Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications

Abstract

In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type differentiation operators with general fractional-order of n − 1 < α ≤ n, n ∈ N . A rule of fractional differentiation under integral sign with general order is necessary and applicable tool for verification by substitution for candidate solutions of inhomogeneous multi-term fractional differential equations. We derive explicit analytical solutions of generalized Bagley-Torvik equations in terms of recently defined bivariate Mittag-Leffler type functions that based on fractional Green's function method and verified solutions by substitution in accordance by applying the fractional Leibniz integral rule. Furthermore, we study an oscillator equation as a special case of differential equations with multi-orders via the Leibniz integral rule.

On a study of Sobolev‐type fractional functional evolution equations

“…The following theorem and its corollary are regarding fractional analogue of the eminent Leibniz integral rule for general order 𝛼 ∈ (n − 1, n], n ∈ N in Riemann-Liouville's sense which is more productive tool for the testing particular solution of inhomogeneous linear multiorder FDEs with variable and constant coefficients is considered by Huseynov et al 49 Theorem 2.1. Let the function K ∶ J × J → R be such that the following assumptions are fulfilled: a.…”

“…In the special cases, Riemann-Riouville-type differentiation under integral sign holds for convolution operator 49 as follows:…”

A new technique for solving Sobolev type fractional multi-order evolution equations

“…The following theorem and its corollary is regarding fractional analogue of the eminent Leibniz integral rule for general order α ∈ (n − 1, n], n ∈ N in Riemann-Liouville's sense which is more productive tool for the testing particular solution of inhomogeneous linear multi-order fractional differential equations with variable and constant coefficients is considered by Huseynov et al [15]. Theorem 2.1.…”

“…In the special cases, Riemann-Riouville type differentiation under integral sign holds for convolution operator [15]:…”

Construction of solutions for delay evolution equations in a Banach space: A delayed Dyson-Phillips series

“…Thus, aside from the uniqueness proof, it remains only to show that U (t) = S(t; τ )x for x ∈ D(A0) is a fundamental solution of (3.1). To do this, with the help of Leibniz integral rule [29], we differentiate last expression as follows:…”