zbMATH — the first resource for mathematics

Variable order differential equations with piecewise constant order-function and diffusion with changing modes. (English) Zbl 1181.35359
Summary: Diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order (with respect to the time derivative) pseudo-differential equations. Depending on the parameters of variable order derivatives short or long range memories may appear when diffusion modes change. These memory effects are classified and studied in detail. Processes that have distinctive regimes of different types of diffusion depending on time are ubiquitous in the nature. Examples include diffusion in a heterogeneous media and protein movement in cell biology.

35S10 Initial value problems for PDEs with pseudodifferential operators
26A33 Fractional derivatives and integrals
45K05 Integro-partial differential equations
35A08 Fundamental solutions to PDEs
35S15 Boundary value problems for PDEs with pseudodifferential operators
33E12 Mittag-Leffler functions and generalizations
Full Text: DOI Link arXiv
[1] Andries, E., Umarov, S. and Steinberg, S., Monte Carlo random walk simula- tions based on distributed order differential equations with application to cell biology. Frac. Calc. Appl. Anal. 9 (2006)(4), 351 - 369. · Zbl 1132.65114 · eudml:11288 · arxiv:math/0606797
[2] Applebaum, D., Lévy Processes and Stochastic Calculus. Cambridge: Cam- bridge Univ. Press 2004.
[3] Billingsley, P., Probability and Measure. New York: John Wiley & Sons 1995. 449 · Zbl 0822.60002
[4] Caputo, M., Linear models of dissipation whose Q is almost frequency inde- pendent. II. Geophys. J. R. Astr. Soc. 13 (1967), 529 - 539.
[5] Chechkin, A. V., Gorenflo, R. and Sokolov, I. M., Retarding subdiffusion and accelerating superdiffusion governed by distributed order fractional diffusion equation. Phys. Rev. E 66 (2002), 046129, 1 - 6.
[6] Chechkin, A. V., Gorenflo, R. and Sokolov, I. M., Fractional diffusion in inho- mogeneous media. J. Physics A: Math. Gen. 38 (2005), L679 - L684.
[7] Djrbashian, M. M., Harmonic Analysis and Boundary Value Problems in the Complex Domain. Basel: Birkhäuser 1993. · Zbl 0798.43001
[8] Dubinski\?ı, Yu. A., On a method of solving partial differential equations (in Russian). Dokl. Akad. Nauk SSSR 258 (1981), 780 - 784; transl. in: Sov. Math. Dokl. 23 (1981), 583 - 587.
[9] Edidin, M., Lipid microdomains in cell surface membranes. Curr. Opin. Struct. Biol. 7 (1997), 528 - 532.
[10] Ghosh, R. N. and Webb, W. W., Automated detection and tracking of indi- vidual and clustered cell surface low density lipoprotein receptor molecules. Biophys. J. 66 (1994), 1301 - 1318.
[11] Gorenflo, R., Luchko, Yu. and Umarov, S., On the Cauchy and multi-point problems for partial pseudo-differential equations of fractional order. Fract. Calc. Appl. Anal. 3 (2000)(3), 249 - 277. · Zbl 1033.35160
[12] Gorenflo, R. and Mainardi, F., Simply and multiply scaled diffusion limits for continuous time random walk. J. Phys. Conf. Ser. 7 (2005), 1 - 16.
[13] Gorenflo, R., Mainardi, F., Moretti, D., Pagnini, G. and Paradisi, P., Discrete random walk models for space-time fractional diffusion. Chemical Phys. 284 (2002), 521 - 541. · Zbl 0986.82037
[14] Hoh, W., Pseudo differential operators with negative definite symbols of vari- able order. Rev. Mat. Iberoamericana 16 (2000)(2), 219 - 241. · Zbl 0977.35151 · doi:10.4171/RMI/274 · eudml:39606
[15] Jacob, N. and Leopold H.-G., Pseudo-differential operators with variable order of differentiation generating Feller semigroups. Integr. Equ. Oper. Theory 17 (1993), 544 - 553. · Zbl 0793.35139 · doi:10.1007/BF01200393
[16] Liu, F., Shen, S., Anh, V. and Turner, I., Analysis of a discrete non- Markovian random walk approximation for the time fractional diffusion equa- tion. ANZIAM J. 46 (2005), C488 - C504.
[17] Lorenzo, C. F. and Hartley, T. T., Variable order and distributed order frac- tional operators. Nonlin. Dynam. 29 (2002), 57 - 98. · Zbl 1018.93007 · doi:10.1023/A:1016586905654
[18] Meerschaert, M. and Scheffler, H.-P., Stochastic model for ultraslow diffusion. Stochastic Process. Appl. 116 (2006)(9), 1215 - 1235. · Zbl 1100.60024 · doi:10.1016/j.spa.2006.01.006
[19] Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Reports 339 (2000), 1 - 77. · Zbl 0984.82032 · doi:10.1016/S0370-1573(00)00070-3
[20] Metzler, R. and Klafter, J., The restaurant at the end of random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Physics. A: Math. Gen. 37 (2004), R161 - R208. · Zbl 1075.82018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.