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The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order.
Cole, K.S., Electrical conductance of biological systems, Electrical excitation in nerves, In: Proceedings Symposium on Quantitative Biology
1933
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Gemant, A., On fractional differentials, Phil. Mag
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Cole, K.S. and Cole, R.H., Dispersion and absorption in dielectrics, I. Alternating current characteristics, J. Chemical Physics
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Scott-Blair, G.W., Veinoglou, B. C. and Caffyn, J.E., Limitations of the Newtonian time scale in relation to non-equilibrium rheological states and a theory of quasi-properties, Proc. R. Soc. London A
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Gerasimov, A., A generalization of linear laws of deformation and its applications to problems of internal friction, Prikl. Matem. i Mekh
1948
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Pollard, H., The completely monotonic character of the Mittag-Leffler function E α ( − x ) , E_{\alpha}(-x)\,, Bull. Amer. Math. Soc
1948
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Rabotnov, Yu. N., Equilibrium of an elastic medium with after effect, Prikl. Matem. i Mekh
1948
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Scott-Blair, G.W., Survey of General and Applied Rheology
1949
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Scott-Blair, G.W. and Caffyn, J.E., An application of the theory of quasi-properties to the treatment of anomalous stress-strain relations, Phil. Mag
1949
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Gemant, A., Frictional Phenomena
1950
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Gross, B., Mathematical Structure of the Theories of Viscoelasticity
1953
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Courant, R., Differential and Integral Calculus
1954
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Erdélyi, A., Magnus, W., Oberhettinger, F and Tricomi, F.G., Higher Transcendental Functions
1955
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Bland, D.R., The Theory of Linear Viscoelasticity
1960
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Sansone, G. and Gerretsen, J., Lectures on the Theory of Functions of a Complex Variable
1960
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Gel’fand, I.M. and Shilov, G.E., Generalized Functions
1964
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Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions
1965
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Caputo, M., Linear models of dissipation whose Q is almost frequency independent, Annali di Geofisica
1966
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Dzherbashyan, M.M., Integral Transforms and Representations of Functions in the Complex Plane
1966
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Rabotnov, Yu. N., Creep Problems in Structural Members
1966
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Caputo, M., Linear models of dissipation whose Q Q is almost frequency independent, Part II, Geophys. J. R. Astr. Soc
1967
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Dzherbashian, M.M. and Nersesian, A.B., Fractional derivatives and the Cauchy problem for differential equations of fractional order, Izv. Acad. Nauk Armjanskvy SSR, Matematika
1968
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Caputo, M., Elasticità e Dissipazione
1969
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Meshkov, S.I. and Rossikhin, Yu.A., Sound wave propagation in a viscoelastic medium whose hereditary properties are determined by weakly singular kernels, In: Rabotnov, Yu.N. (Editor), Waves in Inelastic Media
1970
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Caputo, M. and Mainardi, F., A new dissipation model based on memory mechanism, Pure and Applied Geophysics (Pageoph)
1971
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Caputo, M. and Mainardi, F., Linear models of dissipation in anelastic solids, Rivista del Nuovo Cimento
1971
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Pipkin, A.C., Lectures on Viscoelastic Theory
1972
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Caputo, M., Vibrations of an infinite viscoelastic layer with a dissipative memory, J. Acoust. Soc. Am
1974
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Doetsch, G., Introduction to the Theory and Application of the Laplace Transformation
1974
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Oldham, K.B. and J. Spanier, J., The Fractional Calculus
1974
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Molinari, A., Viscoélasticité linéaire et function complètement monotones, Journal de Mécanique
1975
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Caputo, M., Vibrations of an infinite plate with a frequency independent Q Q , J. Acoust. Soc. Am
1976
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Rabotnov, Yu. N., Elements of Hereditary Solid Mechanics
1977
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Bagley, R.L., Applications of Generalized Derivatives to Viscoelasticity
1979
Cited alongside, same era.
Caputo, M., A model for the fatigue in elastic materials with frequency independent Q Q , J. Acoust. Soc. Am
1979
Cited alongside, same era.
Caputo, M., Elastic radiation from a source in a medium with an almost frequency independent Q Q , J. Phys. Earth
1981
Cited alongside, same era.
Graffi, D., Mathematical models and waves in linear viscoelasticity, In: F. Mainardi (Editor), Wave Propagation in Viscoelastic Media
1982
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Bagley, R.L. and Torvik, P.J., A theoretical basis for the application of fractional calculus, J. Rheology
1983
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Koeller, R.C., Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech
Heymans, N., Hierarchical models for viscoelasticity: dynamic behaviour in the linear range, Rheol. Acta
1996
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Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals
1996
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Pritz, T., Analysis of four-parameter fractional derivative model of real solid materials, J. Sound and Vibration
1996
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Schneider, W.R., Completely monotone generalized Mittag-Leffler functions, Expositiones Mathematicae
1996
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Gaul, L. and Schanz, M., Calculation of transient response of viscoelastic solids based on inverse transformation, Meccanica
1997
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1984
Cited alongside, same era.
Torvik, P.J. and Bagley, R.L., On the appearance of the fractional derivatives in the behavior of real materials, J. Appl. Mech
1984
Cited alongside, same era.
Caputo, M., Generalized rheology and geophysical consequences, Tectonophysics
1985
Cited alongside, same era.
Gaul, L., Bohlen, S. and Kempfle, S., Transient and forced oscillations of systems with constant hysteretic damping, Mechanics Research Communications
1985
Cited alongside, same era.
Bagley, R.L. and Torvik, P.J., On the fractional calculus model of viscoelastic behavior, J. Rheology
1986
Cited alongside, same era.
Caputo, M., Linear and non linear inverse rheologies of rocks, Tectonophysics
1986
Cited alongside, same era.
Koeller, R.C., Polynomial operators, Stieltjes convolution and fractional calculus in hereditary mechanics, Acta Mech
1986
Cited alongside, same era.
Gorenflo, R., Fractional calculus: some numerical methods, In: Carpinteri, A. and Mainardi, F. (Editors) Fractals and Fractional Calculus in Continuum Mechanics
1997
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Gorenflo, R. and Mainardi, F., Fractional calculus: Integral and differential equations of fractional order, In: A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics
1997
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Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, In: A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics
1997
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Miller, K.S. and Samko, S.G., A note on the complete monotonicity of the generalized Mittag-Leffler function, Real Anal. Exchange
1997
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Rossikhin, Yu.A. and Shitikova, M.V., Applications of fractional calculus to dynamic problems of linear and nonlinear fractional mechanics of solids, Appl. Mech. Review
1997
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Caputo, M., 3-dimensional physically consistent diffusion in anisotropic media with memory, Rend. Mat. Acc. Lincei
1998
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Pritz, T., Frequency dependences of complex moduli and complex Poisson’s ratio of real solid materials, J. Sound and Vibration
1998
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Caputo, M., Lectures on Seismology and Rheological Tectonics
1999
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Friedrich, Chr., Schiessel, H. and Blumen, A., Constitutive behavior modeling and fractional derivatives, In: Siginer, D.A., Kee, D. and Chhabra, R.P. (Editors), Advances in the Flow and Rheology of Non-Newtonian Fluids
1999
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Podlubny, I., Fractional Differential Equations
1999
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Pritz, T., Verification of local Kramers-Kronig relations for complex modulus by means of fractional derivative model, J. Sound and Vibration
1999
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Butzer, P. and Westphal, U., Introduction to fractional calculus, In: Hilfer, H. (Editor), Fractional Calculus, Applications in Physics
2000
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Hilfer, R. (Editor), Fractional Calculus, Applications in Physics
2000
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Hilfer, R., Fractional time evolutions, In: Hilfer, H. (Editor), Fractional Calculus, Applications in Physics
2000
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Metzler, R. and Klafter, J., The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports
2000
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Miller, K.S. and Samko, S.G., Completely monotonic functions, Integr. Transf. and Spec. Funct
2001
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Carcione, J. M., Cavallini, F., Mainardi, F. and Hanyga, A. (2002). Time-domain seismic modelling of constant- Q Q wave propagation using fractional derivatives, Pure and Appl. Geophys (PAGEOPH)
2002
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Gorenflo, R., Loutschko, J. and Luchko, Yu., Computation of the Mittag-Leffler function E α , β ( z ) E_{\alpha,\beta}(z) and its derivatives, Fractional Calculus and Applied Analysis
2002
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Kempfle, S., Schäfer, I. and Beyer, H., Fractional calculus via functional calculus: theory and applications, Nonlinear Dynamics
2002
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Metzler, R. and Klafter, J., From stretched exponential to inverse power-law: fractional dynamics, Cole-Cole relaxation processes, and beyond, J. Non-Crystalline Solids
2002
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Sokolov, I, Klafter, J. and A. Blumen, A., Fractional kinetics, Physics Today
2002
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West, B.J., Bologna, M. and Grigolini, P., Physics of Fractal Operators
2003
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Mainardi, F., Physical and mathematical aspects of fractional calculus in linear viscoelasticity, In: A. Le Méhauté, J.A. Tenreiro Machado, J.C. Trigeassou, J. Sabatier (Editors), Proceedings the 1-st IFAC Workshop on Fractional Differentiation and its applications
2004
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Hanyga, A., Viscous dissipation and completely monotonic relaxation moduli, Rheologica Acta
2005
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Seybold, H.J. and Hilfer, R., Numerical results for the generalized Mittag-Leffler function, Fractional Calculus and Applied Analysis
2005
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Zaslavsky, G.M., Hamiltonian Chaos and Fractional Dynamics
2005
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Heymans, N. and Podlubny, I., Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheol. Acta
2006
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Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J., Theory and Applications of Fractional Differential Equations
2006
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Magin, R.L., Fractional Calculus in Bioengineering
2006
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Podlubny, I., Mittag-Leffler function
2006
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Bagley, R.L., On the equivalence of the Riemann-Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials, Fractional Calculus and Applied Analysis
2007
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Rossikhin, Yu.A. and Shitikova, M.V., Comparative analysis of viscoelastic models involving fractional derivatives of different orders, Fractional Calculus and Applied Analysis
2007
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Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity
2008
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