Research Papers

Fractional Differential Equations With Dependence on the Caputo–Katugampola Derivative

[+] Author and Article Information
Ricardo Almeida

Center for Research and Development in
Mathematics and Applications (CIDMA),
Department of Mathematics,
University of Aveiro,
Aveiro 3810–193, Portugal
e-mail: ricardo.almeida@ua.pt

Agnieszka B. Malinowska

Faculty of Computer Science,
Bialystok University of Technology,
Białystok 15-351, Poland
e-mail: a.malinowska@pb.edu.pl

Tatiana Odzijewicz

Department of Mathematics
and Mathematical Economics,
Warsaw School of Economics,
Warsaw 02-554, Poland
e-mail: tatiana.odzijewicz@sgh.waw.pl

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 12, 2015; final manuscript received July 29, 2016; published online September 16, 2016. Assoc. Editor: Brian Feeny.

J. Comput. Nonlinear Dynam 11(6), 061017 (Sep 16, 2016) (11 pages) Paper No: CND-15-1334; doi: 10.1115/1.4034432 History: Received October 12, 2015; Revised July 29, 2016

In this paper, we present a new type of fractional operator, the Caputo–Katugampola derivative. The Caputo and the Caputo–Hadamard fractional derivatives are special cases of this new operator. An existence and uniqueness theorem for a fractional Cauchy-type problem, with dependence on the Caputo–Katugampola derivative, is proved. A decomposition formula for the Caputo–Katugampola derivative is obtained. This formula allows us to provide a simple numerical procedure to solve the fractional differential equation (FDE).

Copyright © 2016 by ASME
Your Session has timed out. Please sign back in to continue.


Kilbas, A. A. , Srivastava, H. M. , and Trujillo, J. J. , 2006, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands.
Podlubny, I. , 1999, Fractional Differential Equations Mathematics in Science and Engineering, Vol. 198, Academic Press, San Diego, CA.
Samko, S. G. , Kilbas, A. A. , and Marichev, O. I. , 1993, Fractional Integrals and Derivatives [translated from the 1987 Russian original], Gordon and Breach, Yverdon, Switzerland.
Katugampola, U. N. , 2011, “ New Approach to a Generalized Fractional Integral,” Appl. Math. Comput., 218(3), pp. 860–865.
Katugampola, U. N. , 2014, “ A New Approach to Generalized Fractional Derivatives,” Bull. Math. Anal. Appl., 6(4), pp. 1–15.
Katugampola, U. N. , 2016, “ Existence and Uniqueness Results for a Class of Generalized Fractional Differential Equations,” arXiv:1411.5229
Gambo, Y. Y. , Jarad, F. , Baleanu, D. , and Abdeljawad, T. , 2014, “ On Caputo Modification of the Hadamard Fractional Derivatives,” Adv. Differ. Equations, 2014(10), pp. 1–12.
Jarad, F. , Abdeljawad, T. , and Baleanu, D. , 2012, “ Caputo-Type Modification of the Hadamard Fractional Derivatives,” Adv. Differ. Equations, 2012(142), pp. 1–8.
Kilbas, A. A. , and Marzan, S. A. , 2004, “ Cauchy Problem for Differential Equation With Caputo Derivative,” Fractional Calculus Appl. Anal., 7(3), pp. 297–321.
Atanacković, T. M. , and Stankovic, B. , 2008, “ On a Numerical Scheme for Solving Differential Equations of Fractional Order,” Mech. Res. Commun., 35(7), pp. 429–438. [CrossRef]
Pooseh, S. , Almeida, R. , and Torres, D. F. M. , 2013, “ Numerical Approximations of Fractional Derivatives With Applications,” Asian J. Control, 15(3), pp. 698–712. [CrossRef]
Pooseh, S. , Almeida, R. , and Torres, D. F. M. , 2012, “ Expansion Formulas in Terms of Integer-Order Derivatives for the Hadamard Fractional Integral and Derivative,” Numer. Funct. Anal. Optim., 33(3), pp. 301–319. [CrossRef]
Ford, N. J. , and Morgado, M. L. , 2011, “ Fractional Boundary Value Problems: Analysis and Numerical Methods,” Fractional Calculus Appl. Anal., 14(4), pp. 554–567.
Ford, N. J. , and Morgado, L. M. , 2012, “ Distributed Order Equations as Boundary Value Problems,” Comput. Math. Appl., 64(10), pp. 2973–2981. [CrossRef]
Gracia, J. L. , and Stynes, M. , 2015, “ Central Difference Approximation of Convection in Caputo Fractional Derivative Two-Point Boundary Value Problems,” J. Comput. Appl. Math., 273(C), pp. 103–115. [CrossRef]
Sousa, E. , 2014, “ An Explicit High Order Method for Fractional Advection Diffusion Equations,” J. Comput. Phys., 278, pp. 257–274. [CrossRef]
Yan, Y. , Pal, K. , and Ford, N. J. , 2014, “ Higher Order Numerical Methods for Solving Fractional Differential Equations,” BIT Numer. Math., 54(2), pp. 555–584. [CrossRef]
Yang, Q. , Liu, F. , and Turner, I. , 2010, “ Numerical Methods for Fractional Partial Differential Equations With Riesz Space Fractional Derivatives,” Appl. Math. Modell., 34(1), pp. 200–218. [CrossRef]
Atanacković, T. M. , Janevb, M. , Pilipovicc, S. , and Zoricab, D. , 2014, “ Convergence Analysis of a Numerical Scheme for Two Classes of Non-Linear Fractional Differential Equations,” Appl. Math. Comput., 243, pp. 611–623.
Garrappa, R. , 2007, “ Some Formulas for Sums of Binomial Coefficients and Gamma Functions,” Int. Math. Forum, 2(13–16), pp. 725–733. [CrossRef]
Tricomi, F. G. , and Erdélyi, A. , 1951, “ The Asymptotic Expansion of a Ratio of Gamma Functions,” Pac. J. Math., 1(1), pp. 133–142. [CrossRef]


Grahic Jump Location
Fig. 1

For α=0.5 and ρ=0.6

Grahic Jump Location
Fig. 2

For α=0.7 and ρ=0.2

Grahic Jump Location
Fig. 3

For α=0.4 and ρ=1.5

Grahic Jump Location
Fig. 8

For α=0.5 and ρ=5

Grahic Jump Location
Fig. 4

For α=0.5 and ρ=0.6

Grahic Jump Location
Fig. 5

For α=0.7 and ρ=0.2

Grahic Jump Location
Fig. 6

For α=0.4 and ρ=1.5

Grahic Jump Location
Fig. 7

For α=0.9 and ρ=1.5



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In