Fractional Calculus: An Introduction for Physicists

By Ralf Metzler. 2012

718 citations recorded by the source corpus.

Subfields: Mathematical Physics, Modeling and Simulation, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/a608e8c62d0396669c61e07a34e9ef36d09479f5

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Citations in the corpus

Science metrics on fractional calculus development since 1966 (2013, background): “Figure 2 (left) depicts the evolution of Da [19, 12, 56, 75, 57, 99, 60, 87, 100, 100, 77, 68, 85, 95, 69, 70, 71, 72, 78, 23, 102, 13, 96, 64, 114, 36, 79, 73, 91, 63, 84, 80, 105, 109, 4, 111, 82, 22, 27, 28, 110, 81, 67, 32, 29, 86, 52, 33, 117, 106, 55, 58, 112, 113, 108, 6, 15, 2, 41, 59, 10, 4”
The Chronicles of Fractional Calculus (2017, neutral): “2009 [6, 90, 122, 27, 89, 77, 155] [109, 17] 2010 [111, 31, 133, 165, 46] [98, 106, 63] 2011 [156, 144, 87, 65, 126, 92, 3, 187, 207, 78, 44, 93] [86, 18]”
A Gronwall inequality and the Cauchy-type problem by means of ψ-Hilfer operator (2017, background): “The emergence of new fractional integrals and derivatives, makes the wide number of definitions becomes increasingly larger and clears its numerous applications [1, 2, 3].”
A class of fractional differential hemivariational inequalities with application to contact problem (2018, background): “The fractional calculus, as a natural generalization of the classical integer order calculus, provides a precise description of some physical phenomena for viscoelastic materials, for example, fractional Kelvin–Voigt constitutive laws and fractional Maxwell model [16,42,51].”
A new truncated $M$-fractional derivative type unifying some fractional derivative types with classical properties (2017, background): “One of the main reasons is that the simplest differential equations have the ability to model more complex natural systems [1, 3, 11].”
Review of Some Promising Fractional Physical Models (2013, background): “It should be noted the books on fractional differential equation by Podlubny [7] and an introduction to fractional calculus for physicists by Herrmann [8].”
On the ψ-Hilfer fractional derivative (2017, background): “These integrals and fractional derivatives have a different kernel and this makes the number of definitions [1, 2, 15, 16, 17, 18, 19] wide.”
Stability of the fractional Volterra integro‐differential equation by means of ψ‐Hilfer operator (2018, background): “The solution of such fractional differential equations, in general, allows a better evaluation of the results in several fields of science, for example in engineering, physics and medicine, among others [2, 4, 9, 10, 11].”
On the existence and stability for noninstantaneous impulsive fractional integrodifferential equation (2018, background): “With the diffusion of the fractional calculus and with new emergencies of fractional derivatives and integrals, we aim to propose and study results of existence, unity and stability in order to consolidate good works in the area, in order to strengthen the theory of stability within the unified math”
Initial-boundary-value problems for the one-dimensional time-fractional diffusion equation (2011, background): “The recent book [4] deals with applications of fractional differential equations in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy, and quantum field theory.”
Fractional wave equation and damped waves (2012, neutral): “and [4] for applications of fractional differential equations in classical mechanics, quantum mechanics, nuclear physics, hadron spectroscopy, and quantum field theory.”
Fractional Calculus: Quo Vadimus? (Where are we Going?) (2015, background): “One example is the fractional extension of SO(n), which besides rotational also describes vibrational degrees of freedom (see [27] and references therein).”
Classes of operators in fractional calculus: A case study (2021, background): “The definitions (5)–(6) of Prabhakar derivatives, of Riemann–Liouville and Caputo type, respectively, both involve combining ordinary derivatives with the operator aI n−β,−ρ α,ω where n − 1 ≤ Re(β) < n and so 0 < Re(n − β) ≤ 1.”
Advanced Local Fractional Calculus and Its Applications (2012, neutral): “[228] R.”