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Identification problems related to cylindrical dielectrics **in presence of polarization**
On some properties of the Mittag-Leffler function $\mathbf{E_\alpha(-t^\alpha)}$, completely monotone for $\mathbf{t> 0}$ with $\mathbf{0<\alpha<1}$
1. | Department of Physics and Astronomy, University of Bologna, and INFN, Via Irnerio 46, Bologna, I-40126, Italy |
References:
[1] |
G. A. Baker, Essentials of Padè Approximants,, Academic Press, (1975).
|
[2] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods,, World Scientific, (2012).
doi: 10.1142/9789814355216. |
[3] |
L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations,, Electron. J. Probab., 15 (2010), 684.
doi: 10.1214/EJP.v15-762. |
[4] |
E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics,, Eur. Phys. J., 193 (2011), 161. Google Scholar |
[5] |
M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism,, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134.
|
[6] |
M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids,, Riv. Nuovo Cimento (Ser. II), 1 (1971), 161.
doi: 10.1007/BF02820620. |
[7] |
K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics, II. Direct current characteristics,, J. Chemical Physics, 10 (1942), 98.
doi: 10.1063/1.1723677. |
[8] |
H. T. Davis, The Theory of Linear Operators,, The Principia Press, (1936). Google Scholar |
[9] |
K. Diethelm, The Analysis of Fractional Differential Equations,, Springer, (2004).
doi: 10.1007/978-3-642-14574-2. |
[10] |
M. M. Dzherbashyan,, Integral Transforms and Representations of Functions in the Complex Plane,, Nauka, (1966). Google Scholar |
[11] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. III. Based, (1955).
|
[12] |
W. Feller, An Introduction to Probability Theory and its Applications,, Vol. II, (1971).
|
[13] |
A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus,, {First Annual Report, (2002), 2002. Google Scholar |
[14] |
R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives,, Fract. Calc. Appl. Anal., 5 (2002), 491.
|
[15] |
R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, in Fractals and Fractional Calculus in Continuum Mechanics, (1997), 223.
|
[16] |
B. Gross, On creep and relaxation,, J. Appl. Phys., 18 (1947), 212.
doi: 10.1063/1.1697606. |
[17] |
R. Hilfer (editor), Fractional Calculus, Applications in Physics,, World Scientific, (2000). Google Scholar |
[18] |
E. Hille and J. D. Tamarkin, On the theory of linear integral equations,, Ann. Math., 31 (1930), 479.
doi: 10.2307/1968241. |
[19] |
A. A. Kilbas, A . A. Koroleva and S. V. Rogosin, Multi-parametric Mittag-Leffler functions and their extensions,, Fract. Calc. Appl. Anal., 16 (2013), 378.
doi: 10.2478/s13540-013-0024-9. |
[20] |
A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type,, Differential and Integral Equations, 8 (1995), 993.
|
[21] |
A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications,, Chapman and Hall/CRC, (2004).
doi: 10.1201/9780203487372. |
[22] |
A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006).
|
[23] |
V. Kiryakova, Generalized Fractional Calculus and Applications,, Longman & J. Wiley, (1994).
|
[24] |
V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus,, Comp. Math. Appl., 59 (2010), 1885.
doi: 10.1016/j.camwa.2009.08.025. |
[25] |
V. Kiryakova and Y. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis,, In American Institute of Physics - Conf. Proc., 1301 (2010), 597.
doi: 10.1063/1.3526661. |
[26] |
J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances,, World Scientific, (2012).
|
[27] |
R. L. Magin, Fractional Calculus in Bioengineering,, Begell House Publishers, (2006). Google Scholar |
[28] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010).
doi: 10.1142/9781848163300. |
[29] |
F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey,, Fract. Calc. Appl. Anal., 10 (2007), 269.
|
[30] |
O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions,, Theory and Algorithmic Tables, (1983).
|
[31] |
A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists,, Springer, (2008).
doi: 10.1007/978-0-387-75894-7. |
[32] |
A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines,, Wiley Eastern Ltd, (1978).
|
[33] |
A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications,, Springer Verlag, (2010).
doi: 10.1007/978-1-4419-0916-9. |
[34] |
K. S. Miller and S. G. Samko, Completely monotonic functions,, Integral Transforms and Special Functions, 12 (2001), 389.
doi: 10.1080/10652460108819360. |
[35] |
I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).
|
[36] |
I. Podlubny, Mittag-Leffler function,, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, (2006). Google Scholar |
[37] |
H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_\alpha (-x)$,, Bull. Amer. Math. Soc., 54 (1948), 1115.
doi: 10.1090/S0002-9904-1948-09132-7. |
[38] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications,, Gordon and Breach, (1993).
|
[39] |
T. Sandev, R. Metzler and Z. Tomovski, Velocity and displacement correlation functions for fractional generalized Langevin equations,, Fract. Calc. Appl. Anal., 15 (2012), 426.
doi: 10.2478/s13540-012-0031-2. |
[40] |
G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable,, Vol. I. Holomorphic Functions, (1960).
|
[41] |
R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications,, 2-nd ed., (2012).
doi: 10.1515/9783110269338. |
[42] |
T. Simon, Comparing Fréchet and positive stable laws,, Electron. J. Probab., 19 (2014), 1.
doi: 10.1214/EJP.v19-3058. |
[43] |
H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications,, South Asian Publishers, (1982).
|
[44] |
A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions,, Sbornik Mathematics, 198 (2007), 1011.
doi: 10.1070/SM2007v198n07ABEH003871. |
[45] |
V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,, Springer, (2010).
doi: 10.1007/978-3-642-14003-7. |
[46] |
Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions,, Integral Transforms and Special Functions, 21 (2010), 797.
doi: 10.1080/10652461003675737. |
[47] |
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers,, Springer, (2013).
doi: 10.1007/978-3-642-33911-0. |
[48] |
R. Wong and Y.-Q Zhao, Exponential asymptotics of the Mittag-Leffler function,, Constructive Approximation, 18 (2002), 355.
doi: 10.1007/s00365-001-0019-3. |
[49] |
C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse,, E-print , (2013). Google Scholar |
show all references
References:
[1] |
G. A. Baker, Essentials of Padè Approximants,, Academic Press, (1975).
|
[2] |
D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calculus. Models and Numerical Methods,, World Scientific, (2012).
doi: 10.1142/9789814355216. |
[3] |
L. Beghin and E. Orsingher, Poisson-type processes governed by fractional and higher-order recursive differential equations,, Electron. J. Probab., 15 (2010), 684.
doi: 10.1214/EJP.v15-762. |
[4] |
E. Capelas de Oliveira, F. Mainardi and J. Vaz Jr, Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics,, Eur. Phys. J., 193 (2011), 161. Google Scholar |
[5] |
M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism,, Pure and Appl. Geophys. (PAGEOPH), 91 (1971), 134.
|
[6] |
M. Caputo and F. Mainardi, Linear models of dissipation in anelastic solids,, Riv. Nuovo Cimento (Ser. II), 1 (1971), 161.
doi: 10.1007/BF02820620. |
[7] |
K. S. Cole and R. H. Cole, Dispersion and absorption in dielectrics, II. Direct current characteristics,, J. Chemical Physics, 10 (1942), 98.
doi: 10.1063/1.1723677. |
[8] |
H. T. Davis, The Theory of Linear Operators,, The Principia Press, (1936). Google Scholar |
[9] |
K. Diethelm, The Analysis of Fractional Differential Equations,, Springer, (2004).
doi: 10.1007/978-3-642-14574-2. |
[10] |
M. M. Dzherbashyan,, Integral Transforms and Representations of Functions in the Complex Plane,, Nauka, (1966). Google Scholar |
[11] |
A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions,, Vol. III. Based, (1955).
|
[12] |
W. Feller, An Introduction to Probability Theory and its Applications,, Vol. II, (1971).
|
[13] |
A. Freed, K. Diethelm and Y. Luchko, Fractional-order Viscoelasticity (FOV): Constitutive Development using the Fractional Calculus,, {First Annual Report, (2002), 2002. Google Scholar |
[14] |
R. Gorenflo, J. Loutchko and Y. Luchko, Computation of the Mittag-Leffler function and its derivatives,, Fract. Calc. Appl. Anal., 5 (2002), 491.
|
[15] |
R. Gorenflo and F. Mainardi, Fractional calculus: Integral and differential equations of fractional order,, in Fractals and Fractional Calculus in Continuum Mechanics, (1997), 223.
|
[16] |
B. Gross, On creep and relaxation,, J. Appl. Phys., 18 (1947), 212.
doi: 10.1063/1.1697606. |
[17] |
R. Hilfer (editor), Fractional Calculus, Applications in Physics,, World Scientific, (2000). Google Scholar |
[18] |
E. Hille and J. D. Tamarkin, On the theory of linear integral equations,, Ann. Math., 31 (1930), 479.
doi: 10.2307/1968241. |
[19] |
A. A. Kilbas, A . A. Koroleva and S. V. Rogosin, Multi-parametric Mittag-Leffler functions and their extensions,, Fract. Calc. Appl. Anal., 16 (2013), 378.
doi: 10.2478/s13540-013-0024-9. |
[20] |
A. A. Kilbas and M. Saigo, On solution of integral equations of Abel-Volterra type,, Differential and Integral Equations, 8 (1995), 993.
|
[21] |
A. A. Kilbas and M. Saigo, $H$-Transforms. Theory and Applications,, Chapman and Hall/CRC, (2004).
doi: 10.1201/9780203487372. |
[22] |
A. A. Kilbas, H. M Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,, Elsevier, (2006).
|
[23] |
V. Kiryakova, Generalized Fractional Calculus and Applications,, Longman & J. Wiley, (1994).
|
[24] |
V. Kiryakova, The multi-index Mittag-Leffler functions as important class of special functions of fractional calculus,, Comp. Math. Appl., 59 (2010), 1885.
doi: 10.1016/j.camwa.2009.08.025. |
[25] |
V. Kiryakova and Y. Luchko, The multi-index Mittag-Leffler functions and their applications for solving fractional order problems in applied analysis,, In American Institute of Physics - Conf. Proc., 1301 (2010), 597.
doi: 10.1063/1.3526661. |
[26] |
J. Klafter, S. C. Lim and R. Metzler (Editors), Fractional Dynamics, Recent Advances,, World Scientific, (2012).
|
[27] |
R. L. Magin, Fractional Calculus in Bioengineering,, Begell House Publishers, (2006). Google Scholar |
[28] |
F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity,, Imperial College Press, (2010).
doi: 10.1142/9781848163300. |
[29] |
F. Mainardi and R. Gorenflo, Time-fractional derivatives in relaxation processes: A tutorial survey,, Fract. Calc. Appl. Anal., 10 (2007), 269.
|
[30] |
O. I. Marichev, Handbook of Integral Transforms of Higher Transcendental Functions,, Theory and Algorithmic Tables, (1983).
|
[31] |
A. M. Mathai and H. J. Haubold, Special Functions for Applied Scientists,, Springer, (2008).
doi: 10.1007/978-0-387-75894-7. |
[32] |
A. M. Mathai and R. K. Saxena, The H-Function with Applications in Statistics and Other Disciplines,, Wiley Eastern Ltd, (1978).
|
[33] |
A. M. Mathai, R. K. Saxena and H. J. Haubold, The H-Function: Theory and Applications,, Springer Verlag, (2010).
doi: 10.1007/978-1-4419-0916-9. |
[34] |
K. S. Miller and S. G. Samko, Completely monotonic functions,, Integral Transforms and Special Functions, 12 (2001), 389.
doi: 10.1080/10652460108819360. |
[35] |
I. Podlubny, Fractional Differential Equations,, Academic Press, (1999).
|
[36] |
I. Podlubny, Mittag-Leffler function,, Matlab-Code that calculates the Mittag-Leffler function with desired accuracy, (2006). Google Scholar |
[37] |
H. Pollard, The completely monotonic character of the Mittag-Leffler function $E_\alpha (-x)$,, Bull. Amer. Math. Soc., 54 (1948), 1115.
doi: 10.1090/S0002-9904-1948-09132-7. |
[38] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications,, Gordon and Breach, (1993).
|
[39] |
T. Sandev, R. Metzler and Z. Tomovski, Velocity and displacement correlation functions for fractional generalized Langevin equations,, Fract. Calc. Appl. Anal., 15 (2012), 426.
doi: 10.2478/s13540-012-0031-2. |
[40] |
G. Sansone and J. Gerretsen, Lectures on the Theory of Functions of a Complex Variable,, Vol. I. Holomorphic Functions, (1960).
|
[41] |
R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications,, 2-nd ed., (2012).
doi: 10.1515/9783110269338. |
[42] |
T. Simon, Comparing Fréchet and positive stable laws,, Electron. J. Probab., 19 (2014), 1.
doi: 10.1214/EJP.v19-3058. |
[43] |
H. M. Srivastava, K. C. Gupta and S. P. Goyal, The H-Functions of One and Two Variables with Applications,, South Asian Publishers, (1982).
|
[44] |
A. P. Starovoitov and N. A. Starovoitova, Padè approximants of the Mittag-Leffler functions,, Sbornik Mathematics, 198 (2007), 1011.
doi: 10.1070/SM2007v198n07ABEH003871. |
[45] |
V. E. Tarasov, Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media,, Springer, (2010).
doi: 10.1007/978-3-642-14003-7. |
[46] |
Z. Tomovski, R. Hilfer and H. M. Srivastava, Fractional and operational calculus with generalized fractional derivative operators and Mittag-Leffler type functions,, Integral Transforms and Special Functions, 21 (2010), 797.
doi: 10.1080/10652461003675737. |
[47] |
V. V. Uchaikin, Fractional Derivatives for Physicists and Engineers,, Springer, (2013).
doi: 10.1007/978-3-642-33911-0. |
[48] |
R. Wong and Y.-Q Zhao, Exponential asymptotics of the Mittag-Leffler function,, Constructive Approximation, 18 (2002), 355.
doi: 10.1007/s00365-001-0019-3. |
[49] |
C. Zeng and Y.-Q. Chen, Global Padè approximations for the generalized Mittag-Leffler function and its inverse,, E-print , (2013). Google Scholar |
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