This report looks at the published literary sources on methods and approaches, which are based on fractional calculus (that is, calculus of integrals and derivatives of any arbitrary real or complex order) to solve the continuous and discrete mechanics problems. The problems solutions and comparative analysis results of fractional and classical models are presented. The report’s goal is to show an efficiency of using fractional calculus to describe the mechanical processes.
SamkoSGKilbasAAMarichevOI. Fractional integrals and derivatives: theory and applications. Tokyo, Paris, Berlin and Langhorne (Pennsylvania): Gordon and Breach Science Publishers, 1993.
2.
OldhamKBSpanierJ. The fractional calculus. New York, London: Academic Press, 1974.
3.
MillerKRossB. An introduction to the fractional calculus and fractional differential equations. New York: Wiley, 1993.
4.
RossB. Brief history and exposition of the fundamental theory of fractional calculus (Lecture Notes in Mathematics). Berlin: Springer, 1975.
5.
McBrideAC. Fractional calculus and integral transforms of generalized functions (Research Notes in Mathem., vol. 31). San Francisco, CA, London: Pitman, 1979.
6.
KiryakovaV. Generalized fractional calculus and applications (Pitman Research Notes in Mathematics, 301). Harlow: Longman Sci and Tech., 1994.
7.
PodlubnyI. Fractional differential equations. San Diego, CA: Mathematics in Sciences and Engineering, 1999.
8.
ButzerPLWestphalU. An introduction to fractional calculus. In: HilferR (ed.) Applications of fractional calculus in physics. Singapore: World Scientific Publishing Co. Pte Ltd, 2000.
9.
HilferR. Fractional time evolution. In: HilferR (ed.) Applications of fractional calculus in physics. Singapore: World Scientific Publishing Co. Pte Ltd, 2000.
10.
DjrbashianMMNersesianAB. Fractional derivatives and the Cauchy problem for differential equations of fractional order. Izv Acad Nauk Armjansk SSR1968; 3: 3–29.
11.
DjrbashianMM. Harmonic analysis and boundary value problems in the complex domain. In: RoccoAWestBJ (eds) Operator theory: advances and applications. Basel, Switzerland: Birkhäuser, 1993.
12.
MandelbrotBB. The fractal geometry of nature. San Francisco, CA: W.H. Freeman and Co, 1982.
13.
XieHKwasniewskiMA. Fractals in rock mechanics (Geomechanics Research Series). Balkema, Rotterdam: Taylor & Francis, 1993.
14.
SukmonoS. Fractal geometry of the Sumatra active fault system and its geodynamical implications. J Geodyn1996; 22: 1–9.
15.
BourdinLOdzijewiczTTorresDFM. Existence of minimizers for fractional variational problems containing Caputo derivatives. Adv Dyn Syst Appl2013; 8: 3–12.
16.
BastosNRO. Ferreira, RAC, and Torres, DFM. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin Dyn Syst2011; 29: 417–437.
17.
BastosNROFerreiraRACTorresDFM. Discrete-time fractional variational problems. Signal Process2011; 91: 513–524.
18.
FerreiraRACTorresDFM. Fractional $h$-difference equations arising from the calculus of variations. Appl Anal Discrete Math2011; 5: 110–121.
19.
KolwankarKM. Fractional differentiability of nowhere differentiable functions and dimensions. PhD Thesis, University of Pune, 1997.
20.
WestBJBolognaMGrigoliniP. Physics of fractal operators. New York: Springer, 2003.
MandelbrotBB. Fractals, form, chance and dimension. San Francisco, CA: W.H.Freeman and Co., 1977.
23.
FederJ. Fractals. New York: Plenum, 1988.
24.
TakayasuH. Fractals in the physical science. Manchester: Manchester University Press, 1990.
25.
FalconerK. Fractal geometry: mathematical foundations and applications. Chichester: Wiley, 1990.
26.
BagleyRLTorvikPJ. On the fractional calculus model of viscoelastic behavior. J Rheol1986; 30: 133–155.
27.
TatomFB. The relationship between fractional calculus and fractals. Fractals1995; 3: 217–229.
28.
BorodichFM. Fractals and fractal scaling in fracture mechanics. Int J Fracture1999; 95: 239–259.
29.
BorodichFM. Parametric homogeneity and non-classical self-similarity. I. Mathematical background. Acta Mech1998; 131: 27–45.
30.
BorodichFMGalanovBA. Non-direct estimations of adhesive and elastic properties of materials by depth-sensing indentation. Proc R Soc Lond A2008; 464: 2098, 2759–2776.
31.
BorodichFMOnishenkoDA. Similarity and fractality in the modeling of the modeling of roughness by a multilevel profile with hierarchical structure. Solid Struct1999; 36: 2585–2612.
32.
SaichevAIZaslavskyGM. Fractional kinetic equations: solutions and applications. Chaos1997; 7: 753–764.
33.
GorenfloRMainardiF. Fractional calculus and stable probability distributions. Arch Mech1998; 50: 377–388.
34.
BensonDA. The fractional advection-dispersion equation: development and application. PhD Thesis. University of Nev. Reno, 1998.
35.
ChavesAS. A fractional diffusion equation to describe Le'vy flights. Phys Lett1998; 239: 13–16.
TurcotteDL. The relationship of fractals in geophysics to “The New Science”. Chaos Soliton Fractal2004; 19: 255–258.
40.
BazantZPYavariA. Is the cause of size effect on structural strength fractal or energetic-statistical?Eng Fract Mech2005; 72: 1-31.
41.
ZhuravkovMAStagurovaOVKovalevaMA. Geomechanical monitoring of the rock massif. Minsk: Unikap, 2002 (in Russian).
42.
KurleniaMVOparinVNVostrikovVI. About formation of elastic wave packages at pulse excitement of block media. Waves of pendular type Uμ. In: Proceedings of the USSR Academy of Sciences, 333, 1993 (in Russian).
43.
CowanGAPinesDMeltzerD. Complexity: metaphors, models and reality. Menlo Park, CA: Addison-Wesley, 1994.
44.
MandelbrotBB. Topics on fractals in mathematics and physics. In: LouisHY. (eds.) Challenges for the 21th century fundamental sciences: mathematics and theoretical physics. Singapore: World Scientific Publishing Co. Pte Ltd, 2000.
45.
GhilM. Turbulence and predictability in geophysical fluid dynamics and climate dynamics. Amsterdam: North-Holland, 1985.
46.
MeakinP. Fractals, scaling and growth far from equilibrium (Cambridge Nonlinear Science Series 5). Cambridge: Cambridge University Press, 1998.
47.
BatchelorGK. Perspectives in fluid dynamics. A collective introduction to current research. Cambridge: Cambridge University Press, 2000.
48.
FrischU. Turbulence. Cambridge: Cambridge University Press, 1995.
49.
PierJ-PBostonB. Development of mathematics 1950-2000. Berlin: Birkhauser, 2000.
RieweF. Mechanics with fractional derivatives. Phys Rev E1997; 55: 3581–3592.
52.
FloresEOslerTJ. The tautochrone under arbitrary potentials using fractional derivatives. Am J Phys1999; 67: 718–722.
53.
AgrawalOP. Formulation of Euler-Lagrange equations for fractional variational problems. J Math Anal Appl2002; 272: 368–379.
54.
AlmeidaRMalinowskaABTorresDFM. A fractional calculus of variations for multiple integrals with application to vibrating string. J Math Phys2010; 51: 033503.
AlmeidaRTorresDFM. Calculus of variations with fractional derivatives and fractional integrals. Appl Math Lett2009; 22: 1816–1820.
57.
AlmeidaRTorresDFM. Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives. Commun Nonlinear Sci Numer Simul2011; 16: 1490–1500.
58.
FredericoGSFTorresDFM. A formulation of Noether’s theorem for fractional problems of the calculus of variations. J Math Anal Appl2007; 334: 834–846.
59.
FredericoGSFTorresDFM. Fractional Noether’s theorem in the Riesz-Caputo sense. Appl Math Comput2010; 217: 1023–1033.
60.
OdzijewiczTMalinowskaABTorresDFM. Fractional variational calculus with classical and combined Caputo derivatives. Nonlinear Anal2012; 75: 1507–1515.
61.
OdzijewiczTMalinowskaABTorresDFM. Generalized fractional calculus with applications to the calculus of variations. Comput Math Appl2012; 64: 3351–3366.
62.
OdzijewiczTMalinowskaABTorresDFM. Fractional calculus of variations in terms of a generalized fractional integral with applications to physics. Abstr Appl Anal2012; Art. ID 871912 .
63.
WesterlandS. Causality. Report No 940426. University of Kalmar, 1994.
64.
BrownEB. Measurement of molecular diffusion in solution by multiphoton fluorescence photobleaching recovery. Biophys J1999; 77: 2837–2849.
65.
RoccoA. Fractional calculus and the evolution of fractal phenomena. Phys Stat Mech Appl1999; 265: 536–546.
NuttingPG. A new general law deformation. J Franklin Inst1921; 191: 678–685.
69.
SchiesselHFriedrichCBlumenA. Applications to problems in polymer physics and rheology. In: HilferR (ed.) Applications of Fractional Calculus in Physics. Singapore: World Scientific Publishing Co. Pte Ltd, 2000, 331–376.
70.
GlockleWGNonnenmacherTF. Fractional integral operators and fox function in the theory of viscoelasticity. Macromolecules1991; 24: 6426–6434.
71.
NonnenmacherTFMetzlerR. On the Riemann-Liouville fractional calculus and some recent applications. Fractals1995; 3: 557–566.
72.
FriedrichC. Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta1991; 30: 151–158.
73.
SchiesselH. Generalized viscoelastic models: their fractional equations with solutions. J Phys Math Gen1995; 28: 6567–6584.
74.
GlockleWGNonnenmacherTF. Fox function representations of non-Debye relaxation processes. J Stat Phys1993; 71: 741–757.
75.
HilferR. Fractional calculus and regular variation in thermodynamics. In: HilferR (ed.) Applications of fractional calculus in physics. Singapore: World Scientific Publishing Co. Pte Ltd., 2000, 430–463.
76.
El-SayedAMAGaafarFM. Fractional calculus and some intermediate physical processes. Appl Math Comput2003; 144: 117–126.
77.
AharonyAAsikainenJ. Critical phenomena. In: FamilyFDaoudMHerrmannHJ. (eds.) Scaling and disordered systems. Singapore: World Scientific Publishing Co. Pte Ltd, 2002, 1–79.
78.
ZhuravkovMAStarovoitovEI. Mechanics of continuous medium. Theory of elasticity and plasticity. Minsk: BSU, 2011 (in Russian).
79.
Can MeralF. Magin surface response of a fractional order viscoelastic halfspace to surface and subsurface sources. J Acoust Soc Am2009; 126: 3278–3285.
80.
MaginRL. Fractional calculus in bioengineering. Reding, CT: Begell House, 2006.
81.
RabotnovYN. Elements of hereditary solid mechanics. Moscow: Mir Publishers, 1980.
82.
CraiemDArmentanoD. A fractional derivative model to describe arterial viscoelasticity. Biorheology2007; 44: 251–263.
83.
KissMZVargheseTHallTJ. Viscoelastic characterization of in vitro canine tissue. Phys Med Biol2004; 49: 4207–4218.
84.
GemantA. A method of analyzing experimental results obtained by elasto-viscous bodies. Physics1936; 7: 311–317.
85.
GemantA. On fractional differentials. Philos Mag1938; 25: 540–549.
86.
GerasimovAN. A generalization of linear laws of deformation and its application to problems of internal friction. Akad Nauk SSSR Prikl Mat Mekh1948; 12: 251–260 (in Russian).
87.
SloninskyGL. Laws of mechanical relaxation processes in polymers. J Polymer Sci C Polymer Lett1967; 16: 1667–1672.
88.
FriedrichChSchiesselHBlumenA. Constitutive behaviour modeling and fractional derivatives. In: SiginerDADe KeeDChhabraRP (eds.) Advances in the flow and rheology of non-Newtonian fluids. Amsterdam: Elsevier, 1999, 429–466.
89.
ZhuravkovMA. Mathematical modeling of deformation processes in firm deformable medium (on the example of problems of rocks and massifs mechanics). Minsk: BSU, 2002.
90.
Scott-BlairGWGaffynJE. An application of the theory of quasi-properties to the treatment of anomalous strain-stress relations. Philos Mag1949; 40: 80–94.
91.
CaputoM. Elasticità e Dissipazione. Bologna: Zanichelli, 1969.
92.
CaputoM. Vibrations of an infinite plate with a frequency independent Q. J Acoust Soc Am1976; 60: 634–639.
93.
CaputoMMainardiF. A new dissipation model based on memory mechanism. Pure Appl Geophys1971; 91: 134–147.
94.
StiassnieM. On the application of fractional calculus on the formulation of viscoelastic models. Appl Math Model1979; 3: 300–302.
95.
RousePE. The theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys1953; 21: 1272–1280.
96.
FerryJDLandelRFWilliamsML. Extensions of the Rouse theory of viscoelastic properties to undiluted linear polymers. J Appl Phys1955; 26: 359–362.
97.
BagleyRL. Applications of generalized derivatives to viscoelasticity. PhD Dissertation, Air Force Institute of Technology; also published as Air Force Materials Laboratory TR-79-4103, November, 1979.
98.
BagleyRLTorvikPJ. Fractional calculus – a different approach to the analysis of viscoelastically damped structures. AIAA J1983; 21: 741–748.
99.
RabotnovYN. Some questions of the creep theory. Vestnik MSU1948; 10: 81–91 (in Russian).
100.
RabotnovYN. Balance of the elastic medium with a consequence. Appl Math Mech1948; 12: 53–62 (in Russian).
101.
RabotnovYN. Creep of structural elements. Moskow: Nauka, 1966 (in Russian).
102.
MeshkovSIPachevskayaGNShermergorTD. To the description of internal friction in terms of fractional exponential kernels. Zh Prikl Mekh Fiziki1966; 3: 106–107 (in Russian).
103.
MeshkovSI. Description of internal friction in the memory theory of elasticity using kernels with a weak singularity. J Appl Mech Tech Phys1967; 4: 100–102.
104.
MeshkovSI. The integral representation of fractionally exponential functions and their application to dynamic problems of linear viscoelasticity, J Appl Mech Tech Phys1970; 11: 100–107.
105.
ZelenevVMMeshkovSIRossikhinYuA. Damped vibrations of hereditary – elastic systems with weakly singular kernels. J Appl Mech Tech Phys1970; 11: 290–293.
106.
RossikhinYuA. Dynamic problems of linear viscoelasticity connected with the investigation of retardation and relaxation spectra. PhD Dissertation, Voronezh Polytechnic Institute, Voronezh (in Russian).
107.
GonsovskiVLRossikhinYuA. Stress waves in a viscoelastic medium with a singular hereditary kernel. J Appl Mech Tech Phys1973; 14: 595–597.
108.
KoellerRC. Polynomial operators, Stieltjes convolution, and fractional calculus in hereditary mechanics. Acta Mech1986; 58: 251–264.
109.
RossikhinYAShitikovaMV. Comparative analysis of viscoelastic models involving fractional derivatives of different orders. FC&AA2007; 10: 111–121.
110.
SokolovskiiVV. Elastic plastic wave propagation in bars. Appl Math Mech1948; 12: 261–280.
111.
EfrosAMDanikevskiiAM. Operational calculation and contour integral. Kharkov: ONTU, 1937.
112.
RossikhinYShitikovaM. Applications of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl Mech Rev1997; 50: 15–67.
113.
RossikhinYShitikovaM. A new method for solving dynamic problems of fractional derivative viscoelasticity. Int J Eng Sci2000; 39: 149–176.
114.
KoellerRC. Applications of fractional calculus to the theory of viscoelasticity. J Appl Mech1984; 51: 299–307.
115.
SchiesselHBlumenA. Mesoscopic pictures of the sol-gel transition: ladder models and fractal networks. Macromolecules1995; 28: 4013–4019.
116.
SchiesselHBlumenA. Hierarchical analogues to fractional relaxation equations. J Phys Math Gen1993; 15: 5057–5069.
HeymansNBauwensJ-C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol Acta1994; 33: 210–219.
119.
FriedrichC. Relaxation and retardation functions of the Maxwell model with fractional derivatives. Rheol Acta1991; 30: 151–158.
120.
PritzT. Analysis of four-parameter fractional derivative model of real solid materials. J Sound Vib1996; 195: 103–115.
121.
GlocleWGNonenmacherTF. Fractional integral operators and fox function in the theory of viscoelasticity. Macromolecules1991; 24: 6426–6434.
122.
ZenerCM. Elasticity and unelasticity of metals. Chicago, IL: Chicago University Press, 1948.
123.
BeyerHKempfleS. Definition of physically damping laws with fractional derivatives. ZAMM1995; 75: 623–635.
124.
PadovanJSawickiJT. Diophantine type fractional derivative representation of structural hysteresis. Part I: Formulation. Comput Mech1997; 19: 335–340.
125.
PadovanJSawickiJT. Diophantine type fractional derivative representation of structural hysteresis. Part II: Fitting. Comput Mech1997; 19: 341–355.
126.
PaladeL-IDeSantoJA. Dispersion equations which model high-frequency linear viscoelastic behavior as described by fractional derivative models. Int J Non Lin Mech2001; 36: 13–24.
127.
Hernandez-JimenezA. Relaxation modulus in PMMA and PTFE fitting by fractional Maxwell model. Polymer Test2001; 21: 325–331.
128.
AtanackovicTM. A modified Zener model of a viscoelastic body. Continuum Mech Thermodyn2002; 14: 137–148.
129.
WenchangTWenxiaoPMingyuX. A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int J Non Lin Mech2003; 38: 645–650.
130.
CataniaGSorrentinoS. Analytical modeling and experimental identification of viscoelastic mechanical systems. In: SabatierJ. (eds.) Advances in fractional calculus: theoretical developments and applications in physics and engineering. Dordrecht: Springer, 2007, 403–416.
HwangJSWangJC. Seismic response prediction of high damping rubber bearings fractional derivatives Maxwell model. Eng Struct1998; 20: 849–856.
133.
ParkSW. Analytical modeling of viscoelastic dampers for structural and vibration control. Int J Solid Struct2001; 38: 8065–8092.
134.
KohCGKellyLM. Application of fractional derivatives to seismic analysis of base isolated models. Earthquake Eng Struct Dyn1990; 19: 229–241.
135.
BagleyRL. Power law and fractional calculus model of viscoelasticity. AIAA J1989; 27: 1412–1417.
136.
MakrisN. Complex parameter Kelvin model for elastic foundations. Earthquake Eng Struct Dyn1994; 23: 251–264.
137.
MakrisNConstantinouMC. Fractional derivative Maxwell model for viscous dampers. J Struct Eng ASCE1991; 117: 2708–2724.
138.
FreedADiethelmKLuchkoY. Fractional-order viscoelasticity (FOV): Constitutive development using the fractional calculus: First annual report NASA/TM 2002-211914, NASA’s Glenn Research Center, Ohio, 2002, 1–138.
139.
AdolfssonK. Nonlinear fractional order viscoelasticity at large strains. Nonlin Dyn2004; 38: 233–246.
140.
AnhVVLeonenkoNN. Spectral analysis of fractional kinetic equations with random data. J Statist Phys2001; 104: 1349–1387.
141.
Ruiz-MedinaMDAnguloJMAnhVV. Fractional generalized random fields on bounded domains. Stoch Anal Appl2003; 21: 465–492.
142.
AnhVVHeydeCCLeonenkoNN. Dynamic models of long-memory processes driven by Lévy noise. J Appl Probab2002; 39: 730–747.
143.
BrinsonHFBrinsonLC. Polymer engineering science and viscoelasticity. An introduction. Berlin: Springer, 2008.
144.
FerryJD. Viscoelastic properties of polymers. 3rd ed.New York: Wiley, 1980.
145.
DoiMEdwardsSF. The theory of polymer dynamics. Oxford: Clarendon, 1986.
146.
AlcoutlabiMMartinez-VegaJJ. Modeling of the viscoelastic behavior of amorphous polymers by the differential and integration fractional method: the relaxation spectrum H(τ). Polymer2003; 44: 7199–7208.
147.
DinzartFLipinskiP. Self-consistent approach of the constitutive law of a two-phase visco-elastic material described by fractional derivative models. Arch Mech2010; 62: 135–156.
148.
SassoMPalmieriGAmodioG. Application of fractional derivative models in linear viscoelastic problems. Mech Time Dependent Mater2011; 15: 367–387.
149.
DietrichLTurskiK. Analysis of identification methods for the viscoelastic properties of materials. Eng Trans1992; 40: 501–523.
150.
WarlusSPontonALeslousA. Dynamic viscoelastic properties of silica alkoxide during the sol-gel transition. Eur Phys J E2003; 12: 275–282.
151.
FriedrichCHeymannL. Extension of a model for crosslinking polymer at gel point. J Rheol1988; 32: 235–241.
152.
WarlusSPontonA. A new interpretation for the dynamic behaviour of complex fluids at the sol-gel transition using the fractional calculus. Rheol Acta2009; 48, 51–58.
153.
RoystonTJMansyHASandlerRH. Excitation and propagation of surface waves on a viscoelastic half-space with application to medical diagnosis. J Acoust Soc Am1999; 106: 3678–3686.
MathaiAM. The H-function with applications in statistics and other disciplines. New Delhi: Wiley Eastern Limited, 1978.
156.
SrivastavaHMGuptaKCGoyalSP. The H-functions of one and two variables with applications. New Delhi: South Asian Publishers, 1982.
157.
ErdelyiA. Higher transcendental functions. New York: McGraw-Hill, 1953.
158.
MetzlerRGlockleWGNonenmacherTF. Fractional model equation for anomalous diffusion. Physica1994; 211 A: 13–21.
159.
GorenfloRLuchkoYMainardiF. Wright functions as scale-invariant solutions of the diffusion-wave equation. J Comput Appl Math2000; 118: 175–191.
160.
XuMYTanWC. Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion. Sci China Math Phys Astron Technol Sci2001; 44: 1387–1399.
161.
TanWCXuMY. Unsteady flows of a generalized second grade fluid with fractional derivative model between two parallel plates. Acta Mech Sin2004; 20: 471–476.
162.
TanWCPanWXXuMY. A note on unsteady flows of a viscoelastic fluid with fractional Maxwell model between two parallel plates. Int J Non Lin Mech2003; 38: 645–650.
163.
TanWCXuMY. The impulse motion of flat plate in a general second grade fluid. Mech Res Commun2002; 29: 3–9.
164.
TanWCXianFWeiL. An exact solution of unsteady Coutte flow of generalized second grade fluid. Chin Sci Bull2002; 47: 1783–1785.
165.
TanWCXuMY. Plane surface suddenly set in motion in a viscoelastic fluid with fractional Maxwell model. Acta Mech Sin2002; 18: 342–349.
166.
JinHXuMY. Some notes to “On the axial flow of generalized second order fluid in a pipe”. In: ZhuangFGLiJC (eds.) Recent Advances in Fluid Mechanics, Proceedings of the 4th International Conference on J. Mech, Beijing: Tsinghua Univ. Press and Springer, 2004.
167.
SongDY. Study of rheological characterization of fenugreek gum with modified Maxwell. Chin J Chem Eng2000; 8: 85–88.
168.
TongDKWangRHYangHS. Exact solutions for the flow of non-Newtonian fluid with fractional derivative in an annular pipe. Sci China G2005; 48: 485–495.
169.
TongDKLiuYS. Exact solutions for the unsteady rotational flow of non- Newtonian fluid in an annular pipe. Int J Eng Sci2005; 43: 281–289.
170.
TongDKWangRH. Analysis of the flow of non-Newtonian viscoelastic fluids in fractal reservoir with the fractional derivative. Sci China2004; 47: 424–441.
171.
HuangJQHeGYLiuCQ. Analysis of general second-order fluid flow in double cylinder rheometer. Sci China Math Phys Astron Technol Sci1997; 40: 183–190.
172.
SongDYJiangTQ. Study on the constitutive equation with fractional derivative fir the viscoelastic fluids – modified Jeffreys model and its application. Rheol Acta1998; 27: 512–517.
173.
MetzlerRKlafterJ. The Random walker’s guide to anomalous diffusion: a fractional dynamics approach. Phys Rep2000; 339: 1–77.
174.
DanVNJr.HancockJFBurrageK. Sources of anomalous diffusion on cell membranes: a Monte Carlo study. Biophys J2007; 92: 1917–1987.
175.
CoscoySHuguetEAmblardF. Statistical analysis of sets of random walks: how to resolve their generating mechanism. Bull Math Biol2007; 69: 2467–2492.
176.
RogersLCG. Arbitrage with fractional Brownian motion. Math Finance1997; 7: 95–105.
177.
ZumofenGKlafterJBlumenA. Long time behavior in diffusion and trapping. J Chem Phys1983; 79: 5131–5135.
178.
KlafterJBlumenAZumofenG. Fractal behavior in trapping and reaction: a random walk study. J Stat Phys1984; 36: 561–577.
179.
ScalasEGorenfloRMainardiF. Uncoupled continuous-time random walks: solution and limiting behavior of master equation. Phys Rev E1984; 69: Article Id 011107.
180.
MontrollEWSchlesingerMF. A wonderful world of random walk. In: LebowitzJLMontrollEW (eds.) Nonequilibrium phenomena II: from stochastic to hydrodynamics (Studies in Statistical mechanics, vol.11). Amsterdam: North-Holland, 1984.
LubelskiAKlafterJ. Fluorescence correlation spectroscopy: the case of subdiffusion. Biophys J2009; 96: 2055–2063.
183.
MeerschaertMMZhangYBaeumerB. Tempered anomalous diffusion in heterogeneous systems. Geophys Res Lett2008; 35: 17403–17408.
184.
BurrageK. Modelling and simulation techniques for membrane biology. Briefing Bioinform2007; 8: 234–244.
185.
LenormandRWangB. A stream – tube model for miscibleflow. Transport Porous Media1995; 18: 263–272.
186.
GlimmJSharpDH. A random field model for anomalous diffusion in heterogeneous porous media. J Stat Phys1991; 62: 415–424.
187.
CushmanJHHuBXGinnTR. Nonequlibrium statistical mechanics of preasymptotic dispersion. J Stat Phys1994; 75: 859–878.
188.
MontrollEWWeissGH. Random walks on lattices. J Math Phys1965; 6: 167–181.
189.
SaichevAIZaslavskyGM. Fractional kinetic equations: solutions and applications. Chaos1997; 7: 753–764.
190.
CompteA. Stochastic foundations of fractional dynamics. Phys Rev1996; 53: 4191–4193.
191.
MetzlerRKlafterJSokolovIM. Anomalous transport in external fields: continuous time random walk and fractional diffusion equations extended. Phys Rev1998; 58: 1621–1633.
192.
MainardiF. On the initial value problem for the fractional diffusion-wave equation, waves and stability in continuous media. Ser Adv Math Appl Sci1994; 23: 246–251.
193.
VazquezJL. Barenblatt solutions and asymptotic behaviour for a nonlinear fractional heat equation of porous medium type. J Euro Math Soc2013; http://arxiv.org/pdf/1205.6332v2.pdf.
194.
LeonenkoNNMeerschaertMMSikorskiiA. Fractional Pearson diffusion. J Math Anal Appl2013; 403: 532–546.
195.
LeonenkoNNMeerschaertMMSikorskiiA. Correlation structure of fractional Pearson diffusion. Comput Math Appl2013; 66: 737–745.
196.
LeonenkoNNRuiz-MedinaMDTaqquMS. Fractional elliptic, hyperbolic and parabolic random fields. Electron J Probab2011; 16: 1134–1172.
197.
MeerschaertMMNaneEVellaisamyP. Transient anomalous sub-diffusion on bounded domains. Proc Am Math Soc2013; 141: 699–710.
198.
MeerschaertMMMetzlerRKlafterJ. Fractional calculus, anomalous diffusion, and probability, fractional dynamics. Singapore: World Scientific, 2012, 265–284.
199.
MeerschaertMMZhangYBaeumerB. Tempered anomalous diffusion in heterogeneous systems. Geophys Res Lett2008; 35: L17403.
ZhangY. Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys Rev E2006; 74, 026706–026716.
207.
BaeumerBMeerschaertMM. Fractional diffusion with two time scales. Phys Stat Mech Appl2007; 373: 237–251.
208.
SchumerR. Fractal mobile/immobile solute transport. Water Resour Res2003; 39: 1296–1308.
209.
DiethelmKLuchkoY. Numerical solution of linear multi-term differential equations of fractional order. J Comput Anal Appl2004; 6: 243–263.
210.
DiethelmK. Algorithms for the fractional calculus: a selection of numerical methods. Comput Meth Appl Mech Eng2005; 194: 743–773.
211.
LuchkoY. Anomalous diffusion models and their analysis. Forum der Berliner mathematischen Gesellschaft2011; 19: 53–85.
212.
LuchkoYPunziA. Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations. Int J Geomath2011; 1: 257–276.
213.
DatskoBLuchkoYGafiychukV. Pattern formation in fractional reaction-diffusion systems with multiple homogeneous states. Int J Bifurcat Chaos2012; 22: 1250087.
214.
BorodichFM. Fractal geometry. In: WangJChungY (eds.) Encyclopedia of tribology. New York: Springer, 2013, 1258–1264.
215.
BorodichFM. Fractal nature of surfaces. In: WangJChungY (eds.) Encyclopedia of tribology. New York: Springer, 2013, 1264–1269.
216.
GoldbergerALWestBJ. Fractals in physiology and medicine. Yale J Biol Med1987; 60: 421–435.
217.
PengCK. Long-range anticorrelations and non-Gaussian behavior of the heartbeat. Phys Rev Lett1993; 70: 1343–1346.
GriffinLWestDJWestBJ. Random stride intervals with memory. J Biol Phys2000; 26: 185–202.
223.
WestBJDeeringW. Fractal physiology for physicists: Levy statistics. Phys Rep1994; 246: 1–100.
224.
RaicuVPopescuA. Integrated molecular and cellular biophysics. New York: Springer Science+Business Media B.V., 2008, 234.
225.
WestBJ. Fractal probability density and EEF/ERP time series (Chapter 10). In: IannocconePMKhokhaM (eds.) Fractal geometry in biological systems. Boca Raton, GL: CRC, 1995, 267–316.
226.
PriebeC. The application of fractal analysis to mammographic tissue classification. Canc Lett1994; 77: 183–187.
227.
Buckland-WrightJC. Fractal signature analysis of macroradiographs measures trabecular organization in lumbar vertebrae of postmenopausal women. Calcif Tissue Int1994; 54: 106–112.
228.
MessentEABuckland-WrightJCBlakeGM. Fractal analysis of trabecular bone in knee osteoarthritis (OA) is a more sensitive marker of disease status than bone mineral density (BMD). Calcif Tissue Int2005; 76: 419–425.
229.
GanRZ. Morphometry of the dog pulmonary venous tree. J Appl Physiol1993; 75: 432–440.
230.
BassingthwaighteJBBeardDA. Fractal 15O-labeled water washout from the heart. Circ Res1995; 77: 1212–1221.
231.
HuoYKassabGS. Intraspecific scaling laws of vascular trees. J R Soc. Interface2012; 9: 190–200.
232.
KronoverRM. Fractals and chaos in dynamic systems. In: Foundations of theory. Moscow: Postmarket, 2000, 353 (in Russian).
233.
WestBJ. Fractal physiology and chaos in medicine. Singapore: Word Scientific, 1990.
234.
AhmedEEl-SakaHA. On fractional order models for Hepatitis C. Nonlinear Biomed Phys2010; 4: 1–3.
235.
SinkusR. MR elastography of breast lesions: understanding the solid/liquid duality can improve the specificity of contrast-enhanced MR mammography. Magn Reson Med2007; 58: 1135–1144.
236.
RoystonTJMansyHASandlerRH. Excitation and propagation of surface waves on a viscoelastic half-space with application to medical diagnosis. J Acoust Soc Am1999; 106: 3678–3686.
237.
RoystonTJYaziciogluYLothF. Surface response of a viscoelastic medium to subsurface acoustic sources with application to medical diagnosis. J Acoust Soc Am2003; 113: 1109–1121.
238.
DahariH. Modeling hepatitis C virus dynamics: liver regeneration and critical drug efficacy. J Theor Biol2007; 247: 371–381.
239.
IominA. Fractional transport of cancer cells due to self-entrapment by fission. In: Mathemat. Modeling of Biological Systems. Modeling and Simulation in Science, Engineering and Technology, Part IV, 2007, 193–203.
240.
BellomoNPreziosiL. Modelling and mathematical problems related to tumor evolution and its interaction with the immune system. Math Comp Model2000; 32: 413–452.
241.
PreziosiLTosinA. Multiphase and multiscale trends in cancer modelling. MathematicalModel Nat Phenom2009; 4: 1–70.
242.
PalocarenADrapacaCD. Biomechanical modeling of tumor growth: its relevance to glioma research. Int J Num Anal Model B2012; 3: 94–108.
243.
CherenkevichSNMartinovichGGKhmelnitzkyAI. Biological membranes. Minsk: BSU, 2009 (in Russian).
244.
FujiwaraT. Phospholipids undergo hop diffusion in compartmentalized cell membrane. J Cell Biol2002; 157: 1071–1081.
245.
SmithPRMorrisonIEGWilsonKM. Anomalous diffusion of major histocompatibility complex class I molecules on HeLa cells determined by single particle tracking. Biophys J1999; 76: 3331–3344.
246.
JovinTVazWLC. Rotational and translational diffusion in membranes measured by fluorescence and phosphorescence methods. Method Enzymol1989; 172: 471–573.
247.
WeissMHashimotoHNilssonT. Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. Biophys J2002; 84: 4043–4052.
248.
FuriniSZerbettoFCavalcantiS. Application of the Poisson–Nernst-Planck theory with space dependent diffusion coefficients to KcsA. Biophys J2006; 91: 3162–3169.
249.
SantamariaF. Anomalous diffusion in purkinje cell dendrites caused by spines. Neuron2006; 52: 635–648.
250.
NonnerWEisenbergB. Ion permeation and glutamate residues linked by Poisson–Nernst Planck theory in l-type calcium channels. Biophys J1998; 75: 1287–1305.
251.
FederTJ. Constrained diffusion or immobile fraction on cell surfaces: A new interpretation. Biophys J1996; 70: 2767–2773.
252.
SheetsED. Transient confinement of a glycosylphosphatidylinositol-anchored protein in the plasma membrane. Biochem1997; 36: 12449–12458.
253.
SimsonR. Structural mosaicism on the submicron scale in the plasma membrane. Biophys J1997; 74: 297–308.
254.
SmithPR. Anomalous diffusion of major histocompatability complex class I molecules on HeLa cells determined by single particle tracking. Biophys J1999; 76: 3331–3344.
255.
BanksDSFradinC. Anomalous diffusion of proteins due to molecular crowding. Biophys J2005; 89: 2960–2971.
256.
SchnellSTurnerTE. Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog Biophys Mol Biol2004; 85: 235–260.
257.
ReynoldsA. On the anomalous diffusion characteristics of membrane bound proteins. Phys Lett A2005; 342: 439–442.
258.
SantamariaF. Anomalous diffusion in purkinje cell dendrites caused by spines. Neuron2006; 52: 635–648.
259.
HardungV. Method for measurement of dynamic elasticity and viscosity of caoutchouc-like bodies, especially of blood vessels and other elastic tissues. Helv Physiol Pharmacol Acta1952; 10: 482–498.
260.
FungYC. Biomechanics: mechanical properties of living tissues. New York: Springer, 1981.
261.
JagerIL. Viscoelastic behavior of organic materials: consequences of a logarithmic dependence of force on strain rate. J Biomech2005; 38: 1451–1458.
262.
CraiemDO. Fractional calculus applied to model arterial viscoelasticity. Latin Am Appl Res2008; 38: 141–145.
263.
DjordjevicVD. Fractional derivatives embody essential features of cell rheological behavior. Ann Biomed Eng2003; 31: 692–699.
264.
SukiBBarabasiALLutchenKR. Lung tissue viscoelasticity: a mathematical framework and its molecular basis. J Appl Physiol1994; 76: 2749–2759.
265.
KissMZVargheseTHallTJ. Viscoelastic characterization of in vitro canine tissue. Phys Med Biol2004; 49: 4207–4218.
266.
CraiemDArmentanoRL. A fractional derivative model to describe arterial viscoelasticity. Biorheology2007; 44: 251–263.
267.
DoehringTC. Fractional order viscoelasticity of the aortic valve cusp: an alternative to quasilinear viscoelasticity. J Biomech Eng2005; 127: 700–708.
268.
PaladeLI. Anomalous stability behavior of a properly invariant constitutive equation which generalizes fractional derivative models. Intern J Eng Sci1999; 37: 315–329.
