In mathematics, Katugampola fractional operators are integral operators that generalize the Riemann–Liouville and the Hadamard fractional operators into a unique form.[1][2][3][4] The Katugampola fractional integral generalizes both the Riemann–Liouville fractional integral and the Hadamard fractional integral into a single form and It is also closely related to the Erdelyi–Kober[5][6][7][8] operator that generalizes the Riemann–Liouville fractional integral. Katugampola fractional derivative[2][3][4] has been defined using the Katugampola fractional integral[3] and as with any other fractional differential operator, it also extends the possibility of taking real number powers or complex number powers of the integral and differential operators.

Definitions

These operators have been defined on the following extended-Lebesgue space.

Let be the space of those Lebesgue measurable functions on for which , where the norm is defined by [1]

for and for the case

Katugampola fractional integral

It is defined via the following integrals [1][2][9][10][11]

 

 

 

 

(1)

  for and This integral is called the left-sided fractional integral. Similarly, the right-sided fractional integral is defined by,  

 

 

 

 

(2)

  for and .

These are the fractional generalizations of the -fold left- and right-integrals of the form

and

for

respectively. Even though the integral operators in question are close resemblance of the famous Erdélyi–Kober operator, it is not possible to obtain the Hadamard fractional integrals as a direct consequence of the Erdélyi–Kober operators. Also, there is a corresponding fractional derivative, which generalizes the Riemann–Liouville and the Hadamard fractional derivatives. As with the case of fractional integrals, the same is not true for the Erdélyi–Kober operator.

Katugampola fractional derivative

As with the case of other fractional derivatives, it is defined via the Katugampola fractional integral.[3][9][10][11]

Let and The generalized fractional derivatives, corresponding to the generalized fractional integrals (1) and (2) are defined, respectively, for , by

The half-derivative of the function for the Katugampola fractional derivative.
The half derivative of the function for the Katugampola fractional derivative for and .

and

respectively, if the integrals exist.

These operators generalize the Riemann–Liouville and Hadamard fractional derivatives into a single form, while the Erdelyi–Kober fractional is a generalization of the Riemann–Liouville fractional derivative.[3] When, , the fractional derivatives are referred to as Weyl-type derivatives.

Caputo–Katugampola fractional derivative

There is a Caputo-type modification of the Katugampola derivative that is now known as the Caputo–Katugampola fractional derivative.[12][13] Let and . The C-K fractional derivative of order of the function with respect to parameter can be expressed as

It satisfies the following result. Assume that , then the C-K derivative has the following equivalent form

Hilfer–Katugampola fractional derivative

Another recent generalization is the Hilfer-Katugampola fractional derivative.[14][15] Let order and type . The fractional derivative (left-sided/right-sided), with respect to , with , is defined by

where , for functions in which the expression on the right hand side exists, where is the generalized fractional integral given in (1).

Mellin transform

As in the case of Laplace transforms, Mellin transforms will be used specially when solving differential equations. The Mellin transforms of the left-sided and right-sided versions of Katugampola Integral operators are given by [2][4]

Theorem

Let and Then,

for , if exists for .

Hermite-Hadamard type inequalities

Katugampola operators satisfy the following Hermite-Hadamard type inequalities:[16]

Theorem

Let and . If is a convex function on , then

where .

When , in the above result, the following Hadamard type inequality holds:[16]

Corollary

Let . If is a convex function on , then

where and are left- and right-sided Hadamard fractional integrals.

Recent Development

These operators have been mentioned in the following works:

  1. Fractional Calculus. An Introduction for Physicists, by Richard Herrmann [17]
  2. Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics, Tatiana Odzijewicz, Agnieszka B. Malinowska and Delfim F. M. Torres, Abstract and Applied Analysis, Vol 2012 (2012), Article ID 871912, 24 pages [18]
  3. Introduction to the Fractional Calculus of Variations, Agnieszka B Malinowska and Delfim F. M. Torres, Imperial College Press, 2015
  4. Advanced Methods in the Fractional Calculus of Variations, Malinowska, Agnieszka B., Odzijewicz, Tatiana, Torres, Delfim F.M., Springer, 2015
  5. Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative, Shakoor Pooseh, Ricardo Almeida, and Delfim F. M. Torres, Numerical Functional Analysis and Optimization, Vol 33, Issue 3, 2012, pp 301–319.[19]

References

  1. 1 2 3 Katugampola, Udita N. (2011). "New approach to a generalized fractional integral". Applied Mathematics and Computation. 218 (3): 860–865. arXiv:1010.0742. doi:10.1016/j.amc.2011.03.062. S2CID 27479409.
  2. 1 2 3 4 Katugampola, Udita N. (2011). On Generalized Fractional Integrals and Derivatives, Ph.D. Dissertation, Southern Illinois University, Carbondale, August, 2011.
  3. 1 2 3 4 5 Katugampola, Udita N. (2014), "New Approach to Generalized Fractional Derivatives" (PDF), Bull. Math. Anal. App., 6 (4): 1–15, MR 3298307
  4. 1 2 3 Katugampola, Udita N. (2015). "Mellin transforms of generalized fractional integrals and derivatives". Applied Mathematics and Computation. 257: 566–580. arXiv:1112.6031. doi:10.1016/j.amc.2014.12.067. S2CID 28000114.
  5. Erdélyi, Arthur (1950–51). "On some functional transformations". Rendiconti del Seminario Matematico dell'Università e del Politecnico di Torino. 10: 217–234. MR 0047818.
  6. Kober, Hermann (1940). "On fractional integrals and derivatives". The Quarterly Journal of Mathematics (Oxford Series). 11 (1): 193–211. Bibcode:1940QJMat..11..193K. doi:10.1093/qmath/os-11.1.193.
  7. Fractional Integrals and Derivatives: Theory and Applications, by Samko, S.; Kilbas, A.A.; and Marichev, O. Hardcover: 1006 pages. Publisher: Taylor & Francis Books. ISBN 2-88124-864-0
  8. Theory and Applications of Fractional Differential Equations, by Kilbas, A. A.; Srivastava, H. M.; and Trujillo, J. J. Amsterdam, Netherlands, Elsevier, February 2006. ISBN 0-444-51832-0
  9. 1 2 Thaiprayoon, Chatthai; Ntouyas, Sotiris K; Tariboon, Jessada (2015). "On the nonlocal Katugampola fractional integral conditions for fractional Langevin equation". Advances in Difference Equations. 2015. doi:10.1186/s13662-015-0712-3.
  10. 1 2 Almeida, R.; Bastos, N. (2016). "An approximation formula for the Katugampola integral" (PDF). J. Math. Anal. 7 (1): 23–30. arXiv:1512.03791. Bibcode:2015arXiv151203791A. Archived from the original (PDF) on 2016-03-04. Retrieved 2016-01-02.
  11. 1 2 Katugampola, Udita. "Google Site". Retrieved 11 November 2017. {{cite journal}}: Cite journal requires |journal= (help)
  12. Almeida, Ricardo (2017). "Variational Problems Involving a Caputo-Type Fractional Derivative". Journal of Optimization Theory and Applications. 174 (1): 276–294. arXiv:1601.07376. doi:10.1007/s10957-016-0883-4. S2CID 6350899.
  13. Zeng, Sheng-Da; Baleanu, Dumitru; Bai, Yunru; Wu, Guocheng (2017). "Fractional differential equations of Caputo–Katugampola type and numerical solutions". Applied Mathematics and Computation. 315: 549–554. doi:10.1016/j.amc.2017.07.003.
  14. Oliveira, D.S.; Capelas de Oliveira, E. (2017). "Hilfer-Katugampola fractional derivative". arXiv:1705.07733 [math.CA].
  15. Bhairat, Sandeep P.; Dhaigude, D.B. (2017). "Existence and Stability of Fractional Differential Equations Involving Generalized Katugampola Derivative". arXiv:1709.08838 [math.CA].
  16. 1 2 M. Jleli; D. O'Regan; B. Samet (2016). "On Hermite-Hadamard Type Inequalities via Generalized Fractional Integrals" (PDF). Turkish Journal of Mathematics. 40: 1221–1230. doi:10.3906/mat-1507-79.
  17. Fractional Calculus. An Introduction for Physicists, by Richard Herrmann. Hardcover. Publisher: World Scientific, Singapore; (February 2011) ISBN 978-981-4340-24-3
  18. Odzijewicz, Tatiana; Malinowska, Agnieszka B.; Torres, Delfim F. M. (2012). "Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics". Abstract and Applied Analysis. 2012: 1–24. arXiv:1203.1961. doi:10.1155/2012/871912. S2CID 8270676.
  19. Pooseh, Shakoor; Almeida, Ricardo; Torres, Delfim F. M. (2012). "Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative". Numerical Functional Analysis and Optimization. 33 (3): 301. arXiv:1112.0693. doi:10.1080/01630563.2011.647197. S2CID 119144021.

Further reading

  • Miller, Kenneth S. (1993). Ross, Bertram (ed.). An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley. ISBN 0-471-58884-9.
  • Oldham, Keith B.; Spanier, Jerome (1974). The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order. Mathematics in Science and Engineering. Vol. V. Academic Press. ISBN 0-12-525550-0.
  • Podlubny, Igor (1998). Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering. Vol. 198. Academic Press. ISBN 0-12-558840-2.
  • Herrmann, Richard (2011). Fractional Calculus. An Introduction for Physicists. World Scientific. ISBN 978-981-4340-24-3.
  • Machado, J.T.; Kiryakova, V.; Mainardi, F. (2011). "Recent history of fractional calculus" (PDF). Communications in Nonlinear Science and Numerical Simulations. 16 (3): 1140. Bibcode:2011CNSNS..16.1140M. doi:10.1016/j.cnsns.2010.05.027. hdl:10400.22/4149. Archived from the original (PDF) on 2013-10-20. Retrieved 2016-01-02.

Notes

The CRONE (R) Toolbox, a Matlab and Simulink Toolbox dedicated to fractional calculus, can be downloaded at http://cronetoolbox.ims-bordeaux.fr

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