zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Front-type solutions of fractional Allen-Cahn equation. (English) Zbl 1160.35442
Summary: Super-diffusive front dynamics have been analysed via a fractional analogue of the Allen-Cahn equation. One-dimensional kink shape and such characteristics as slope at origin and domain wall dynamics have been computed numerically and satisfactorily approximated by variational techniques for a set of anomaly exponents $1<\gamma <2$. The dynamics of a two-dimensional curved front has been considered. Also, the time dependence of coarsening rates during the various evolution stages was analysed in one and two spatial dimensions.

MSC:
35K57Reaction-diffusion equations
35K55Nonlinear parabolic equations
35K60Nonlinear initial value problems for linear parabolic equations
35Q35PDEs in connection with fluid mechanics
35B40Asymptotic behavior of solutions of PDE
26A33Fractional derivatives and integrals (real functions)
35S10Initial value problems for pseudodifferential operators
WorldCat.org
Full Text: DOI
References:
[1] Pismen, L. M.: Patterns and interfaces in dissipative dynamics. (2006) · Zbl 1098.37001
[2] Murray, J. D.: Mathematical biology. (1993) · Zbl 0779.92001
[3] Pelcé, P.: Dynamics of curved fronts. (1988) · Zbl 0712.76009
[4] Allen, S. M.; Cahn, J. W.: A microscope theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta metall. 27, No. 6, 1085-1095 (1979)
[5] Fisher, R. A.: The wave of advance of advantageous genes. Ann. eugenics 7, 355-369 (1937) · Zbl 63.1111.04
[6] Kolmogoroff, A. N.; Pertrovsky, I. G.; Piscounoff, N. S.: Study of the diffusion equation with growth of the quantity of matter and its application to a biological problem. Moscow univ. Bull. math. 1, 1-25 (1937)
[7] Balk, A. M.: Anomalous behaviour of a passive tracer in wave turbulence. J. fluid mech. 467, 163-203 (2002) · Zbl 1035.76024
[8] Meerschaert, M. M.; Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. comput. Appl. math. 172, 65-77 (2004) · Zbl 1126.76346
[9] Viswanathan, G. M.; Afanasyev, V.; Buldyrev, S. V.; Murphy, E. J.; Prince, P. A.; Stanley, H. E.: Lévy flight search patterns of wandering albatrosses. Nature 381, 413-415 (1996)
[10] Metzler, R.; Klafter, J.: The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. rep. 339, No. 1, 1-77 (2000) · Zbl 0984.82032
[11] Klafter, J.; Zumofen, G.; Shlesinger, M. F.: F.mallamaceh.e.stanleythe physics of complex systems. The physics of complex systems (1997)
[12] Mancinelli, R.; Vergni, D.; Vulpiani, A.: Superfast front propagation in reaction systems with anomalous diffusion. Europhys. lett. 60, 532 (2002) · Zbl 1058.80004
[13] Del Castillo-Negrete, D.; Carreras, B. A.; Lynch, V. E.: Front dynamics in reaction-diffusion systems with Lévy flights: A fractional diffusion approach. Phys. rev. Lett. 91, No. 1, 018302 (2003)
[14] Brockmann, D.; Hufnagel, L.: Front propagation in reaction-superdiffusion dynamics: taming Lévy flights with fluctuations. Phys. rev. Lett. 98, 178301 (2007)
[15] Heinrich, H.: Non-equilibrium phase transitions with long-range interactions. J. stat. Mech. theory. Exp. 07, P07006 (2007)
[16] Frontera, C.; Vives, E.; Planes, A.: Monte Carlo study of the relation between vacancy diffusion and domain growth in two-dimensional binary alloys. Phys. rev. B 48, No. 13, 9321-9326 (1993)
[17] Marconi, U. Marini Bettolo: Interface pinning and slow ordering kinetics on infinitely ramified fractal structures. Phys. rev. E 57, No. 2, 1290-1301 (1998)
[18] Kawasaki, K.; Ohta, T.: Kink dynamics in one-dimensional non-linear systems. Physica A 116, No. 3, 573-593 (1982)
[19] Nagai, T.; Kawasaki, K.: Molecular dynamics of interacting kinks. I. Physica A 120, No. 3, 587-599 (1983)
[20] Bogdan, K.; Kulczycki, T.; Kwaśnicki, M.: Estimates and structure of ${\alpha}$-harmonic functions. Probab. theory related fields 140, 345-381 (2008) · Zbl 1146.31004