The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type

By Kai Diethelm. 2010

3,335 citations recorded by the source corpus.

Subfields: Applied Mathematics, Modeling and Simulation, Numerical Analysis

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Citations in the corpus

fPINNs: Fractional Physics-Informed Neural Networks (2018, background): “A typical example is the fractional derivatives of our interest, whose chain rule, if any, takes the form of infinite series [50, 51, 52]; see Appendix A for comparison of chain rules for the integer-order and the fractional derivatives.”
Attractivity for fractional evolution equations with almost sectorial operators (2018, background): “Introduction In the past thirty years, theory of ordinary and partial differential equations involving fractional derivatives have been developed very rapidly, see the monographs of Kiryakova [11], Podlubny [18], Kilbas et al. [10], Diethelm [5], Zhou [20] and a series of recent research articles [1”
The Grünwald–Letnikov Fractional-Order Derivative with Fixed Memory Length (2013, background): “1) of [24] and taking into account the fact that the Grünwald–Letnikov operator and the Riemann–Liouville operator are equivalent for a suitable set of functions we conclude that the initial value problem (31) has got a uniquely defined continuous solution x1(t) ∈ C(I1).”
A numerical scheme for the solution of a class of fractional variational and optimal control problems using the modified Jacobi polynomials (2016, methodology): “In the investigation of FVPs, there usually exist two definitions to express the fractional derivatives: Caputo and Riemann– Liouville definitions; readers can refer to Diethelm (2010), Hilfer et al. (2000), Kilbas et al. (2006) and Podlubny (1999) for further study.”
Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative (2016, background): “Although many definitions of a different kind of fractional derivatives have been suggested, we will not touch them in this paper and instead the reader is referred to [9], [13], [23], [22], [35], [37] and [44].”
Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models (2020, background): “It generalizes the notions of integer-order differentiation and n-fold integration (Atangana and Baleanu, 2016; Baleanu et al., 2011, 2012; Diethelm, 2010; Edwards et al., 2002; Gemant, 1936; Gorenflo and Mainardi, 1997; Kilbas et al., 2006; Magin, 2006; Podlubny, 1999; Sabatier et al., 2007; Tenrei”
Finite Difference Methods for Caputo–Hadamard Fractional Differential Equations (2020, background): “Introduction In recent years, fractional differential equations (FDEs) have received increasing attention because of their widespread applications in science and engineering [2,3,5,11,14,15,18,19].”
Variational iteration method for the Burgers' flow with fractional derivatives—New Lagrange multipliers (2013, neutral): “(III) The simplest variational iteration formula (17) can reduce to the Volterra integral equation and the analysis of the convergence and the existence of the solutions can be found in [26].”
Method of upper and lower solutions for fractional differential equations (2012, background): “D is the Caputo fractional derivative of order δ (see [1, 2]); that is,”
Linearized Asymptotic Stability for Fractional Differential Equations (2015, background): “D S] 8 M 2 } where α > 0 denotes the order of the fractional differential equation, then the equilibrium of the nonlinear fractional differential equation is asymptotically stable.”
Generalized thermoelasticity with memory-dependent derivatives involving two temperatures (2016, background): “Diethelm [16] has developed Caputo [17] derivative to be:”
On tempered fractional calculus with respect to functions and the associated fractional differential equations (2021, background): “For broad overviews of these topics, we refer the reader to the well-known textbooks and survey articles such as [1, 2, 3, 4, 5, 6].”
On a Fractional Operator Combining Proportional and Classical Differintegrals (2020, methodology): “In comparison with the classical Riemann–Liouville fractional derivative, the Caputo one requires more natural initial conditions when it is used for fractional differential equations [12].”
Existence and Uniqueness results for a class of Generalized Fractional Differential Equations (2014, methodology): “The proof of this theorem uses the following lemma, which asserts the connection between a FDE and the corresponding Volterra type integral equation [7, 8].”