Fractional Calculus| 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.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 26 apr 2021