AN INTRODUCTION TO FRACTIONAL CALCULUS

By Paul L. Butzer, U. Westphal. 2000

303 citations recorded by the source corpus.

Subfields: Fluid Flow and Transfer Processes

View Paper: https://www.semanticscholar.org/paper/276f35fe3b2ebad0f1aabc91777cea6186042117

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Fluid Flow and Transfer Processes

Citations in the corpus

Threefold Introduction to Fractional Derivatives (2008, background): “Modern authors also use I−α [97], I−α x [23], aD x [85,94,102], d/dx [102,129], d/d(x−a) [92] instead of Da+.”
Entropies based on fractional calculus (2009, background): “at a precise mathematical formulation of fractional calculus was developed thanks to the contributions of mathematicians like N. H. Abel, J. B. Fourier, J. Liouville and B. Riemann, among many others [9]. More recently, some applications of fractional calculus include the ﬁelds of anomalous diﬀusion”
An application of fractional differential equations to risk theory (2019, background): “Definition 5 The Weyl-Liouville fractional derivatives [37, 8] are special cases of the Riemann-Liouville derivatives, whenever a is replaced by −∞ or b is replaced by ∞ in Definition 4.”
Experimental evidence for fractional time evolution in glass forming materials (2002, background): “7] The special case μ = 1 is sometimes called Caputo fractional derivative [34, 35], others attribute it to Liouville [33].”
The fundamental solution of the space-time fractional diffusion equation (2007, background): “According to Samko, Kilbas & Marichev[39] and Butzer & Westphal[3] the ”regularized” fractional derivative was considered by Liouville himself (but then disregarded).”
Finite Element Method for the Space and Time Fractional Fokker-Planck Equation (2008, background): “The use of fractional calculus to deal with engineering and physical problems has become increasingly popular in recent years [2, 7, 21, 33, 36].”
Thermoelasticity that uses fractional heat conduction equation (2009, background): “Essentials of Fractional Calculus In this section, we recall the main idea of fractional calculus [5, 7, 23, 37, 38, 45, 51, 54].”
Discrete random walk models for space-time fractional diffusion (2002, background): “For more details on this fractional derivative (and its extension to higher orders) we refer the interested reader to Gorenflo and Mainardi [22], Podlubny [50] and Butzer and Westphal[2].”
Time-fractional derivatives in relaxation processes: a tutorial survey (2008, methodology): “actional derivative and of its property (1.13) with respect to the Laplace transform. However the form used by Caputo is found in a paper by Liouville himself as recently noted by Butzer and Westphal [10] but Liouville disregarded this notion because he did not recognize its role. As far as we know,”