Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications

By Igor Podlubny. 1999

Citations in the corpus

• fPINNs: Fractional Physics-Informed Neural Networks (2018, neutral): “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.”
• Stability of Caputo fractional differential equations by Lyapunov functions (2015, unknown): “1 in [4], or [13]) t0D q tm(t) = 1 Γ(n− q) d dtn ∫ t t0 (t− s)m(s) ds, t > t0.”
• Key words: Nonlinear Differential-Difference Equations; Exp-Function Method; N-Soliton Solutions (2010, unknown): “Several methods have been used to solve fractional partial differential equations, such as Adomian’s decomposition method (ADM) [6, 7], Fourier transform method [8], Laplace transform method [2, 3, 9], and so on.”
• Spline collocation methods for seismic analysis of multiple degree of freedom systems with visco-elastic dampers using fractional models (2020, unknown): “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”
• Suboptimal control of fractional-order dynamic systems with delay argument (2018, unknown): “There are several definitions for a fractional derivative including Riemann–Liouville, Weyl, Caputo, Marchaud, and Riesz (Samko et al., 1993; Podlubny, 1999; Kilbas et al., 2006).”
• Analog Fractional Order Controller in Temperature and Motor Control Applications (2008, unknown): “The output of a fractional order integrator can be summed with the output of a standard proportional amplifier to implement the PI fractional order controller (FOC) described by Podlubny (1999) and others.”