Journal
MATHEMATICS
Volume 8, Issue 3, Pages -Publisher
MDPI
DOI: 10.3390/math8030360
Keywords
fractional integrals; Caputo fractional derivatives; fractional differential equations; bivariate Mittag-Leffler functions; 26A33; 34A08
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The Caputo fractional derivative has been one of the most useful operators for modelling non-local behaviours by fractional differential equations. It is defined, for a differentiable function f(t), by a fractional integral operator applied to the derivative f ' (t). We define a new fractional operator by substituting for this f ' (t) a more general proportional derivative. This new operator can also be written as a Riemann-Liouville integral of a proportional derivative, or in some important special cases as a linear combination of a Riemann-Liouville integral and a Caputo derivative. We then conduct some analysis of the new definition: constructing its inverse operator and Laplace transform, solving some fractional differential equations using it, and linking it with a recently described bivariate Mittag-Leffler function.
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