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January  2014, 13(1): 217-223. doi: 10.3934/cpaa.2014.13.217

Diffusion effects in a superconductive model

Monica De Angelis 1, and  Gaetano Fiore 2,

1. 

Univ. of Naples Federico II, Dept of Math and Appl, Via Claudio n. 21, 80125 Naples, Italy
2. 

Univ. of Naples Federico II, I.N.F.N., Sez. of Naples, Complesso MSA, V. Cintia, 80126 Naples, Italy

Received  November 2012 Revised  April 2013 Published  July 2013

A superconductive model characterized by a third order parabolic operator $ {\mathcal L}_\varepsilon $ is analyzed. When the viscous terms, represented by higher-order derivatives, tend to zero, a hyperbolic operator $ {\mathcal L}_0 $ appears. Furthermore, if ${\mathcal P}_\varepsilon$ is the Dirichlet initial-boundary value problem for $ {\mathcal L}_\varepsilon$, when ${\mathcal L} _\varepsilon $ turns into ${\mathcal L}_0 , $ ${\mathcal P}_\varepsilon$ turns into a problem ${\mathcal P}_0$ with the same initial-boundary conditions of ${\mathcal P}_\varepsilon $. As long as the higher-order derivatives of the solution of ${\mathcal P}_0$ are bounded, an estimate of solution for the nonlinear problem related to the remainder term $ r, $ is achieved. Moreover, some classes of explicit solutions related to $ {\mathcal P}_0 $ are determined, proving the existence of at least one motion whose derivatives are bounded. The estimate shows that the diffusion effects are bounded even when time tends to infinity.
Keywords: fundamental solution, Laplace transform., Josephson junction, parabolic-hyperbolic behavior, Superconductivity, initial-boundary value problems for higher-order parabolic equations.
Mathematics Subject Classification: Primary: 82D55,74K30; Secondary: 35K35, 35E0.
Citation: Monica De Angelis, Gaetano Fiore. Diffusion effects in a superconductive model. Communications on Pure and Applied Analysis, 2014, 13 (1) : 217-223. doi: 10.3934/cpaa.2014.13.217
References:
[1]

T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the sine-Gordon equation, Journal of Mathematical Physics, 51 (2010), 1-26.

[2]

A. Benabdallah, J. G. Caputo and A. C. Scott, Exponentially tapered josephson flux-flow oscillator, Phy. Rev. B, 54 (1996), 16139.

[3]

A. Benabdallah, J. G. Caputo and A. C. Scott, Laminar phase flow for an exponentially tapered josephson oscillator, J. Apl. Phys., 588 (2000), 3527.

[4]

S. Bondarenko and Nakagawa, SQUID-based magnetic microscope, in "Smart Materials for Ranking Systems," J. France et al (edition), Springer (2006), 195-201.

[5]

T. L. Boyadjiev, E. G. Semerdjieva and Yu. M. Shukrinov, Common features of vortex structure in long exponentially shaped Josephson junctions and Josephson junctions with inhomogeneities, Physica C, 460-462 (2007), 1317-1318.

[6]

G. Carapella, N. Martucciello and G. Costabile, Experimental investigation of flux motion in exponentially shaped Josephson junctions, PHYS REV B, 66 (2002), 134531.

[7]

J. Clarke, "SQUIDs for Everything," Nature Materials, 10 (2011).

[8]

J. Clarke, SQUIDs: Then and Now, chapter in BCS: 50 Years (eds. Leon N Cooper and Dmitri Feldman) World Scientific Publishing Co. Pte. Ltd., Singapore (2010).

[9]

S. A. Cybart et al., Dynes Series array of incommensurate superconducting quantum interference devices, Appl. Phys Lett, 93 (2008), 1-3.

[10]

A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta Appl. Math., 122 (2012), 255-267.

[11]

M. Dehghan and A. Shokri, A numerical method for solution of the two dimensional sine- Gordon equation using the radial basis functions, Mat Comp in Simulation, 79 (2008), 700-715.

[12]

M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50.

[13]

M. De Angelis, Asymptotic analysis for the strip problem related to a parabolic third-order operator, Applied Mathematics Letters, 14 (2001), 425-430.

[14]

M. De Angelis, A priori estimates for excitable models, Meccanica (2013). doi: 10.1007/s11012-013-9763-2.

[15]

M. De Angelis, On exponentially shaped Josephson junctions, Acta appl. Math, 122 (2012), 179-189

[16]

M. De Angelis, On a parabolic operator of dissipative systems,, submitted to Acta appl. Math, (). 

[17]

M. De Angelis and G. Fiore, Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect, J. Math. Anal. Appl., 404 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029.

[18]

M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in "Mathematical Physics Models and Engineering Sciences" (ed. Liguori), Italy, (2008), 191-202.

[19]

M. De Angelis and E. Mazziotti, Non linear travelling waves with diffusion, Rend. Acc. Sc. Fis. Mat. Napoli, 73 (2006), 23-36.

[20]

De Angelis, A. M. Monte and P. M. Renno, On fast and slow times in models with diffusion, Math Models and Methods in Applied Sciences, 12 (2002), 1741-1749.

[21]

M. De Angelis and P. Renno, Asymptotic effects of boundary perturbations in excitable systems, (2013),, submitted to Discrete and Continuous Dynamical Systems - B. Available from: \url{http://arxiv.org/pdf/1304.3891v1.pdf}, (). 

[22]

M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric Mat, 57 (2008), 95-109.

[23]

De Angelis and P. M. Renno, Diffusion and wave behaviour in linear Voigt model, C. R. Mecanique, 330 (2002), 21-26

[24]

Gutman S. Junhohg Ha, Identification problem for damped sine Gordon equation with point sources, J. Math. Anal. Appl., 375 (2011), 648-666. doi: 10.1016/j.jmaa.2010.10.006.

[25]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," The MIT press, England, 2007.

[26]

M. Jaworski, "Fluxon Dynamics in Exponentially Shaped Josephson Junction," Phy. rev. B, 71 (2005), 1-6.

[27]

M. Jaworski, Exponentially tapered Josephson junction: some analytic results, Theor and Math Phys, 144 (2005), 1176-1180.

[28]

J. McCall and Lindsa, Superconductor cables: Advanced capabilities for the smart grid, Utility Automation Engineering TD, 13 (2008), 54.

[29]

J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, N.Y, 2002.

[30]

S. Rionero, Asymptotic behaviour of solutions to a nonlinear third order P.D.E modeling physical phenomena, Boll Unione Mat Ital, 9 (2012), 451-468.

[31]

H. Rogalla and P. H. Kes, "100 Years of Superconductivity," CRC Press, (2012).

[32]

A.C. Scott, "The Nonlinear Universe: Chaos, Emergence, Life," Springer-Verlag, 2007.

[33]

A.C. Scott, "Neuroscience A mathematical Primer," Springer-Verlag, 2002.

show all references

References:
[1]

T. Aktosun, F. Demontis, and C. van der Mee, Exact solutions to the sine-Gordon equation, Journal of Mathematical Physics, 51 (2010), 1-26.

[2]

A. Benabdallah, J. G. Caputo and A. C. Scott, Exponentially tapered josephson flux-flow oscillator, Phy. Rev. B, 54 (1996), 16139.

[3]

A. Benabdallah, J. G. Caputo and A. C. Scott, Laminar phase flow for an exponentially tapered josephson oscillator, J. Apl. Phys., 588 (2000), 3527.

[4]

S. Bondarenko and Nakagawa, SQUID-based magnetic microscope, in "Smart Materials for Ranking Systems," J. France et al (edition), Springer (2006), 195-201.

[5]

T. L. Boyadjiev, E. G. Semerdjieva and Yu. M. Shukrinov, Common features of vortex structure in long exponentially shaped Josephson junctions and Josephson junctions with inhomogeneities, Physica C, 460-462 (2007), 1317-1318.

[6]

G. Carapella, N. Martucciello and G. Costabile, Experimental investigation of flux motion in exponentially shaped Josephson junctions, PHYS REV B, 66 (2002), 134531.

[7]

J. Clarke, "SQUIDs for Everything," Nature Materials, 10 (2011).

[8]

J. Clarke, SQUIDs: Then and Now, chapter in BCS: 50 Years (eds. Leon N Cooper and Dmitri Feldman) World Scientific Publishing Co. Pte. Ltd., Singapore (2010).

[9]

S. A. Cybart et al., Dynes Series array of incommensurate superconducting quantum interference devices, Appl. Phys Lett, 93 (2008), 1-3.

[10]

A. D'Anna, M. De Angelis and G. Fiore, Existence and uniqueness for some 3rd order dissipative problems with various boundary conditions, Acta Appl. Math., 122 (2012), 255-267.

[11]

M. Dehghan and A. Shokri, A numerical method for solution of the two dimensional sine- Gordon equation using the radial basis functions, Mat Comp in Simulation, 79 (2008), 700-715.

[12]

M. De Angelis, On a model of superconductivity and biology, Advances and Applications in Mathematical Sciences, 7 (2010), 41-50.

[13]

M. De Angelis, Asymptotic analysis for the strip problem related to a parabolic third-order operator, Applied Mathematics Letters, 14 (2001), 425-430.

[14]

M. De Angelis, A priori estimates for excitable models, Meccanica (2013). doi: 10.1007/s11012-013-9763-2.

[15]

M. De Angelis, On exponentially shaped Josephson junctions, Acta appl. Math, 122 (2012), 179-189

[16]

M. De Angelis, On a parabolic operator of dissipative systems,, submitted to Acta appl. Math, (). 

[17]

M. De Angelis and G. Fiore, Existence and uniqueness of solutions of a class of third order dissipative problems with various boundary conditions describing the Josephson effect, J. Math. Anal. Appl., 404 (2013), 477-490. doi: 10.1016/j.jmaa.2013.03.029.

[18]

M. De Angelis, A. Maio and E. Mazziotti, Existence and uniqueness results for a class of non linear models, in "Mathematical Physics Models and Engineering Sciences" (ed. Liguori), Italy, (2008), 191-202.

[19]

M. De Angelis and E. Mazziotti, Non linear travelling waves with diffusion, Rend. Acc. Sc. Fis. Mat. Napoli, 73 (2006), 23-36.

[20]

De Angelis, A. M. Monte and P. M. Renno, On fast and slow times in models with diffusion, Math Models and Methods in Applied Sciences, 12 (2002), 1741-1749.

[21]

M. De Angelis and P. Renno, Asymptotic effects of boundary perturbations in excitable systems, (2013),, submitted to Discrete and Continuous Dynamical Systems - B. Available from: \url{http://arxiv.org/pdf/1304.3891v1.pdf}, (). 

[22]

M. De Angelis and P. Renno, Existence, uniqueness and a priori estimates for a non linear integro-differential equation, Ric Mat, 57 (2008), 95-109.

[23]

De Angelis and P. M. Renno, Diffusion and wave behaviour in linear Voigt model, C. R. Mecanique, 330 (2002), 21-26

[24]

Gutman S. Junhohg Ha, Identification problem for damped sine Gordon equation with point sources, J. Math. Anal. Appl., 375 (2011), 648-666. doi: 10.1016/j.jmaa.2010.10.006.

[25]

E. M. Izhikevich, "Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting," The MIT press, England, 2007.

[26]

M. Jaworski, "Fluxon Dynamics in Exponentially Shaped Josephson Junction," Phy. rev. B, 71 (2005), 1-6.

[27]

M. Jaworski, Exponentially tapered Josephson junction: some analytic results, Theor and Math Phys, 144 (2005), 1176-1180.

[28]

J. McCall and Lindsa, Superconductor cables: Advanced capabilities for the smart grid, Utility Automation Engineering TD, 13 (2008), 54.

[29]

J. D. Murray, "Mathematical Biology. I. An Introduction," Springer-Verlag, N.Y, 2002.

[30]

S. Rionero, Asymptotic behaviour of solutions to a nonlinear third order P.D.E modeling physical phenomena, Boll Unione Mat Ital, 9 (2012), 451-468.

[31]

H. Rogalla and P. H. Kes, "100 Years of Superconductivity," CRC Press, (2012).

[32]

A.C. Scott, "The Nonlinear Universe: Chaos, Emergence, Life," Springer-Verlag, 2007.

[33]

A.C. Scott, "Neuroscience A mathematical Primer," Springer-Verlag, 2002.

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