Fractional differential equations with dependence on the Caputo-Katugampola derivative

By Ricardo Almeida, Agnieszka B. Malinowska, Tatiana Odzijewicz. 2016

185 citations recorded by the source corpus.

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 proven. 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.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/2f0e30ca77f02e297fbdb8524b233365b0bafdc2

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In Applied Mathematics

In Modeling and Simulation

Citations in the corpus

New fractional integral unifying six existing fractional integrals (2016, unknown): “Further applications of Katugampola fractional integrals or derivatives are in probability theory [1], variational calculus [2], inequalities [9], Langevin equations [34], Fourier and Laplace transforms [20] fractional differential equations [4] and Numerical analysis [3], among others.”
Existence for fractional Dirichlet boundary value problem under barrier strip conditions (2017, unknown): “5, then, f(t, x,p) > 0 for (t, x,p) ∈ [0, 1]× [−1, 1]× ([4, 5]∪ [−2,−1]), f(t, x,p) 6 0 for (t, x,p) ∈ [0, 1]× [−1, 1]× ([1, 2]∪ [0, 0.”
A Gronwall inequality for a general Caputo fractional operator (2017, background): “Later, in [1], the authors present a Caputo type fractional derivative of order α ∈ (0, 1), and some properties are proven.”
Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals ☆ (2016, background): “A Caputo-type modification of the operator in question can be found in [4].”
Fractional differential equations with a Caputo derivative with respect to a Kernel function and their applications (2018, background): “Moreover, assume that f(t, 0) = 0 for all t, and that for all b > 0 and all continuous functions y ∶ [a,∞) → [b, xa] the following lim t→∞ I a+ f (t, y(t)) = −∞ (11) holds.”
On the generalized fractional derivatives and their Caputo modification (2017, background): “The authors in [5] did define the Caputo version of the generalized fractional derivatives.”