Functional fractional calculus for system identification and controls

Functional Fractional Calculus for System Identification and Controls

Volumn , Issue , 2008, Pages 1-240

  Das, Shantanu ^a  

^a BHABHA ATOMIC RESEARCH CENTRE   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84892075371     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1007/978-3-540-72703-3     Document Type: Book

References (158)

1. A.D. Poularikas, The handbook of formulas and tables for signal processing, The Electrical Engineering Handbook Series, CRC Press and IEEE Press, New York 1999.
2. A. Erdelyi (ed), Tables of Integral Transforms, vol. 1, McGraw-Hill, 1954.
3. M. Gopal, Modern Control System Theory, Second Edition April, 1993, Wiley Eastern Limited.
4. F. F. Kuo, Network Analysis and Synthesis, Willey 1966.
5. Igor Podlubny, Fractional differential equations, Academic Press San Diego 1999.
6. BJ. Lurie, Three parameter tunable Tilt-Integral-Derivative TID Controller, US Patent, 5-371-670, 1994.
7. J.C. Wang, Realization of generalized Wartburg impedance with RC ladder networks and transmission lines, J. of Electrochem. Soc, vol. 134, no. 8 August 1987, pp. 1915-1920.
8. A. Tustin, et al. The design of systems for automatic control of the position of massive objects, The proceedings of Institute of Electrical Engineers, 105C(1), 1958.
9. ß controllers, IEEE Trans. Auto Cont., vol. 44 no. 1, pp 208-214 Jan 1999.
10. I. Podlubny, Matrix approach to discrete fractional calculus, Fractional Calculus and Applied Analysis vol. 3, no. 4 (2000).
11. I Podlubny, The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Inst. Exp. Phys. Slovak Acad. Sci., no. UEF-02-94, 1994, Kocice.
12. Y.Y. Tsao, B. Onaral and H H Sun, An algorithm for determining global parameters of minimum phase systems with fractional power spectra. IEEE Trans. Inst. And Meas (1989), vol. 38, no. 3, pp 723-729.
13. R.L Bagley The thermorheological complex materials, Int J. Engg. Sci, vol. 29, no 7, pp 797-806 (1991).
14. A. Carpinteri and F Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Vienna-New York (1997).
15. R.L Bagley and R A Calico, Fractional order state equations for control of viscoelastic structures, J. Guid. Cont. and Dyn, vol. 14, no 2, pp 304-311, March-April (1991).
16. R L Bagley and P J Torvik, Fractional calculus in transient analysis of viscoelastically damped structures, AIAA Journal, vol. 23, no. 6. pp. 918-925, (1985).
17. 2. IEEE Circuit Theory, vol. CT-11, no 4, pp 494-495 Dec 1964.
18. Carlson and Halijak, Simulation of the fractional derivative operator ^/s and fractional integral operator 1/*/s. Kansas State University Bulletin, vol. 45, pp 1-22 July 1961.
19. 1/n by regular Newton process, IEEE Trans. on Circuit Theory, vol. 11, no. 2, 1964, pp. 210-213.
20. M. Sugi, Y Hirano, Y.F. Miura, K. Saito, Simulation of fractal immitance by analog circuits: an approach to optimized circuits, IEICE Trans. Fundamentals, vol. E82-A, no. 8, 1999, pp. 1627-1635.
21. M. Sugi et al. Co-Variations of AC admittance in the inhomogeneous distributed RC lines. Jpn. J. Appl. Phys. Vol. 39, no 9A, 2000, pp. 5367-5368.
22. J. Dostal: Operational Amplifiers, Butterworth-Heinemann, Boston, 1993.
23. S.C. Dutta Roy, On realization of constant-argument immitances of fractional operator, IEEE Trans. on Circuit Theory, vol. 14, no. 3, 1967, pp. 264-374.
24. S. Manabe, The Non-Integer Integral and its application to Control Systems, ETJ of Japan, vol. 6, no. 3-4, 1961, pp. 83-87.
25. M. Nakagava, K. Sorimachi, Basic characteristics offractance device, IEICE Trans. fundamentals, vol. E75-A, no. 12, 1992, pp. 1814-1818.
26. H W Bode, Network analysis and feed back amplifier design, Van Nostrand New-York 1949.
27. K.B. Oldham and J. Spanier, The replacement of Fick's law by formulation involving semi differentiation. J. Electroanal.Chem, vol. 26, pp.331-341 (1970).
28. D. Matignon. Generalized Fractional Differential and Difference Equation: Stability Properties and Modeling Issues, Proc. of Maths. Theory of Networks and Systems Symposium, Padova, Italy 1998.
29. K.B. Oldham Interrelation of current and concentration at electrodes, J. Appl Electrochem, vol. 21, pp. 1068-1072 (1991).
30. A.M.A El Sayeed Fractional order diffusion wave equation Int. J. of Theor. Phy 1996, vol. 35, pp. 311-322.
31. R L Bagley, Power law and fractional calculus model of viscoelasticity AIAA Journal, vol. 27, no. 10, 1989, pp. 1412-1417.
32. I. Podlubny, Fractional derivatives: A new stage in process modeling and control, 4th International DAAAM Symposium, Brno, Czech Republic, 16-18 Sept, 1993, pp. 263-264.
33. S. Saha Ray and R K Bera, Analytical solution of dynamic system containing fractional derivative of order one-half by Admonian decomposition method. ASME J. of Applied Mechanic July 2005, vol. 72/1.
34. N. Engheta, The fractional curl operator in electromagnetic. Microwave Opt.Tech.Letter vol. 17, pp 86-91, 1998.
35. Naqvi Q A and M Abbas, Fractional duality solutions and corresponding sources. PIER vol. 25, pp 290-294 (2000).
36. Benson, The fractional order governing levy motion. Water resources research, vol. 36 no. 6 pp 1413-1423 June2000.
37. Heymans and Podlubny et al, Physical interpretation of initial conditions for fractional differential equation with Reimann-Liouvelli fractional derivative. Rheological Acta (online) Nov29, 2005.
38. Yoshinoi Akoi, Mihir Sen et al Approximation of transient temperature in complex geometries using fractional derivatives. Department of Aerospace and Mechanical Engineering University of Notre Damme. Technical Note Dec 19, 2005.
39. Adam Loverro, Fractional Calculus history definition and application. Department of Aerospace and Mechanical Engg. University of Notre Damme. Technical Note May 8-2004.
40. Dorcak, Ivo Petras et al, Fractional order state space models. Int. Carpathian Control Conference ICCC Malenovice Czech Rep. May 27-30 2002 pp 193-198.
41. Ivo Petras The fractional order controller method for their synthesis and application. BERG University Koissce Slovak Republic Technical Note 2004.
42. Ivo Petras, Vinagre, Practical Applications of digital fractional order controller to Temperature Control, Acta Montanistica, Slovaca Rocnik 7(2002), 2 pp 131-137.
43. Valimaki et al, Principles of fractional delay filters IEEE Conf on Acoustic Speech Processing (ICASSP) Istanbul, Turkey 5-9 June 2000.
44. P. Launsse, A. Oustaloup et al A restricted complexity controller with CRONE control system design and tuning. University Bordeaux-ENSEIRB 2003 Technical Note.
45. A. Oustaloup et al. Frequency band complex non-integer differentiator characterization and synthesis. IEEE Trans. on circuits and systems-I: Fundamental Theory and Application, vol. 47, no. 1, pp.-25-39, 2000.
46. th IMACS World Congress Paris July 18-22, 1988, vol. 3, pp. 203-208.
47. I. Podlubny et al, Using continued fraction expansion to discretize fractional order derivative, FDTA Non Linear Dynamic 2003. For submission to special issue on Fractional Derivative and their application.
48. th IEEE CDC San Diego December 2004, pp 4985-4990.
49. Chien Chang Tseng (Senior member IEEE), Improved design of digital fractional order differentiator using fractional sample delay TCAS 1-1815, 2004.
50. Ivo Petras, Control of fractional order Chua's system Univ of Koisscs Slovak Technical Note 2000.
51. Bohennan Gary, Analog Realization of fractional order control element revisited Department of Physics Montonna State Univ. Technical Note Oct 27, 2002.
52. Yang Quan Chen Analytical stability bound for class of delayed fractional order dynamic system, IEEE Conf. On Decision and control Florida WeA05-1 2001 pp 1421-1424.
53. Kundert.K, Modeling di-electric absorption in capacitors The Designer Guide Community Technical Note 2004.
54. Tom Hartley, Carl F Lorenzo, H. K. Qammer, Chaos in fractional Chua's circuit, IEEE Trans. On Circuits and Systems-I: Fundamental Theory and Applications, vol. 42, no. 8, 1995, pp. 485-490.
55. Kiran M Kolwankar, Anil D Gangal, Local Fractional Derivatives and fractal functions of several variables, J of Physics 9801010 vol. 1 1998.
56. Kolwankar and Gangal, Fractional differentiability of nowhere differentiable functions and dimensions, Chaos 6(4) pp. 505-513, (1996).
57. K S Miller and Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Willey New York 1993.
58. K.B. Oldham and J. Spanier, Fractional calculus Academic Press San Diego 1974.
59. H M Srivastava. On extension of Mittag-Leffler function, Yokohama Math. J. 1968, vol. 16, no. 2 pp. 77-88.
60. H M Srivastava, A certain family of sub-exponential series, Int. J. Math. Educ. Sci. Tech, vol. 25, no. 2, 00211-216, (1994).
61. S. Westurlund, Dead Matter has memory. Physics Scripta 1991, vol. 43, pp. 174-179.
62. S. Westurlund et al Capacitor theory IEEE Trans on Dielectric and Electrical Insulation 1994, vol. 1, no. 5, pp. 826-839.
63. I. Podlubny Presentation on fractional calculus IEEE chapter control San Diego, September 29, 2005.
64. I. Podlubny, Ivo Petras et al, Realization of fractional order controllers, Acta Montanistica Slovak, Rocnik 8(2003) pp 233-235.
65. Fracncois Dubois Stephanie Mengue Mixed Collocation for fractional differential equations, Numerical Algorithms Dec. 2005 vol. 34, no. 2 pp 303-311.
66. Shequin Zhang, Positive solutions for boundary value problems on non-linear fractional differential equations. Electronic J. of differential equations 2006 no. 36 pp 1-12.
67. Cecilia Reis et al Synthesis of logic circuit using fractional order dynamic fitness function. Transaction on engineering computing and Technology ISSN ENFORMATICA VI 2004 1305-5313 pp 77-80.
68. J.D. Munkhammar Fractional calculus and Taylor-Reimann series. UUDM Thesis Report 2005.
69. V V Anh et al, Fractional differential equation driven by Levy noise. J. Appl. Math and stochastic analysis 16: 2 (2003) pp 97-119
70. M Wallis Computer experiments with fractional Gausian noises PART 3: Mathematical Appendix Water Resource research 5, 1969, pp 328-338.
71. Vingare, Ivo Petras, Podlubny, et al, Using fractional order adjustment rules and fractional order reference model in model reference adaptive control, Non linear Dynamics 29, pp 269-279 (2002).
72. Kostadin Trenceveski et al. On fractional derivative of some function of exponential type. Ser Mat. University Beograd PUBL. ELEKTROTEHN FAK. Ser. Mat. 13(2002) pp 77-84.
73. Manabe. Early development of fractional order control. ASME DETC2003/VIB-48370 pp 1-8.
74. Xiaorang Li et al. On concept of local fractional differentiation. Department of applied mathematics University of Western Ontario London 2004 Lecture Notes.
75. Wang Jifeng, Yunkai et al. Frequency domain analysis and application for fractional order control system J of Phy. Conference Series 13(2005) pp 268-273.
76. C Yang et al. A conceptually effective predictor corrector method for simulating fractional order dynamic control system. ANZIAM. J. 47(EMAC2005) pp C168-C184.
77. Rafael Barcena et al Auto tuning of fractional order hold circuit IEEE Int. Conf on Control applications Sept 5-7, 2001, Mexico City pp 7-12.
78. I Podlubny. Geometric and Physical meaning of fractional derivative. Technical Uviversity Kosice Slovak.vol. 5 No. 4 (2002) ISSN 1311-045 Fractional Calculus and Applied Analysis 5(4) pp 367-386.
79. Mauro Bologna. Short Introduction to fractional calculus. Univ. Tarpeca Chile 2005. Lecture Notes pp 41-54.
80. R. K. Saxena Fractional calculus and fractional differential equations Lecture Note (chapter-3) pp 1-39, Jai Narayan Vyas University Rajasthan 2006.
81. R. K. Saxena, A. M. Mathai, H J Haubold, On Generalized Fractional Kinetic Equations Int J of Math. Phys. 0406046 v1 22-06-2004.
82. Juan Bisquert, Albert Compte, Theory of electrochemical impedance of anomalous diffusion, J of Electro analytical Chemistry 499(2001) pp 112-120.
83. A. Erdelyi On some functional transformation Univ Potitec Torino 1950.
84. Karabi Biswas, Siddharta Sen, Pranab Kumar Dutta, Modeling of a capacitive probe in a polarizable medium, Sensors actuators: Phys. 120 (2005) pp 115-122.
85. K. Biswas, S Sen, P K Dutta, A constant phase element sensor for monitoring microbial growth, Sensors and Actuators SNB9063 (2005) pp 1-6.
86. Shantanu Das, B B Biswas, Fractional divergence for neutron flux profile in nuclear reactors, International Journal Nuclear Energy Science and Technology. Vol.3, No.2 (2007) pp 139-159.
87. Shantanu Das, Efficient control of Nuclear Plants IAEA-TM-Control & Instrumentation 2007.
88. Shantanu Das, B. B. Biswas, Total Energy Utilization from Nuclear Source, PORT-2006, Nuclear Energy for New Europe Slovenia 2006.
89. Shantanu Das, et al. Ratio control with logarithmic logics in new P&P control algorithm for a true fuel-efficient reactor. Int. J. Nuclear Energy Science & Technology. Vol. 3, No. 1, 2007 pp. 1-18.
90. Ivo Petras, Vinagre, Practical application of digital fractional order controller to temperature control, Acta Montansitica Slovak 2002.
91. Chen, Vinagre, A new type of IIR digital fractional order differentiator, Signal Processing. 83(2003) pp 2359-2365
92. Chyi Hwang and Cheng, A numerical algorithm for stability testing of fractional delay systems, Elsevier Automatca 42(2006) pp 825-831.
93. Jack Leszcynski et al A numerical method solution for ordinary differential equation of fractional order. Technical university of Czestochowa Poland Lecture Notes 2005.
94. Kuo Diethelm and Ford, Analysis of fractional differential equations University of Manchester. Numerical Analysis Report No 377 pp 1-18 (2003).
95. K Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Electronics transactions on Numerical Analysis, ISSN 1068-9613, vol. 5, March 1997, pp. 1-6.
96. Barcena et al Discrete control for computer hard disc by fractional order hold devise. University of Pais Vasc Spain. 2000 (pp 1-34)
97. Shih-Tong, H M Srivastava et al A certain family of fractional differential equations, Ta i-wanese Journal of Mathematics Sept-2000, vol. 4, no. 3, pp 417-426.
98. Chen, Vinagre, Podlubny et al On fractional order disturbance observer DETC03-VIB48371 ASME Design Engineering Technical Conference Chicago USA Sept 2-6, 2003. pp. 1-8.
99. Chen, Vinagre, Podlubny, Fractional order disturbance observer for robust vibration suspension, Non Linear Dynamics 38:pp. 355-367 (2004).
100. Lubomir Dorcak Numerical Models for simulation of fractional order control system Slovak Accedemy of Science 1994. UER04-94, pp 1-15.
101. Vinagre, Ivo Petras, Two direct Tustin discretization methods for fractional order differentiation/integration. J. Franklin Institute: 340 (2003) pp 349-362.
102. Lubomir Dorcak, Ivo Petras et al Comparision of the methods for discrete approximations of fractional order operator Acta Montanisca Slovak Rocnik 8(2003) cislo-4, pp 236-239.
103. Tom Hartley and Carl Lorenzo Fractional order system identification by continuous order distribution Elsevier Signal Processing 2003. pp 2287-2300.
104. S G Samko, A A Kilbas, O I Maritcheva On Fractional Integrals and derivatives: Theory and Applications, Gordon and Breach New York (1993).
105. Saha Ray S, Exact solution for time fractional diffusion wave equation by decomposition method. Physics Scripta (2007) pp 53-61.
106. Saha Ray S, Bera R K, An approximate solution of non-linear fractional differential equation by Adomian's Decomposition, Analytical Applied Math Computer 167 pp 561-71.
107. Saha Ray S Bera R K, Analytical solution of Bagley Torvik equation by Adomian's decomposition method, Appl. Math. Comp. 168 pp 389-410.
108. Agarwal O P, Solution for fractional diffusion wave equation defined in bounded domain, Non-Linear Dynamics 29 pp 145-155 (2002).
109. Ramiro S, Barbosa et al, Time domain design of fractional differintegrators using least square, Signal Processing 86(2006) pp. 2567-2581.
110. α fractional control of robotic time delays systems, Mechanics Research Communications 33(2006) pp. 269-279.
111. Lynch V E, et al, Numerical methods of solution of partial differential equations of fractional order, J of Comput. Phy. 192(2003) pp 406-421.
112. Sierociuk Dominik, Fractional Kalman Filter algorithm for states, parameters and order of fractional system estimation, XV Krajowa Konferencja Automatyki, Warszawa, Polska (2005) pp 1-14.
113. Al-Alouni M A, A class of second order integrators and low pass differentiators, IEEE Trans. on circuit and systems 1: Fundamental Theory and Applications, 42(4) pp. 220-223 (1995)
114. Al-Alouni M A, Filling the gap between the bilinear and backward difference transforms: an interactive design approach, Int. J of Electrical Engineering Education, 34(4) pp. 331-337 (1997).
115. F. Riew, Mechanics with fractional derivatives, Phys. Rev. E, vol. 55, no. 3, pp 3581-3592, (1997).
116. H.E. Roman P A Alemany, Continuous time random walks and fractional diffusion equation, J Phys A: Math. Gen, vol. 27, pp. 3407-3410 (1994).
117. Scott Blair G W, Psychoreology: links between the past and the present, Journal of Texture Studies, vol. 5, pp. 3-12, (1974).
118. G W Scott Blair, Measurement of Mind and Matter, Dennis Dobson London 1950.
119. W. Wyss, The fractional diffusion equation, J. Math. Phys, vol. 27, no. 11, pp. 2782-2785 (1986).
120. W G Glockle, T F Nonnenmacher, Fractional integrators and Fox functions in the thory of viscoelelasticity, Macromolecules, vol. 24, pp. 6426-6436, (1991).
121. M Axtell, M E Bise, Fractional calculus applications in control systems, Proc of IEEE Nat. Aerospace and Electronics Conf, New York 1990, pp. 563-566.
122. H.H. Sun A.A. Abdelwahab B. Onaral, Linear Approximation of Transfer function with a pole of fractional power. IEEE Trans on Automatic Control vol. AC-29, no. 5, pp. 441-444, May 1984.
123. A. Charef, H H. Sun, Y.Y. Tsao, B. Onaral, Fractal systems as represented by singularity function, IEEE Trans. on Automatic Control, vol. 37, no. 9, pp. 1465-1470, Sept 1992.
124. H.E. Jones, B.A. Shenoi, Maximally flat lumped element approximation to fractional operator immitance function, IEEE Trans. on Circuits & Systems, vol. 17, no. pp. 125-128, 1970.
125. M. Moshrefi-Torbati & J.K. Hammond, Physical and geometrical interpretation of fractional operators, Journal of Franklin Institute, vol. 335, no. 6, pp. 1077-1086, 1998.
126. M. Nakagawa & K. Sorimachi, Basic characteristics of fractance device, IEICE Trans fundamentals, vol. E-75-A, no. 12, pp. 1814-1818.
127. M. Ichise, Y. Nagayanagi, T. Kojima, An analog simulation of non-integer order transfer function analysis for electrode process, J. Electroanal. Chem. Vol. 33, no. 1, pp. 253-265, 1971.
128. Bohennan Gary et al. Electrical components with fractional order impedances, Patent US 20060267595, November 2006.
129. V. Kirayakova, Generalized Fractional Calculus and Applications, Pitman Research Notes in Maths. No. 301, Longman, Harlow, 1994.
130. F. Mainardi, Fractional relaxation-oscillation and fractional diffusion wave phenomena, Chaos Solitons and Fractals, vol. 7, 1996, pp. 1467-1477.
131. R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, Fractals and Fractional Calculus in Continuum Mechanics Springer-Verlag, Vienna-New York, 1997.
132. R. Gorenflo, Yu. Luchko and S. Rogosin, Mittag-Leffler Type Functions: Notes on Growth Properties and Distribution of Zeros, no. A-97-04. Department of Mathematics and Informatics, Free University of Berlin, 1997.
133. R. Gorenflo, Fractional Calculus: Some numerical methods, Fractals and Fractional Calculus in Continuum Mechanics Springer-Verlag, Vienna-New York, 1997.
134. M. Caputo and F. Mainardi, A new dissipation model based on memory mechanism, Pure and applied Geophysics, vol. 91, no. 8, 1971, pp. 134-147.
135. M. Caputo, The rheology of an anelastic medium studied by means of the observation of splitting of its eigenfrequencies, J. Acoust. Soc. Am.vol. 86, no. 5. 1989 pp. 1984-1987.
136. Bush. V. Operational Circuit Analysis Wiley New-York (1929)
137. LePage. W R Complex variables and the Laplace transform for Engineers Dover, New-York (1961).
138. Oberhettinger.F and Badii L, Tables of Laplace transforms, Springer-Verlag Berlin (1973).
139. Robotnov Y N, Elements of Hereditary Solid Mechanics, English MIR Publishers Moscow (1980).
140. Heaviside. O. (1922) Electromagnetic Theory, Chelsea Edition New-York (1971).
141. Glockle W G and Nonnenmacher T F, Fractional Integral Operators and Fox Functions in the Theory of Viscoelasticity, Macromolecules, vol. 24, pp. 6426-6434 (1991)
142. V Churchill Ruel, W Brown James and F Verhey Roger, Complex Variables & Applications, Third Edition Mc Graw Hill Kogakusha Ltd. (1974).
143. M.E. Rayes-Melo, J.J. Martinez-Vega, C.A. Guerrero-Salazar, U. Ortiz-Mendez, Modeling of relaxation phenomena in organic dielectric materials, applications of differential and integral operator of fractional order, Journal of Optoelectronics and Advanced materials, Vol. 6, No. 3, September, 2004 pp1037-1043.
144. Marco Raberto, Enrico Scalas, Francesco Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study. Elsevier ARXIV: COND-MAT/0203596V1 28 (2006) pp1-8
145. Enrico Scalas, Rudolf Gorenflo, Francesco Mainardi, Marco Raberto, Revisiting the derivation of the fractional diffusion equation, Elsevier ARXIV: COND-MAT/0210166 V2 11.10.2002 (2002).
146. Rina Schumer, David.A. Benson, Mark.M. Merschaert, Stephen.W. Wheatcraft, Eulerian derivation of fractional advection-dispersion equation, J. of contaminant hydrology 48(2001) Elsevier pp69-88.
147. Chien-Cheng Tseng, Soo-Chang-Pei, Shih-Cheng Hsia, Computation of fractional derivatives using Fourier transforms and digital FIR differentiator, Elsevier Signal Processing 80(2000) pp151-159.
148. Andrea Roceo, Bruce. J. West, Fractional calculus and evolution of fractal phenomena, Elsevier Physica A 265(1999) pp535-546.
149. R. Hifler, Fractional Diffusion based on Riemann-Liouvelli Fractional Derivatives, J. Phys. Chem B Vol. 104 (2000) p3914.
150. H.M. Srivastava, S. Owa, Some characterization and distortion theorems involving fractional calculus, generalized hyper geometric functions, Hadamard products, Linear Operators and certain subclasses of Analytical functions, Nagoya Math. J. Vol.106 (1987) pp1-28.
151. Rina Schumer, David.A. Benson, Multiscaling fractional advection dispersion equation and their solution, Water Resource Research Vol.39, No.1, pp12-1-12-11. (2003)
152. R. Gorenflo, F. Mainardi, Fractional oscillators and Mittag-Leffler function, Technical Report Preprint A-14/96, University of Berlin, (1996).
153. H. Bateman, Higher transcendental function, Vol. 3, Mc Graw Hill New-York (1954).
154. A. Erdelyi, Asymptotic Expansions, Dover (1954).
155. R. Hifler, Applications of fractional calculus in Physics, World Scientific Singapore (2000).
156. R. Hifler, L. Anton, Fractional master equation and fractal time random walks, Phys Rev. E51 (1995) R818-R851.
157. J. Klafter, A. Blumen, M. F. Shlesinger Stochastic pathways to anomalous diffusion, Phys Rev A 35 (1987).
158. I.M. Gel'fand, G.E Shilov, Generalized functions, Vol.1 Academic Press New-York (