[Paper Review] Fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives and their applications

Ismail T. Huseynov, Arzu Ahmadova+1 more|arXiv (Cornell University)|Dec 18, 2020

Fractional Differential Equations Solutions36 references13 citations

TL;DR

This paper establishes fractional Leibniz integral rules for Riemann-Liouville and Caputo fractional derivatives of general order $ n-1 < \alpha \leq n $, enabling verification by substitution of candidate solutions to inhomogeneous multi-term fractional differential equations. The key contribution is the derivation of explicit analytical solutions for generalized Bagley-Torvik and oscillator equations using fractional Green's functions and bivariate Mittag-Leffler functions, validated via the new integral rules.

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

Motivation & Objective

• To address the lack of systematic development in fractional Leibniz integral rules for general fractional orders beyond $ \alpha \in (0,1) $.
• To develop a generalized fractional Leibniz rule applicable to Riemann-Liouville and Caputo derivatives of order $ n-1 < \alpha \leq n $.
• To provide a rigorous verification framework for candidate solutions of inhomogeneous multi-term fractional differential equations using differentiation under the integral sign.
• To derive explicit analytical solutions for generalized Bagley-Torvik and oscillator equations using fractional Green's function methods.
• To establish connections between solutions expressed in bivariate Mittag-Leffler functions and known special functions like Fox-Wright and multi-parameter Mittag-Leffler functions.

Proposed method

• Derivation of a generalized fractional Leibniz integral rule for Riemann-Liouville and Caputo derivatives of arbitrary order $ \alpha \in (n-1, n] $, extending classical results.
• Application of the fractional Leibniz rule to verify candidate solutions of multi-term fractional differential equations via substitution.
• Use of fractional Green's functions to construct explicit solutions for generalized Bagley-Torvik and oscillator equations.
• Employment of bivariate Mittag-Leffler functions $ E_{2,1,2} $, $ E_{2,1,1} $, etc., as fundamental solution kernels in integral representations.
• Utilization of Laplace integral transform techniques to derive and validate solutions in terms of Mittag-Leffler-type functions.
• Application of Pascal’s rule for binomial coefficients to simplify and verify higher-order derivative expressions in the verification process.

Experimental results

Research questions

• RQ1 How can the classical Leibniz rule for differentiation under the integral sign be generalized to fractional-order derivatives of arbitrary order $ \alpha \in (n-1, n] $?
• RQ2 What is the explicit form of the fractional Leibniz integral rule for Riemann-Liouville and Caputo derivatives beyond the $ \alpha \in (0,1) $ case?
• RQ3 Can the proposed fractional Leibniz rule be used to verify analytical solutions of inhomogeneous multi-term fractional differential equations?
• RQ4 What are the explicit analytical solutions of the generalized Bagley-Torvik and oscillator equations in terms of bivariate Mittag-Leffler functions?
• RQ5 How do the derived solutions relate to known special functions such as Fox-Wright and multi-parameter Mittag-Leffler functions?

Key findings

• The paper derives a generalized fractional Leibniz integral rule for Riemann-Liouville and Caputo derivatives of order $ \alpha \in (n-1, n] $, extending prior results limited to $ \alpha \in (0,1) $.
• Explicit analytical solutions are obtained for the generalized Bagley-Torvik equation using fractional Green's functions and bivariate Mittag-Leffler functions $ E_{2,1,2} $.
• The solution to the oscillator equation is verified using the new fractional Leibniz rule, confirming consistency with the differential equation structure.
• The derived solutions are shown to coincide with Fox-Wright type functions and $ l $-th derivatives of two-parameter Mittag-Leffler functions, establishing connections to known special functions.
• Verification by substitution confirms that the candidate solution $ y(t) = \int_0^t (t-s)E_{2,1,2}(\lambda(t-s)^2, \mu(t-s))g(s)ds $ satisfies the second-order fractional differential equation.
• The method successfully validates solutions through direct differentiation under the integral sign using the new fractional Leibniz rule, including boundary terms and derivative identities.