[Paper Review] Fractional Calculus: Integral and Differential Equations of Fractional Order

Rudolf Gorenflo, Francesco Mainardi|ArXiv.org|May 25, 2008

Fractional Differential Equations Solutions41 references1,118 citations

TL;DR

This paper provides a comprehensive yet accessible introduction to Riemann-Liouville fractional calculus, focusing on fractional integral and differential equations using Laplace transform techniques. It establishes the analytical solutions of key equations—Abel-type integral equations and relaxation/oscillation equations of fractional order—demonstrating the central role of the Mittag-Leffler function, whose properties are detailed in an appendix.

ABSTRACT

We introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analytical solutions of the most simple linear integral and differential equations of fractional order. We show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

Motivation & Objective

• To present fractional calculus in a way accessible to applied scientists, avoiding excessive mathematical rigor.
• To derive analytical solutions of linear fractional integral and differential equations using Laplace transforms.
• To highlight the fundamental role of the Mittag-Leffler function in solving fractional-order equations.
• To bridge theoretical fractional calculus with physical applications in continuum mechanics, particularly viscoelasticity and relaxation phenomena.
• To provide a self-contained reference on fractional operators and their solutions, supported by an appendix on Mittag-Leffler functions.

Proposed method

• Uses the Riemann-Liouville definition of fractional integration and differentiation for $\alpha > 0$.
• Applies Laplace transforms to fractional integral and differential operators to derive solution techniques.
• Derives solution formulas for Abel-type integral equations of the first and second kind.
• Solves fractional relaxation and oscillation differential equations using Laplace transform pairs.
• Relies on the Mittag-Leffler function $E_{\alpha,\beta}(z)$ as the core solution structure for fractional differential equations.
• Provides explicit Laplace transform pairs involving $s^{-\alpha}$ and rational functions of $s^{1/2}$, linking to error functions in special cases.

Experimental results

Research questions

• RQ1 How can Laplace transforms be systematically applied to solve fractional integral and differential equations?
• RQ2 What is the analytical structure of solutions to Abel-type integral equations of fractional order?
• RQ3 How do fractional relaxation and oscillation equations differ from their integer-order counterparts in behavior and solution form?
• RQ4 What role does the Mittag-Leffler function play in the solutions of fractional differential equations?
• RQ5 In what physical contexts do fractional-order differential equations naturally arise, and how are they solved?

Key findings

• The solution to the Abel integral equation of the second kind is expressed in terms of the Mittag-Leffler function $E_{\alpha,\beta}(z)$, confirming its central role in fractional calculus.
• For $\alpha = 1/2$, the Laplace transform pair $\frac{1}{s^{1/2}(s^{1/2} \pm \lambda)} \div e_{1/2}(t; \pm \lambda) = e^{\lambda^2 t} \text{erfc}(\pm \lambda \sqrt{t})$ is derived, linking fractional operators to error functions.
• The solution to the fractional relaxation equation involves the Mittag-Leffler function $E_{\alpha}(-\lambda t^\alpha)$, which generalizes exponential decay to power-law behavior.
• The fractional oscillation equation yields solutions involving $E_{\alpha,\beta}(-\lambda t^\alpha)$, showing damped oscillatory behavior with long-term power-law decay.
• The Laplace transform technique allows derivation of exact solutions for linear fractional differential equations with constant coefficients, using known transform pairs.
• The Mittag-Leffler function emerges naturally in viscoelastic models, as shown by Caputo and Mainardi, confirming its physical relevance in relaxation processes.