Fractional Calculus  

In General > s.a. analysis / differential forms; functional analysis [functional derivatives]; vector calculus.
* History: In a letter to Guillaume de l'Hôpital dated 1695, Leibniz asked what meaning one might assign to a non-integer-order differential; Scientists who followed up and developed the theory include Jacques Hadamard, Paul Pierre Lévy, Joseph Liouville, Bernhard Riemann, and Hermann Weyl.
* Properties: One of their general properties is non-locality for non-integer order; They are related to (multi)fractals, and have applications in engineering (viscoelastic materials), time series, kinetics of chaotic systems, polymer science, biophysics; They can behave as dissipative terms in dynamical systems.
@ Books: Oldham & Spanier 74 [classic]; Hilfer ed-00 [intro and applications]; Baleanu et al 16.
@ For physicists: Uchaikin 13; Herrmann 14 [e1 r: PT(12)feb]; He IJTP(14) [and fractal spacetime, tutorial].
@ General references: in Gel'fand & Shilov 64; in Bateman 54, v2, chXIII; Nigmatullin TMP(92); Kolwankar a1307-proc [rev], CFC-a1312; Geisinger JMP(14) [proof of Weyl's law]; Mijena & Nane SPA-a1409 [spacetime fractional stochastic partial differential equations]; Giusti ND(18)-a1710 [broader class of fractional operators]; Zine & Torres FrFr(20)-a2008 [stochastic fractional calculus, and stochastic processes].
@ Fractional derivatives: Kobelev a1202-wd [new type]; Oliveira & Capelas de Oliveira a1704 [Hilfer-Katugampola]; Gladkina et al a1801 [expansion in terms of integer derivative series]; Mingarelli JPA(18)-a1803; Bagarello RMJM-a1912; Capelas de Oliveira et al a1912; Makris a2006 [of the Dirac delta function].
@ Properties: Weberszpil BSCAM(15)-a1405 [modified fractional chain rule]; Shchedrin et al a1803 [generalized product and chain rules].
@ Examples: Shchedrin et al SciPP(18)-a1711 [exact results for a class of elementary functions].
@ Interpretation, mathematical applications: Rutman TMP(95); Podlubny FCAA(02)m.CA/01 [and integration]; Cottone & Di Paola PEM(09)-a1301 [probabilistic characterization of random variables]; > s.a. differential equations [fractional]; fractals [relationship].
> Online resources: see MathPages page; MathWorld page; Wikipedia page.

Physics Applications > s.a. description of chaos; variational principles.
* In quantum theory: The space-fractional Schrödinger equation provides a natural extension of the standard Schrödinger equation when the Brownian trajectories in Feynman path integrals are replaced by Lévy flights.
@ Classical mechanics: Mainardi in(97)-a1201 [viscoelastic bodies, etc]; Fa PhyA(05) [and dissipation, falling body]; Kobelev Chaos(06)-a1101 [post-Newtonian mechanics]; Stanislavsky EPJB(06)-a1111 [Hamiltonian, and fractional oscillators]; Atanacković et al JPA(08) [Euler-Lagrange equations], JPA(10) [generalized Hamilton principle]; Tarasov 11 [fractional dynamics]; Laskin a1302 [fractional classical mechanics]; Korichi & Meftah JMP(14) [statistical mechanics].
@ Wave equations: Mainardi FBMG-a1202 [wave propagation in viscoelastic media]; Näsholm & Holm a1202-conf [acoustic]; Luchko a1311-conf [multi-dimensional]; > s.a. types of wave equations.
@ Field theory: Baleanu & Muslih PS(05) [fractional Lagrangian]; Muslih & Baleanu NCB(05) [quantum field theory]; Baleanu & Vacaru CEJP(11)-a1007 [fractional configurations in gravity theories and Lagrange mechanics]; Calcagni FrontP(18)-a1801-proc [multifractional calculus and quantum gravity]; Behtouei et al a2004 [electromagnetism]; > s.a. dirac fields; klein-gordon fields; modified electromagnetism.
@ Cosmology: Roberts a0909; Shchigolev MPLA(21)-a2104 [accelerated expansion].
@ Quantum mechanics: Muslih & Rabei MPLA(05); Muslih et al PS(06) [Hamiltonian formulation and path-integral quantization]; Weberszpil et al a1206-conf [coarse-grained formulation]; Godinho et al IJTP(14)-a1208 [and non-commutativity]; Longhi OL(15)-a1501 [application of the fractional Schrödinger equation in optics]; Moniz & Jalalzadeh Math(20)-a2003, Rasouli et al MPLA(21) [quantum cosmology].
@ Quantum field theory: Kleinert EPL(12)-a1210 [strongly-interacting many-particle systems]; Calcagni a2102 [scalar and gravitational fields].
@ Other physics applications: Hilfer ed-00; Kobelev m.CA/00 [on multifractal sets], ht/00 [and multifractal spacetime]; > s.a. hamilton-jacobi theory.
> Physical systems: see classical systems [non-conservative]; diffusion; dissipative systems; Feynman-Kac Formula; fokker-planck equation; gauge transformations; hamiltonian systems; lagrangian systems; markov process; MOND; Relaxation Phenomena.

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