Fractional Calculus: Integral and Differential Equations of Fractional Order

By Rudolf Gorenflo, Francesco Mainardi. 2008

1,587 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/700b20517249c7cc168b67b0ff73a816663b3c5e

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Citations in the corpus

Hadamard Type Inequalities for m-Convex and ({\alpha},m)-Convex Functions via Fractional Integrals (2012, background): “Properties of this operator can be found in the references [12]-[14].”
AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER (2008, background): “More recently fractional calculus has been applied to continuum and statistical mechanics for viscoelasticity problems, Brownian motion and fractional diffusion-wave equations and many physical phenomena [1-12].”
Review of Some Promising Fractional Physical Models (2013, background): “As a simple example of quantum system, which is described by (90), we can consider the linear fractional oscillator that is an object of numerous investigations [108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120] because of different applications.”
Ostrowski type fractional integral inequalities for MT-convex functions (2015, background): “Fractional calculus [7, 16, 19] was introduced at the end of the nineteenth century by Liouville and Riemann, the subject of which has become a rapidly growing area and has found applications in diverse fields ranging from physical sciences and engineering to biological sciences and economics.”
On fractional Duhamel's principle and its applications (2010, background): “Recall that for the Laplace transform of D +g(t) and D α ∗ g(t), where α ∈ (m − 1,m], the following formulas are valid [12]: L[D +g](s) = s L[g](s)− m−1”
Generalized Taylor's formula (2007, background): “Furthermore, the Caputo fractional derivative is considered recently in many physical and engineering problems [8] more than the Riemann–Liouville fractional derivatives, because it allows traditional initial and boundary conditions to be included in the formulation of the problem.”
The application of continuous-time random walks in finance and economics (2006, background): “(17) in terms of the Mittag-Leffler function of parameter b [9,65,66]: CðtÞ 1⁄4 Ebð tÞ, (21) defined by the following power series in the complex plane:”
Continuous-time random walk and parametric subordination in fractional diffusion (2007, neutral): “For the mathematical details the interested reader is referred to [13,41] on the Caputo derivative, and to [44] on the Feller potentials.”