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March  2020, 13(3): 575-607. doi: 10.3934/dcdss.2020032

Memorized relaxation with singular and non-singular memory kernels for basic relaxation of dielectric vis-à-vis Curie-von Schweidler & Kohlrausch relaxation laws

Scientist RCSDS E & I Group BARC Mumbai and Honorary Senior Research Professor CMPRC Dept. of Physics, Jadavpur University, Kolkata, India

Received  August 2018 Revised  January 2019 Published  March 2019

We have constructed the basic dielectric relaxations evolution expression for relaxing current as convolution operation of the chosen memory kernel and rate of change of applied voltage. We have studied types of memory kernels singular, non-singular and combination of singular and non-singular (mixed) decaying functions. With these, we form constitutive equations for relaxation dynamics of dielectrics; i.e. capacitor. We observe that though mathematically we can use non-singular kernels yet this does not give presently much useful practical or physically realizable results and interpretations. We relate our observations to relaxation currents given via Curie-von Schweidler and Kohlraush laws. The Curie-von Schweidler law gives singular function power law as basic relaxation current in dielectric relaxation; whereas the Kohlraush law is Electric field relaxation in dielectric as non-Debye function taken as stretched exponential i.e. non-singular function. These two laws are used since late nineteenth century for various dielectric relaxation and characterization studies. Here we arrive at general constitutive equation for capacitor and with each type of memory kernel we give corresponding impedance function and in some cases equivalent circuit representation for the capacitor element. We classify these systems as Curie-von Shweidler type for system with singular memory kernel function and Kohlraush type for system evolved via using non-singular function or mixed functions as memory kernel. We note that use of singular memory kernel gives constituent relations and impedance functions that are experimentally verified in large number of cases of dielectric studies. Therefore, we have a question, does natural relaxation dynamics for dielectrics have a singular memory kernel, and the relaxation current function is singular in nature? Is it the singular relaxation function for capacitor dynamics with singular memory kernel remains universal law for dielectric relaxation? However, we are not questioning researchers modeling relaxation of dielectric via non-singular functions, yet we are hinting about complexity and lack of interpretability of basic constituent equation of dielectric relaxation dynamics thus obtained via considering non-singular and mixed memory kernels; perhaps due to insufficient experimental evidences presently. However, the method employed in this study is general method. This method can be used to form memorized constituent equations for other systems (say Radioactive Decay/Growth, Diffusion and Wave phenomena) from basic evolution equation, i.e. effect function as convolution of memory kernel with cause function.

Citation: Shantanu Das. Memorized relaxation with singular and non-singular memory kernels for basic relaxation of dielectric vis-à-vis Curie-von Schweidler & Kohlrausch relaxation laws. Discrete and Continuous Dynamical Systems - S, 2020, 13 (3) : 575-607. doi: 10.3934/dcdss.2020032
References:
[1]

M. Abramowitz and I. A. Stegun, (eds.), Handbook of Mathematical Functions, Nat. Bur. Stand. Appl. Math. Ser 55. U.S. Govt. Printing Office Washington D C, 1964.

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[3]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[4]

P. Bak, How Nature Works: The Science of Self-Organized Criticality, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-5426-1.

[5]

M. N. Berberan-SantosE. N. Bodunov and B. Valeur, Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential), Chemical Physics, 315 (2005), 171-182. 

[6] A. R. Blythe, Electrical Properties of Polymer, Cambridge University Press, 1979. 
[7]

E. N. Bodunov, Yu. A. Antonov and A. L. Simões Gamboa, On the origin of stretched exponential (Kohlrausch) relaxation kinetics in the room, The Journal of Chemical Physics, 146 (2017), 114102.

[8]

J. W. Borough, W. G. Brammer and J. Burnham, Degradation of PVDF Capacitors during accelerated tests, Report Huges Aircraft Company, Culver City, CA, 219–223, 1980.

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ Appl., 1 (2015), 73-85. 

[10]

S. Chiniba and R. Tobazeon, Long term break-down of polypropylene films in different media, Conference Record of the 1985 International Conference on Properties and Application of Dielectric Materials, Xian, China, IEEE, 1985,566–584.

[11]

S. Das, Functional Fractional Calculus, 2nd Edition Springer-Verlag, Germany, 2011. doi: 10.1007/978-3-642-20545-3.

[12]

S. Das, Revisiting the Curie-von Schweidler law for dielectric relaxation and derivation of distribution function for relaxation rates as Zipf's power law and manifestation of fractional differential equation for capacitor, Journal of Modern Physics, 8 (2017), Article ID: 80594, 25 pages. doi: 10.4236/jmp.2017.812120.

[13]

S. Das, Kindergarten of Fractional Calculus, (Book-under print at Cambridge Scholars Publishing UK- of lecture notes on fractional calculus course at Dept. of Physics Jadavpur University, Phys. Dept. St Xaviers Univ. Calcutta and Dept. of Appl. Mathematics Calcutta University etc.)

[14]

S. Das, A new look at formulation of charge storage in capacitors and application to classical capacitor and fractional capacitor theory, Asian Journal of Research and Reviews in Physics, 1 (2018), 1–18, Article No.AJR2P.43738.

[15]

S. Das and N. C. Pramanik, Micro-structural roughness of electrodes manifesting as temporal fractional order differential equation in super-capacitor transfer characteristics, International Journal of Mathematics and Computation, 20 (2013), 94-113. 

[16]

A. S. ElwakilA. AllaguiB. J. Maundy and C. A. Psycchalinos, A low frequency oscillator using super-capacitor, AEU – Int. J Electron Commun., 70 (2016), 970-973.  doi: 10.1016/j.aeue.2016.03.020.

[17]

T. J. FreebonB. Maundy and A. S. Elwakil, Measurement of super-capacitor fractional order model parameters from voltage excited step response, IEEE J Emerging Selected Topics in Circuits and Systems, 3 (2013), 367-76. 

[18]

T. J. FreebonB. Maundy and A. S. Elwakil, Fractional order models of super-capacitors, batteries, fuel-cell: A survey, Mater Renew Sustain Energy, 4 (2015), 1-7. 

[19]

A. Giusti, A Comment on some new definitions of fractional derivative, arXiv: 1710.06852v4 [math.CA] 24 Apr 2018.

[20]

A. Giusti and I. Colombaro, Prabhakar-like fractional viscoelasticity, Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 138-143.  doi: 10.1016/j.cnsns.2017.08.002.

[21]

J. F. Gómez-Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels, The European Physical Journal plus, 133 (2018), 1-21. 

[22]

F. Gómez-AguilarJ. Rosales and M. Guía, Analytical and numerical solutions of electrical circuits described by fractional derivatives, Applied Mathematical Modeling, 40 (2016), 9079-9094.  doi: 10.1016/j.apm.2016.05.041.

[23]

M. GuíaF. Gomez-Aguilar and J. Rosales, RLC electrical circuit of non-integer order, Central European Journal of Physics, 11 (2013), 1361-1365. 

[24]

M. GuíaF. Gomez-Aguilar and J. Rosales, Analysis on the time and frequency domain for the RC electric circuit of fractional order, Central European Journal of Physics, 11 (2013), 1366-1371. 

[25]

S. HazraT. DuttaS. Das and S. Tarafdar, Memory of electric field in Laponite and how it affects crack formation: Modeling through generalized calculus, Langmuir, 33 (2017), 8468-8475.  doi: 10.1021/acs.langmuir.7b02034.

[26]

F. S. HowellR. A. BoseP. B. Macedo and C. T. Moynihan, Electrical relaxation in a glass-forming molten salt, The Journal of Physical Chemistry, 78 (1974), 639-648. 

[27]

J. Hristov, Electrical Circuits of Non-Integer Order: Introduction To an Emerging Interdisciplinary Area With Examples, Springer, 2018.

[28]

N. Jordan Jameson, M. H. Azarian and M. Pecht, Thermal Degradation of Polyimide Insulation and its Effect on Electromagnetic Coil Impedance, Proceedings of the Society for Machinery Failure Prevention Technology 2017 Annual Conference, 2017.

[29]

C. Jaques, Recherches sur le pouvoir inducteur specifique et la conductibilite des corps cristallises, Annales de Chimie et de Physique, 17 (1889), 384-434. 

[30] A. K. Jonscher, Dielectric Relaxation in Solids, Chelsea dielectric Press, London, 1983.  doi: 10.1088/0022-3727/32/14/201.
[31]

M. KumarS. Ghosh and S. Das, Frequency dependent piecewise fractional order modeling of ultra-capacitors using hybrid optimization and fuzzy clustering, Journal of Power Sources, 335 (2015), 98-104. 

[32]

M. KumarS. Ghosh and S. Das, Charge discharge energy efficiency analysis of ultra-capacitor with fractional order dynamics using hybrid optimization and its experimental validation, International Journal of Electronics & Communications (AEU), 78 (2017), 2714-2780. 

[33]

E. K. LenziA. A. Tateishi and H. V. Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. 

[34]

V. F. Morales-Delgado, J. F. G'omez-Aguilar, M. A. Taneco-Hernández and R. F. Escobar-Jim'enez, A novel fractional derivative with variable- and constant-order applied to a mass-spring-damper system, Eur. Phys. J. Plus, 133 (2018), 78. doi: 10.1140/epjp/i2018-11905-4.

[35]

C. T. Moynihan, L. P. Boesch and N. L. Laberge, Decay functions for Electric field Relaxation in vitreous ionic conductors, Physics and Chemistry of Glasses, 14 (1973).

[36]

I. J. Nagrath and M. Gopal, Control System Engineering (Second Edition), Wiley Eastern Limited New Age International Limited, 1982.

[37] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974. 
[38]

M. D. Ortigueira and Machado J. Tenreiro, A critical analysis of the Caputo-Fabrizio operator, Communications In Nonlinear Science and Numerical Simulation, 59 (2018), 608-611.  doi: 10.1016/j.cnsns.2017.12.001.

[39]

A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010558.

[40]

D. M. W. Powers, Application and explanation of Zipf's Law, Association for Computational Linguistics, (1998), 151–160. doi: 10.3115/1603899.1603924.

[41]

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffer function in the kernel, Yokohama Mathematical Journal, 19 (1971), 7-15. 

[42]

K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[43]

T. Sandev, Generalized Langevin equation and the Prabhakar derivative, Mathematics, 5 (2017), 66.

[44]

E. R. von Schweidler, Studien über die Anomalien im Verhalten der Dielektrika (Studies on the anomalous behavior of dielectrics), Annalen der Physik, 329 (1907), 711-770. 

[45]

D. Sornette, Mechanism of Power laws without self-organization, International Journal of Modern Physics, C13 (2001), 133-136. 

[46]

K. Thorborg, Power Electronics, Chalmer Tech. Univ. Goteborg, Sweeden, 1985.

[47]

S. Westerlund, Dead matter has memory, Physica Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.

[48]

S. Westerlund and L. Ekstam, Capacitor theory, IEEE Trans on Dielectrics and Insulation, 1 (1994), 826-839.  doi: 10.1109/94.326654.

[49]

G. William and D. C. Watts, Non-symmetrical dielectric relaxation behavior arising from a simple empirical decay function, Transactions on Faraday Society, 66 (1970).

[50]

G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, (eds.), Handbook of Mathematical Functions, Nat. Bur. Stand. Appl. Math. Ser 55. U.S. Govt. Printing Office Washington D C, 1964.

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Physica A: Statistical Mechanics and its Applications, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.

[3]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769.  doi: 10.2298/TSCI160111018A.

[4]

P. Bak, How Nature Works: The Science of Self-Organized Criticality, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4757-5426-1.

[5]

M. N. Berberan-SantosE. N. Bodunov and B. Valeur, Mathematical functions for the analysis of luminescence decays with underlying distributions 1. Kohlrausch decay function (stretched exponential), Chemical Physics, 315 (2005), 171-182. 

[6] A. R. Blythe, Electrical Properties of Polymer, Cambridge University Press, 1979. 
[7]

E. N. Bodunov, Yu. A. Antonov and A. L. Simões Gamboa, On the origin of stretched exponential (Kohlrausch) relaxation kinetics in the room, The Journal of Chemical Physics, 146 (2017), 114102.

[8]

J. W. Borough, W. G. Brammer and J. Burnham, Degradation of PVDF Capacitors during accelerated tests, Report Huges Aircraft Company, Culver City, CA, 219–223, 1980.

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ Appl., 1 (2015), 73-85. 

[10]

S. Chiniba and R. Tobazeon, Long term break-down of polypropylene films in different media, Conference Record of the 1985 International Conference on Properties and Application of Dielectric Materials, Xian, China, IEEE, 1985,566–584.

[11]

S. Das, Functional Fractional Calculus, 2nd Edition Springer-Verlag, Germany, 2011. doi: 10.1007/978-3-642-20545-3.

[12]

S. Das, Revisiting the Curie-von Schweidler law for dielectric relaxation and derivation of distribution function for relaxation rates as Zipf's power law and manifestation of fractional differential equation for capacitor, Journal of Modern Physics, 8 (2017), Article ID: 80594, 25 pages. doi: 10.4236/jmp.2017.812120.

[13]

S. Das, Kindergarten of Fractional Calculus, (Book-under print at Cambridge Scholars Publishing UK- of lecture notes on fractional calculus course at Dept. of Physics Jadavpur University, Phys. Dept. St Xaviers Univ. Calcutta and Dept. of Appl. Mathematics Calcutta University etc.)

[14]

S. Das, A new look at formulation of charge storage in capacitors and application to classical capacitor and fractional capacitor theory, Asian Journal of Research and Reviews in Physics, 1 (2018), 1–18, Article No.AJR2P.43738.

[15]

S. Das and N. C. Pramanik, Micro-structural roughness of electrodes manifesting as temporal fractional order differential equation in super-capacitor transfer characteristics, International Journal of Mathematics and Computation, 20 (2013), 94-113. 

[16]

A. S. ElwakilA. AllaguiB. J. Maundy and C. A. Psycchalinos, A low frequency oscillator using super-capacitor, AEU – Int. J Electron Commun., 70 (2016), 970-973.  doi: 10.1016/j.aeue.2016.03.020.

[17]

T. J. FreebonB. Maundy and A. S. Elwakil, Measurement of super-capacitor fractional order model parameters from voltage excited step response, IEEE J Emerging Selected Topics in Circuits and Systems, 3 (2013), 367-76. 

[18]

T. J. FreebonB. Maundy and A. S. Elwakil, Fractional order models of super-capacitors, batteries, fuel-cell: A survey, Mater Renew Sustain Energy, 4 (2015), 1-7. 

[19]

A. Giusti, A Comment on some new definitions of fractional derivative, arXiv: 1710.06852v4 [math.CA] 24 Apr 2018.

[20]

A. Giusti and I. Colombaro, Prabhakar-like fractional viscoelasticity, Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 138-143.  doi: 10.1016/j.cnsns.2017.08.002.

[21]

J. F. Gómez-Aguilar, Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels, The European Physical Journal plus, 133 (2018), 1-21. 

[22]

F. Gómez-AguilarJ. Rosales and M. Guía, Analytical and numerical solutions of electrical circuits described by fractional derivatives, Applied Mathematical Modeling, 40 (2016), 9079-9094.  doi: 10.1016/j.apm.2016.05.041.

[23]

M. GuíaF. Gomez-Aguilar and J. Rosales, RLC electrical circuit of non-integer order, Central European Journal of Physics, 11 (2013), 1361-1365. 

[24]

M. GuíaF. Gomez-Aguilar and J. Rosales, Analysis on the time and frequency domain for the RC electric circuit of fractional order, Central European Journal of Physics, 11 (2013), 1366-1371. 

[25]

S. HazraT. DuttaS. Das and S. Tarafdar, Memory of electric field in Laponite and how it affects crack formation: Modeling through generalized calculus, Langmuir, 33 (2017), 8468-8475.  doi: 10.1021/acs.langmuir.7b02034.

[26]

F. S. HowellR. A. BoseP. B. Macedo and C. T. Moynihan, Electrical relaxation in a glass-forming molten salt, The Journal of Physical Chemistry, 78 (1974), 639-648. 

[27]

J. Hristov, Electrical Circuits of Non-Integer Order: Introduction To an Emerging Interdisciplinary Area With Examples, Springer, 2018.

[28]

N. Jordan Jameson, M. H. Azarian and M. Pecht, Thermal Degradation of Polyimide Insulation and its Effect on Electromagnetic Coil Impedance, Proceedings of the Society for Machinery Failure Prevention Technology 2017 Annual Conference, 2017.

[29]

C. Jaques, Recherches sur le pouvoir inducteur specifique et la conductibilite des corps cristallises, Annales de Chimie et de Physique, 17 (1889), 384-434. 

[30] A. K. Jonscher, Dielectric Relaxation in Solids, Chelsea dielectric Press, London, 1983.  doi: 10.1088/0022-3727/32/14/201.
[31]

M. KumarS. Ghosh and S. Das, Frequency dependent piecewise fractional order modeling of ultra-capacitors using hybrid optimization and fuzzy clustering, Journal of Power Sources, 335 (2015), 98-104. 

[32]

M. KumarS. Ghosh and S. Das, Charge discharge energy efficiency analysis of ultra-capacitor with fractional order dynamics using hybrid optimization and its experimental validation, International Journal of Electronics & Communications (AEU), 78 (2017), 2714-2780. 

[33]

E. K. LenziA. A. Tateishi and H. V. Ribeiro, The role of fractional time-derivative operators on anomalous diffusion, Frontiers in Physics, 5 (2017), 1-9. 

[34]

V. F. Morales-Delgado, J. F. G'omez-Aguilar, M. A. Taneco-Hernández and R. F. Escobar-Jim'enez, A novel fractional derivative with variable- and constant-order applied to a mass-spring-damper system, Eur. Phys. J. Plus, 133 (2018), 78. doi: 10.1140/epjp/i2018-11905-4.

[35]

C. T. Moynihan, L. P. Boesch and N. L. Laberge, Decay functions for Electric field Relaxation in vitreous ionic conductors, Physics and Chemistry of Glasses, 14 (1973).

[36]

I. J. Nagrath and M. Gopal, Control System Engineering (Second Edition), Wiley Eastern Limited New Age International Limited, 1982.

[37] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, 1974. 
[38]

M. D. Ortigueira and Machado J. Tenreiro, A critical analysis of the Caputo-Fabrizio operator, Communications In Nonlinear Science and Numerical Simulation, 59 (2018), 608-611.  doi: 10.1016/j.cnsns.2017.12.001.

[39]

A. D. Polyanin and A. V. Manzhirov, Handbook of Integral Equations, 2nd Edition, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781420010558.

[40]

D. M. W. Powers, Application and explanation of Zipf's Law, Association for Computational Linguistics, (1998), 151–160. doi: 10.3115/1603899.1603924.

[41]

T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffer function in the kernel, Yokohama Mathematical Journal, 19 (1971), 7-15. 

[42]

K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[43]

T. Sandev, Generalized Langevin equation and the Prabhakar derivative, Mathematics, 5 (2017), 66.

[44]

E. R. von Schweidler, Studien über die Anomalien im Verhalten der Dielektrika (Studies on the anomalous behavior of dielectrics), Annalen der Physik, 329 (1907), 711-770. 

[45]

D. Sornette, Mechanism of Power laws without self-organization, International Journal of Modern Physics, C13 (2001), 133-136. 

[46]

K. Thorborg, Power Electronics, Chalmer Tech. Univ. Goteborg, Sweeden, 1985.

[47]

S. Westerlund, Dead matter has memory, Physica Scripta, 43 (1991), 174-179.  doi: 10.1088/0031-8949/43/2/011.

[48]

S. Westerlund and L. Ekstam, Capacitor theory, IEEE Trans on Dielectrics and Insulation, 1 (1994), 826-839.  doi: 10.1109/94.326654.

[49]

G. William and D. C. Watts, Non-symmetrical dielectric relaxation behavior arising from a simple empirical decay function, Transactions on Faraday Society, 66 (1970).

[50]

G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley, 1949.

Figure 1.  Response current $ {\rm i}(t) = {\rm C}t^{-\alpha } $ to unit step voltage excitation for dielectric relaxation with singular power law memory kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $
Figure 2.  Response current $ {\rm i}(t) = {\rm C}(1+\lambda t)^{-\alpha } $ to unit step voltage excitation for dielectric relaxation with non-singular power law memory kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $
Figure 3.  Response current $ {\rm i}(t) = {\rm C}E_{\alpha } (-\lambda t^{\alpha }) $ to unit step voltage excitation for dielectric relaxation with non-singular Mittag-Leffler kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $
Figure 4.  Response current to unit step voltage excitation for dielectric relaxation with non-singular stretched exponential kernel for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $
Figure 5.  Response current to unit step voltage excitation for dielectric relaxation with derivative of stretched exponential for various memory indexes: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $
Figure 6.  Response current to unit step voltage excitation for dielectric relaxation with non-singular and singular memory kernels with memory index as $ \alpha $ 0.5: Y-Axis current $ {\rm i}(t) $ and X-Axis time $ t $ : (ⅱ) - $ {\rm k}(t) = t^{-0.5} $, (ⅲ) - $ {\rm k}(t) = (1+t)^{-0.5} $, (ⅳ) - $ {\rm k}(t) = {\it E}_{0.5} (-t^{0.5} ) $, (ⅴ)- $ {\rm k}(t) = {\rm e}^{-t} $, (vi)-$ {\rm k}(t) = {\rm e}^{-t^{0.5}} $ and (ⅶ)-$ {\rm k}(t) = 0.5t^{-0.5} {\rm e}^{-{\it t}^{0.5}} $
Table 1.  Summary of results of various singular and Non-singular Memory Kernels
S.NoMemory kernel ${\rm k}(t)$ Type Constitutive equation of capacitor Impedance function in Laplace domain
1Delta Function
${\rm C}\delta (t)$
Singular ${\rm i}(t) = {\rm Cv}^{(1)}(t)$
${\rm i}(t) = {\rm C}\left({}_{0} I_{t}^{0} {\rm v}^{(1)}(t)\right)$
${\rm Z}(s)=\frac{1}{s{\rm C}}$
2Power Law
${\rm C}{\it t}^{-\alpha}$
$0<\alpha <1$
Singular ${\rm i}(t) = {\rm C}_{\alpha} {\rm v}^{(\alpha)}(t)$
${\rm C}_{\alpha} = {\rm C}\left(\Gamma (1-\alpha)\right)$
${\rm i}(t) = {\rm C}_{\alpha} \left({}_{0} I_{t}^{1-\alpha} {\rm v}^{(1)}(t)\right)$
${\rm Z}(s)=\frac{1}{s^{\alpha} {\rm C}_{\alpha }}$
3Non-singular Power Law
${\rm C}(1+\lambda t)^{-\alpha } $
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=1}^{\infty}w_n\left({}_{0}I_{t}^{n}{\rm v}^{(1)}(t)\right)$
$w_1=1, w_2=-\alpha \lambda, $
$w_3=(\alpha)(\alpha +1)\lambda^{2}, \ldots$
${\rm Z}(s)=\frac{1}{{\rm C}\left(\sum _{n=1}^{\infty }w_{n} s^{1-n}\right)}$
4Mittag-Leffler
${\rm C}E_{\alpha} (-\lambda t^{\alpha})$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0} I_{t}^{\alpha n+1} {\rm v}^{(1)} (t)\right)$
${w_n=(-1)^{n}\lambda^{n}}$
${\rm v}(t)=\frac{1}{{\rm C}} {\rm i}(t)+\frac{\lambda}{{\rm C}} \left({}_{0} I_{t}^{\alpha } {\rm i}(t)\right)$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
${\rm Z}(s)=\frac{1}{{\rm C}} +\frac{1}{\left({\frac{{\rm C}}{\lambda}}\right)s^{\alpha}}$
5Exponential
${\rm Ce}^{-\lambda t}$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0}I_{t}^{n+1}{\rm v}^{(1)}(t)\right)$
$w_{n}=(-1)^{n}\lambda^{n}$
${\rm v}(t)=\frac{1}{{\rm C}}{\rm i}(t)+\frac{\lambda}{{\rm C}}\int_0^t{\rm i}(\tau){\rm d}\tau$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-n}}$
${\rm Z}(s)=\frac{1}{{\rm C}}+\frac{1}{s\left({\frac{{\rm C}}{\lambda}}\right)}$
6Stretched-Exponential
${\rm Ce}^{-(\lambda t)^{\alpha}}$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{\alpha n+1}\left[{\rm v}^{(1)} ({\it t})\right]\right)$
${\it w}_{n} =(-1)^{n} \left({\frac{\lambda^{\alpha n} \Gamma(\alpha n+1)}{n!}}\right)$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
7Derivative of
Stretched
Exponential
${\rm C}\alpha\lambda^{\alpha} t^{\alpha-1}{\rm e}^{-(\lambda {\it t})^{\alpha}}$
$0<\alpha <1$
Singular ${\rm i}(t)={\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{(n+1)\alpha} {\rm v}^{(1)} ({\it t})\right)$
${\rm C}_{\lambda}=\alpha\lambda^{\alpha} {\rm C}$
${\it w}_{n}=(-1)^{n} \left({\textstyle\frac{\lambda^{\alpha n} \Gamma((n+1)\alpha)}{n!}}\right)$
${\rm Z}(s)=$
$\frac{1}{{\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} s^{1-(n+1)\alpha}}$
S.NoMemory kernel ${\rm k}(t)$ Type Constitutive equation of capacitor Impedance function in Laplace domain
1Delta Function
${\rm C}\delta (t)$
Singular ${\rm i}(t) = {\rm Cv}^{(1)}(t)$
${\rm i}(t) = {\rm C}\left({}_{0} I_{t}^{0} {\rm v}^{(1)}(t)\right)$
${\rm Z}(s)=\frac{1}{s{\rm C}}$
2Power Law
${\rm C}{\it t}^{-\alpha}$
$0<\alpha <1$
Singular ${\rm i}(t) = {\rm C}_{\alpha} {\rm v}^{(\alpha)}(t)$
${\rm C}_{\alpha} = {\rm C}\left(\Gamma (1-\alpha)\right)$
${\rm i}(t) = {\rm C}_{\alpha} \left({}_{0} I_{t}^{1-\alpha} {\rm v}^{(1)}(t)\right)$
${\rm Z}(s)=\frac{1}{s^{\alpha} {\rm C}_{\alpha }}$
3Non-singular Power Law
${\rm C}(1+\lambda t)^{-\alpha } $
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=1}^{\infty}w_n\left({}_{0}I_{t}^{n}{\rm v}^{(1)}(t)\right)$
$w_1=1, w_2=-\alpha \lambda, $
$w_3=(\alpha)(\alpha +1)\lambda^{2}, \ldots$
${\rm Z}(s)=\frac{1}{{\rm C}\left(\sum _{n=1}^{\infty }w_{n} s^{1-n}\right)}$
4Mittag-Leffler
${\rm C}E_{\alpha} (-\lambda t^{\alpha})$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0} I_{t}^{\alpha n+1} {\rm v}^{(1)} (t)\right)$
${w_n=(-1)^{n}\lambda^{n}}$
${\rm v}(t)=\frac{1}{{\rm C}} {\rm i}(t)+\frac{\lambda}{{\rm C}} \left({}_{0} I_{t}^{\alpha } {\rm i}(t)\right)$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
${\rm Z}(s)=\frac{1}{{\rm C}} +\frac{1}{\left({\frac{{\rm C}}{\lambda}}\right)s^{\alpha}}$
5Exponential
${\rm Ce}^{-\lambda t}$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n}\left({}_{0}I_{t}^{n+1}{\rm v}^{(1)}(t)\right)$
$w_{n}=(-1)^{n}\lambda^{n}$
${\rm v}(t)=\frac{1}{{\rm C}}{\rm i}(t)+\frac{\lambda}{{\rm C}}\int_0^t{\rm i}(\tau){\rm d}\tau$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-n}}$
${\rm Z}(s)=\frac{1}{{\rm C}}+\frac{1}{s\left({\frac{{\rm C}}{\lambda}}\right)}$
6Stretched-Exponential
${\rm Ce}^{-(\lambda t)^{\alpha}}$
$0<\alpha <1$
Non-Singular ${\rm i}(t)={\rm C}\sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{\alpha n+1}\left[{\rm v}^{(1)} ({\it t})\right]\right)$
${\it w}_{n} =(-1)^{n} \left({\frac{\lambda^{\alpha n} \Gamma(\alpha n+1)}{n!}}\right)$
${\rm Z}(s)=\frac{1}{{\rm C}\sum_{n=0}^{\infty}w_{n} s^{-\alpha n}}$
7Derivative of
Stretched
Exponential
${\rm C}\alpha\lambda^{\alpha} t^{\alpha-1}{\rm e}^{-(\lambda {\it t})^{\alpha}}$
$0<\alpha <1$
Singular ${\rm i}(t)={\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} \left({}_{0} I_{t}^{(n+1)\alpha} {\rm v}^{(1)} ({\it t})\right)$
${\rm C}_{\lambda}=\alpha\lambda^{\alpha} {\rm C}$
${\it w}_{n}=(-1)^{n} \left({\textstyle\frac{\lambda^{\alpha n} \Gamma((n+1)\alpha)}{n!}}\right)$
${\rm Z}(s)=$
$\frac{1}{{\rm C}_{\lambda} \sum_{n=0}^{\infty}w_{n} s^{1-(n+1)\alpha}}$
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