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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).

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Figures

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. 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

Grahic Jump Location
Fig. 8

For α=0.5 and ρ=5

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