations with Caputo derivatives described in [7]. For instance, we will deal with the fractional oscillation equation, the fractional relaxation equation and, specially, its particular case of the Basset problem. We also expose how to compute these solutions for the Riemann-Liouville case.

MSC 2010: Primary 26A33; 34A08; Secondary 76-XX
Key Words and Phrases: fractional differential equations; fractional integral equations; Basset problem

Acknowledgements

The research of R. Rodríguez-López was partially supported by grant number MTM2016-75140-P (AEI/FEDER, UE); and also by grant number MTM2013-43014-P [Ministerio de Economía y Competitividad, co-financed by the European Community fund FEDER].

The research of D. Cao Labora was partially supported by a PhD scholarship from Xunta de Galicia until 15/10/2017 and, since then, by a PhD scholarship from Ministerio de Educación, Cultura y Deporte (FPU).

Furthermore, we want to thank Professor Francesco Mainardi for providing us many documents and information on the topic of the Basset problem. This help allowed us to build new approaches in the resolution of some concrete problems which involve fractional differential equations.

References

[1] R. Ashurov, A. Cabada, B. Turmetov, Operator method for construction of solutions of linear fractional differential equations with constant coefficients. Fract. Calc. Appl. Anal. 19, No 1 (2016), 229–252; 10.1515/fca-2016-0013; https://www.degruyter.com/view/j/fca.2016.19.issue-1/issue-files/fca.2016.19.issue-1.xml.Suche in Google Scholar

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[3] A.B. Basset, On the descent of a sphere in a viscous liquid. Quart. J. Math. 41 (1910), 369–381.10.1038/083521a0Suche in Google Scholar

[4] D. Cao Labora and R. Rodréguez-López, From fractional order equations to integer order equations. Fract. Calc. Appl. Anal. 20, No 6 (2017), 1405–1423; 10.1515/fca-2017-0074; https://www.degruyter.com/view/j/fca.2017.20.issue-6/issue-files/fca.2017.209.issue-6.xml.Suche in Google Scholar

[5] G. Devillanova and G. Marano, A free fractional viscous oscillator as a forced standard damped vibration. Fract. Calc. Appl. Anal. 19, No 2 (2016), 319–356; 10.1515/fca-2016-0018; https://www.degruyter.com/view/j/fca.2016.19.issue-2/issue-files/fca.2016.19.issue-2.xml.Suche in Google Scholar

[6] K. Diethelm and N.J. Ford, Numerical solution of the Bagley-Torvik equation. BIT42, No 3 (2002), 490–507; 10.1023/A:1021973025166.Suche in Google Scholar

[7] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order. Revision at arXiv:0805.3823v1 of A. Carpinteri, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics. Springer Verlag, Vienna & New York (1997), 223–276.10.1007/978-3-7091-2664-6_5Suche in Google Scholar

[8] A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam (2006).Suche in Google Scholar

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[11] S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993).Suche in Google Scholar

[12] S. Staněk, Periodic problem for the generalized Basset fractional differential equation. Fract. Calc. Appl. Anal. 18, No 5 (2015), 1277–1290; 10.1515/fca-2015-0073; https://www.degruyter.com/view/j/fca.2015.18.issue-5/issue-files/fca.2015.18.issue-5.xml.Suche in Google Scholar

Received: 2017-10-22
Published Online: 2018-3-13
Published in Print: 2018-2-23

© 2018 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
https://doi.org/10.1515/fca-2018-0011
Schlagwörter für diesen Artikel
fractional differential equations; fractional integral equations; Basset problem

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA–volume 21–1–2018)
  4. Survey Paper
  5. From continuous time random walks to the generalized diffusion equation
  6. Survey Paper
  7. Properties of the Caputo-Fabrizio fractional derivative and its distributional settings
  8. Research Paper
  9. Exact and numerical solutions of the fractional Sturm–Liouville problem
  10. Research Paper
  11. Some stability properties related to initial time difference for Caputo fractional differential equations
  12. Research Paper
  13. On an eigenvalue problem involving the fractional (s, p)-Laplacian
  14. Research Paper
  15. Diffusion entropy method for ultraslow diffusion using inverse Mittag-Leffler function
  16. Research Paper
  17. Time-fractional diffusion with mass absorption under harmonic impact
  18. Research Paper
  19. Optimal control of linear systems with fractional derivatives
  20. Research Paper
  21. Time-space fractional derivative models for CO2 transport in heterogeneous media
  22. Research Paper
  23. Improvements in a method for solving fractional integral equations with some links with fractional differential equations
  24. Research Paper
  25. On some fractional differential inclusions with random parameters
  26. Research Paper
  27. Initial boundary value problems for a fractional differential equation with hyper-Bessel operator
  28. Research Paper
  29. Mittag-Leffler function and fractional differential equations
  30. Research Paper
  31. Complex spatio-temporal solutions in fractional reaction-diffusion systems near a bifurcation point
  32. Research Paper
  33. Differential and integral relations in the class of multi-index Mittag-Leffler functions
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