On a more general fractional integration by parts formulae and applications

The integration by part comes from the product rule of classical differentiation and integration. The concept was adapted in fractional differential and integration and has several applications in control theory. However, the formulation in fractional calculus is the classical integral of a fractional derivative of a product of a fractional derivative of a given function f and a function g. We argue that, this formulation could be done using only fractional operators; thus, we develop fractional integration by parts for fractional integrals, Riemann–Liouville, Liouville–Caputo, Caputo–Fabrizio and Atangana–Baleanu fractional derivatives. We allow the left and right fractional integrals of order α>0 to act on the integrated terms instead of the usual integral and then make use of the fractional type Leibniz rules to formulate the integration by parts by means of new generalized type fractional operators with binomial coefficients defined for analytic functions. In the case α=1, our formulae of fractional integration by parts results in previously obtained integration by parts in fractional calculus. The two disciplines or branches of mathematics are built differently, while classical differentiation is built with the concept of rate of change of a given function, a fractional differential operator is a convolution.