September  2014, 3(3): 541-555. doi: 10.3934/eect.2014.3.541

Shocks and acceleration waves in modern continuum mechanics and in social systems

1. 

Department of Mathematical Sciences, University of Durham, Durham DH1 3LE

Received  May 2013 Revised  November 2013 Published  August 2014

The use of discontinuity surface propagation (e.g. shock waves and acceleration waves) is well known in modern continuum mechanics and yields a very useful means to obtain important information about a fully nonlinear theory with no approximation whatsoever. A brief review of some of the recent uses of such discontinuity surfaces is given and then we mention modelling of some social problems where the same mathematical techniques may be used to great effect. We specifically show how to develop and analyse models for evolution of one language overtaking use of another leading to possible extinction of the former language. Then we analyse shock transmission in a model for the evolutionary transition from the human period when hunter-gatherers transformed into farming. Finally we address modelling discontinuity waves in the context of diffusion of an innovation.
Citation: Brian Straughan. Shocks and acceleration waves in modern continuum mechanics and in social systems. Evolution Equations and Control Theory, 2014, 3 (3) : 541-555. doi: 10.3934/eect.2014.3.541
References:
[1]

K. Aoki, M. Shida and N. Shigesada, Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers, Theor. Popul. Biol., 50 (1996), 1-17. doi: 10.1006/tpbi.1996.0020.

[2]

E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction-diffusion model for the hantavirus infection, Math. Meth. Appl. Science, 31 (2002), 481-499. doi: 10.1002/mma.929.

[3]

N. Bellomo and A. Bellouquid, On the modelling of crowd dynamics, looking at the beautiful shapes of swarms, Netw. Heterog. Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[4]

N. Bellomo, B. Piccoli and A. Tosin, Modelling crowd dynamics from a complex system viewpoint, Math. Models Meth. Appl. Science, 22 (2012), 29 pp. doi: 10.1142/S0218202512300049.

[5]

J. J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Compartmental modelling of social dynamics with generalised peer incidence, Math. Models Meth. Appl. Science, 24 (2014), 719-750. doi: 10.1142/S0218202513500656.

[6]

J. J. Bissell and B. Straughan, Discontinuity waves as tipping points, Discrete and Continuous Dynamical Systems B, 19 (2014), 1911-1934.

[7]

C. I. Christov, On frame indifferent formulation of the Maxwell - Cattaneo model of finite - speed heat conduction, Mech. Res. Comm., 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003.

[8]

I. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/4/043027.

[9]

I. Christov and P. M. Jordan, Shock and traveling wave phenomena on an externally damped, non-linear string, Int. J. Nonlinear Mech., 44 (2009), 511-519. doi: 10.1016/j.ijnonlinmec.2008.12.004.

[10]

I. Christov and P. M. Jordan, On the propagation of second - sound in nonlinear media: shock, acceleration and traveling wave results, J. Thermal Stresses, 33 (2010), 1109-1135. doi: 10.1080/01495739.2010.517674.

[11]

I. Christov, P. M. Jordan and C. I. Christov, Nonlinear acoustic propagation in homentropic perfect gases: a numerical study, Phys. Lett. A, 353 (2006), 273-280. doi: 10.1016/j.physleta.2005.12.101.

[12]

I. Christov, P. M. Jordan and C. I. Christov, Modelling weakly nonlinear acoustic wave propagation, Quart. Jl Mech. Appl. Math., 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017.

[13]

C. Ciarcià, P. Falsaperla, A. Giacobbe and G. Mulone, A mathematical model of anorexia and bulimia, Manuscript, (2013).

[14]

M. Ciarletta, B. Straughan and V. Tibullo, Christov-Morro theory for non-isothermal diffusion, Nonlinear Anal. Real World Appl., 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014.

[15]

C. Currò, M. Sugiyama, H. Suzumura and G. Valenti, Weak shock waves in isotropic solids at finite temperatures up to the melting point, Continuum Mech. Thermodyn., 18 (2007), 395-409. doi: 10.1007/s00161-006-0033-6.

[16]

C. Currò, G. Valenti, M. Sugiyama and S. Taniguchi, Propagation of an acceleration wave in layers of isotropic solids at finite temperatures, Wave Motion, 46 (2009), 108-121. doi: 10.1016/j.wavemoti.2008.09.003.

[17]

M. Fabrizio, A Cahn-Hilliard model for social integration, in Modelling Social Problems and Health, (eds. J. J. Bissell, C. C. S. Caiado, S. E. Curtis, M. Goldstein and B. Straughan), Wiley, 2013.

[18]

M. Fabrizio, F. Franchi and B. Straughan, On a model for thermo-poroacoustic waves, Int. J. Engng. Sci., 46 (2008), 790-798. doi: 10.1016/j.ijengsci.2008.01.016.

[19]

M. Fabrizio and A. Morro, Electromagnetism of Continuous Media, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198527008.001.0001.

[20]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[21]

M. Gentile and B. Straughan, Acceleration waves in double porosity elasticity, Int. J. Engng. Sci., 73 (2013), 10-16. doi: 10.1016/j.ijengsci.2013.07.006.

[22]

M. Gentile and B. Straughan, Hyperbolic diffusion with Christov-Morro theory, Math. Comput. Simulat., (2013). doi: 10.1016/j.matcom.2012.07.010.

[23]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy-dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[24]

H. F. Huo and N. N. Song, Global stability for a binge drinking model with two stages, Discrete Dyn. Nat. Soc., 2012 (2012), 15 pages. doi: 10.1155/2012/829386.

[25]

N. Hussaini and M. Winter, Travelling waves for an epidemic model with non-smooth treatment rates, J. Stat. Mech., 2010 (2010), P11019. doi: 10.1088/1742-5468/2010/11/P11019.

[26]

P. M. Jordan, Growth and decay of acoustic acceleration waves in Darcy-type porous media, Proc. Roy. Soc. London A, 461 (2005), 2749-2766. doi: 10.1098/rspa.2005.1477.

[27]

P. M. Jordan, Growth and decay of shock and acceleration waves in a traffic flow model with relaxation, Phys. D, 207 (2005), 220-229. doi: 10.1016/j.physd.2005.06.002.

[28]

P. M. Jordan, Finite amplitude acoustic travelling waves in a fluid that saturates a porous medium: Acceleration wave formation, Phys. Lett. A, 355 (2006), 216-221. doi: 10.1016/j.physleta.2006.02.033.

[29]

P. M. Jordan, Growth, decay and bifurcation of shock amplitudes under the type-II flux law, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2783-2798. doi: 10.1098/rspa.2007.1895.

[30]

P. M. Jordan, On the growth and decay of transverse acceleration waves on a nonlinear, externally damped string, J. Sound and Vibration, 311 (2008), 597-607. doi: 10.1016/j.jsv.2007.09.024.

[31]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena, Math. Comput. Simulation, 80 (2009), 202-211. doi: 10.1016/j.matcom.2009.06.004.

[32]

P. M. Jordan, A note on poroacoustic travelling waves under Forchheimer's law, Phys. Lett. A, 377 (2013), 1350-1357. doi: 10.1016/j.physleta.2013.03.041.

[33]

P. M. Jordan and A. Puri, Qualitative results for solutions of the steady Fisher - KPP equation, Appl. Math. Lett., 15 (2002), 239-250. doi: 10.1016/S0893-9659(01)00124-0.

[34]

A. S. Kalula and F. Nyabadza, A theoretical model for substance abuse in the presence of treatment, S. Afr. J. Sci., 108 (2012), 96-107. doi: 10.4102/sajs.v108i3/4.654.

[35]

A. Kandler and J. Steele, Ecological models of language competition, Biological Theory, 3 (2008), 164-173. doi: 10.1162/biot.2008.3.2.164.

[36]

A. Kandler, R. Unger and J. Steele, Language shift, bilingualism and the future of Britain's Celtic languages, Phil. Trans. Royal Soc. London B, 365 (2010), 3855-3864. doi: 10.1098/rstb.2010.0051.

[37]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Étude de l'équations de la diffusion avec croissance de la quantité de matière et son application a un prolème biologique, Bull. Univ. Moskou, Ser. Int., Sec. A, 1 (1937), 1-25.

[38]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Letters, 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019.

[39]

A. Marasco, On the first-order speeds in any direction of acceleration waves in prestressed second - order isotropic, compressible and homogeneous materials, Math. Comput. Modelling, 49 (2009), 1644-1652. doi: 10.1016/j.mcm.2008.07.037.

[40]

A. Marasco, Second - order effects on the wave propagation in elastic, isotropic, incompressible and homogeneous media, Int. J. Engng. Sci., 47 (2009), 499-511. doi: 10.1016/j.ijengsci.2008.08.009.

[41]

A. Marasco and A. Romano, On the acceleration waves in second - order elastic, isotropic, compressible and homogeneous materials, Math. Comput. Modelling, 49 (2009), 1504-1518. doi: 10.1016/j.mcm.2008.06.005.

[42]

A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comput. Modelling, 52 (2010), 1869-1876. doi: 10.1016/j.mcm.2010.07.021.

[43]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006.

[44]

G. Mulone and B. Straughan, Modelling binge drinking, International Journal of Biomathematics, 5 (2012), 14 pp. doi: 10.1142/S1793524511001453.

[45]

F. Nyabadza, S. Mukwembi and B.G. Rodrigues, A graph theoretical perspective of a drug abuse epidemic model, Physica A - Statistical Methods and Applications, 390 (2011), 1723-1732. doi: 10.1016/j.physa.2011.01.014.

[46]

F. Nyabadza, J. B. H. Njagarah and R. J. Smith, Modelling the dynamics of crystal meth ('tik') abuse in the presence of drug-supply chains in South Africa, Bull. Math. Biol., 75 (2013), 24-48. doi: 10.1007/s11538-012-9790-5.

[47]

P. Paoletti, Acceleration waves in complex materials, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 637-659. doi: 10.3934/dcdsb.2012.17.637.

[48]

T. Ruggeri and M. Sugiyama, Hyperbolicity, convexity and shock waves in one-dimensional crystalline solids, J. Phys. A, 38 (2005), 4337-4347. doi: 10.1088/0305-4470/38/20/003.

[49]

V. D. Sharma and R. Venkatramani, Evolution of weak shocks in one dimensional planar and non - planar gas dynamics, Int. J. Non-linear Mechanics, 47 (2012), 918-926. doi: 0.1186/2251-7235-7-14.

[50]

B. Straughan, Stability and Wave motion in Porous Media, volume 165, Appl. Math. Sci., Springer, New York, 2008.

[51]

B. Straughan, Acoustic waves in a Cattaneo-Christov gas, Phys. Lett. A, 374 (2010), 2667-2669. doi: 10.1016/j.physleta.2010.04.054.

[52]

B. Straughan, Heat Waves, volume 177, Appl. Math. Sci., Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.

[53]

B. Straughan, Gene-culture shock waves, Phys. Lett. A, 377 (2013), 2531-2534. doi: 10.1016/j.physleta.2013.07.025.

[54]

B. Straughan, Stability and uniqueness in double porosity elasticity, Int. J. Engng. Sci., 65 (2013), 1-8. doi: 10.1016/j.ijengsci.2013.01.001.

[55]

G. Valenti, C. Curro and M. Sugiyama, Acceleration waves analysed by a new continuum model of solids incorporating microscopic thermal vibrations, Continuum Mech. Thermodyn., 16 (2004), 185-198. doi: 10.1007/s00161-003-0150-4.

[56]

C. E. Walters, B. Straughan and J. Kendal, Modelling alcohol problems: Total recovery, Ric. Mat., 62 (2013), 1-18. doi: 10.1007/s11587-012-0138-0.

[57]

W. Wang, P. Fergola, S. Lombardo and G. Mulone, Mathematical models of innovation diffusion with stage structure, Appl. Math. Model., 30 (2006), 129-146. doi: 10.1016/j.apm.2005.03.011.

show all references

References:
[1]

K. Aoki, M. Shida and N. Shigesada, Travelling wave solutions for the spread of farmers into a region occupied by hunter-gatherers, Theor. Popul. Biol., 50 (1996), 1-17. doi: 10.1006/tpbi.1996.0020.

[2]

E. Barbera, C. Currò and G. Valenti, A hyperbolic reaction-diffusion model for the hantavirus infection, Math. Meth. Appl. Science, 31 (2002), 481-499. doi: 10.1002/mma.929.

[3]

N. Bellomo and A. Bellouquid, On the modelling of crowd dynamics, looking at the beautiful shapes of swarms, Netw. Heterog. Media, 6 (2011), 383-399. doi: 10.3934/nhm.2011.6.383.

[4]

N. Bellomo, B. Piccoli and A. Tosin, Modelling crowd dynamics from a complex system viewpoint, Math. Models Meth. Appl. Science, 22 (2012), 29 pp. doi: 10.1142/S0218202512300049.

[5]

J. J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Compartmental modelling of social dynamics with generalised peer incidence, Math. Models Meth. Appl. Science, 24 (2014), 719-750. doi: 10.1142/S0218202513500656.

[6]

J. J. Bissell and B. Straughan, Discontinuity waves as tipping points, Discrete and Continuous Dynamical Systems B, 19 (2014), 1911-1934.

[7]

C. I. Christov, On frame indifferent formulation of the Maxwell - Cattaneo model of finite - speed heat conduction, Mech. Res. Comm., 36 (2009), 481-486. doi: 10.1016/j.mechrescom.2008.11.003.

[8]

I. Christov and P. M. Jordan, Shock bifurcation and emergence of diffusive solitons in a nonlinear wave equation with relaxation, New J. Phys., 10 (2008). doi: 10.1088/1367-2630/10/4/043027.

[9]

I. Christov and P. M. Jordan, Shock and traveling wave phenomena on an externally damped, non-linear string, Int. J. Nonlinear Mech., 44 (2009), 511-519. doi: 10.1016/j.ijnonlinmec.2008.12.004.

[10]

I. Christov and P. M. Jordan, On the propagation of second - sound in nonlinear media: shock, acceleration and traveling wave results, J. Thermal Stresses, 33 (2010), 1109-1135. doi: 10.1080/01495739.2010.517674.

[11]

I. Christov, P. M. Jordan and C. I. Christov, Nonlinear acoustic propagation in homentropic perfect gases: a numerical study, Phys. Lett. A, 353 (2006), 273-280. doi: 10.1016/j.physleta.2005.12.101.

[12]

I. Christov, P. M. Jordan and C. I. Christov, Modelling weakly nonlinear acoustic wave propagation, Quart. Jl Mech. Appl. Math., 60 (2007), 473-495. doi: 10.1093/qjmam/hbm017.

[13]

C. Ciarcià, P. Falsaperla, A. Giacobbe and G. Mulone, A mathematical model of anorexia and bulimia, Manuscript, (2013).

[14]

M. Ciarletta, B. Straughan and V. Tibullo, Christov-Morro theory for non-isothermal diffusion, Nonlinear Anal. Real World Appl., 13 (2012), 1224-1228. doi: 10.1016/j.nonrwa.2011.10.014.

[15]

C. Currò, M. Sugiyama, H. Suzumura and G. Valenti, Weak shock waves in isotropic solids at finite temperatures up to the melting point, Continuum Mech. Thermodyn., 18 (2007), 395-409. doi: 10.1007/s00161-006-0033-6.

[16]

C. Currò, G. Valenti, M. Sugiyama and S. Taniguchi, Propagation of an acceleration wave in layers of isotropic solids at finite temperatures, Wave Motion, 46 (2009), 108-121. doi: 10.1016/j.wavemoti.2008.09.003.

[17]

M. Fabrizio, A Cahn-Hilliard model for social integration, in Modelling Social Problems and Health, (eds. J. J. Bissell, C. C. S. Caiado, S. E. Curtis, M. Goldstein and B. Straughan), Wiley, 2013.

[18]

M. Fabrizio, F. Franchi and B. Straughan, On a model for thermo-poroacoustic waves, Int. J. Engng. Sci., 46 (2008), 790-798. doi: 10.1016/j.ijengsci.2008.01.016.

[19]

M. Fabrizio and A. Morro, Electromagnetism of Continuous Media, Oxford University Press, Oxford, 2003. doi: 10.1093/acprof:oso/9780198527008.001.0001.

[20]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen., 7 (1937), 355-369. doi: 10.1111/j.1469-1809.1937.tb02153.x.

[21]

M. Gentile and B. Straughan, Acceleration waves in double porosity elasticity, Int. J. Engng. Sci., 73 (2013), 10-16. doi: 10.1016/j.ijengsci.2013.07.006.

[22]

M. Gentile and B. Straughan, Hyperbolic diffusion with Christov-Morro theory, Math. Comput. Simulat., (2013). doi: 10.1016/j.matcom.2012.07.010.

[23]

A. E. Green and P. M. Naghdi, Thermoelasticity without energy-dissipation, J. Elasticity, 31 (1993), 189-208. doi: 10.1007/BF00044969.

[24]

H. F. Huo and N. N. Song, Global stability for a binge drinking model with two stages, Discrete Dyn. Nat. Soc., 2012 (2012), 15 pages. doi: 10.1155/2012/829386.

[25]

N. Hussaini and M. Winter, Travelling waves for an epidemic model with non-smooth treatment rates, J. Stat. Mech., 2010 (2010), P11019. doi: 10.1088/1742-5468/2010/11/P11019.

[26]

P. M. Jordan, Growth and decay of acoustic acceleration waves in Darcy-type porous media, Proc. Roy. Soc. London A, 461 (2005), 2749-2766. doi: 10.1098/rspa.2005.1477.

[27]

P. M. Jordan, Growth and decay of shock and acceleration waves in a traffic flow model with relaxation, Phys. D, 207 (2005), 220-229. doi: 10.1016/j.physd.2005.06.002.

[28]

P. M. Jordan, Finite amplitude acoustic travelling waves in a fluid that saturates a porous medium: Acceleration wave formation, Phys. Lett. A, 355 (2006), 216-221. doi: 10.1016/j.physleta.2006.02.033.

[29]

P. M. Jordan, Growth, decay and bifurcation of shock amplitudes under the type-II flux law, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2783-2798. doi: 10.1098/rspa.2007.1895.

[30]

P. M. Jordan, On the growth and decay of transverse acceleration waves on a nonlinear, externally damped string, J. Sound and Vibration, 311 (2008), 597-607. doi: 10.1016/j.jsv.2007.09.024.

[31]

P. M. Jordan, Some remarks on nonlinear poroacoustic phenomena, Math. Comput. Simulation, 80 (2009), 202-211. doi: 10.1016/j.matcom.2009.06.004.

[32]

P. M. Jordan, A note on poroacoustic travelling waves under Forchheimer's law, Phys. Lett. A, 377 (2013), 1350-1357. doi: 10.1016/j.physleta.2013.03.041.

[33]

P. M. Jordan and A. Puri, Qualitative results for solutions of the steady Fisher - KPP equation, Appl. Math. Lett., 15 (2002), 239-250. doi: 10.1016/S0893-9659(01)00124-0.

[34]

A. S. Kalula and F. Nyabadza, A theoretical model for substance abuse in the presence of treatment, S. Afr. J. Sci., 108 (2012), 96-107. doi: 10.4102/sajs.v108i3/4.654.

[35]

A. Kandler and J. Steele, Ecological models of language competition, Biological Theory, 3 (2008), 164-173. doi: 10.1162/biot.2008.3.2.164.

[36]

A. Kandler, R. Unger and J. Steele, Language shift, bilingualism and the future of Britain's Celtic languages, Phil. Trans. Royal Soc. London B, 365 (2010), 3855-3864. doi: 10.1098/rstb.2010.0051.

[37]

A. Kolmogoroff, I. Petrovsky and N. Piscounoff, Étude de l'équations de la diffusion avec croissance de la quantité de matière et son application a un prolème biologique, Bull. Univ. Moskou, Ser. Int., Sec. A, 1 (1937), 1-25.

[38]

J. Liu and T. Zhang, Global behaviour of a heroin epidemic model with distributed delays, Appl. Math. Letters, 24 (2011), 1685-1692. doi: 10.1016/j.aml.2011.04.019.

[39]

A. Marasco, On the first-order speeds in any direction of acceleration waves in prestressed second - order isotropic, compressible and homogeneous materials, Math. Comput. Modelling, 49 (2009), 1644-1652. doi: 10.1016/j.mcm.2008.07.037.

[40]

A. Marasco, Second - order effects on the wave propagation in elastic, isotropic, incompressible and homogeneous media, Int. J. Engng. Sci., 47 (2009), 499-511. doi: 10.1016/j.ijengsci.2008.08.009.

[41]

A. Marasco and A. Romano, On the acceleration waves in second - order elastic, isotropic, compressible and homogeneous materials, Math. Comput. Modelling, 49 (2009), 1504-1518. doi: 10.1016/j.mcm.2008.06.005.

[42]

A. Morro, Evolution equations and thermodynamic restrictions for dissipative solids, Math. Comput. Modelling, 52 (2010), 1869-1876. doi: 10.1016/j.mcm.2010.07.021.

[43]

G. Mulone and B. Straughan, A note on heroin epidemics, Math. Biosci., 218 (2009), 138-141. doi: 10.1016/j.mbs.2009.01.006.

[44]

G. Mulone and B. Straughan, Modelling binge drinking, International Journal of Biomathematics, 5 (2012), 14 pp. doi: 10.1142/S1793524511001453.

[45]

F. Nyabadza, S. Mukwembi and B.G. Rodrigues, A graph theoretical perspective of a drug abuse epidemic model, Physica A - Statistical Methods and Applications, 390 (2011), 1723-1732. doi: 10.1016/j.physa.2011.01.014.

[46]

F. Nyabadza, J. B. H. Njagarah and R. J. Smith, Modelling the dynamics of crystal meth ('tik') abuse in the presence of drug-supply chains in South Africa, Bull. Math. Biol., 75 (2013), 24-48. doi: 10.1007/s11538-012-9790-5.

[47]

P. Paoletti, Acceleration waves in complex materials, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 637-659. doi: 10.3934/dcdsb.2012.17.637.

[48]

T. Ruggeri and M. Sugiyama, Hyperbolicity, convexity and shock waves in one-dimensional crystalline solids, J. Phys. A, 38 (2005), 4337-4347. doi: 10.1088/0305-4470/38/20/003.

[49]

V. D. Sharma and R. Venkatramani, Evolution of weak shocks in one dimensional planar and non - planar gas dynamics, Int. J. Non-linear Mechanics, 47 (2012), 918-926. doi: 0.1186/2251-7235-7-14.

[50]

B. Straughan, Stability and Wave motion in Porous Media, volume 165, Appl. Math. Sci., Springer, New York, 2008.

[51]

B. Straughan, Acoustic waves in a Cattaneo-Christov gas, Phys. Lett. A, 374 (2010), 2667-2669. doi: 10.1016/j.physleta.2010.04.054.

[52]

B. Straughan, Heat Waves, volume 177, Appl. Math. Sci., Springer, New York, 2011. doi: 10.1007/978-1-4614-0493-4.

[53]

B. Straughan, Gene-culture shock waves, Phys. Lett. A, 377 (2013), 2531-2534. doi: 10.1016/j.physleta.2013.07.025.

[54]

B. Straughan, Stability and uniqueness in double porosity elasticity, Int. J. Engng. Sci., 65 (2013), 1-8. doi: 10.1016/j.ijengsci.2013.01.001.

[55]

G. Valenti, C. Curro and M. Sugiyama, Acceleration waves analysed by a new continuum model of solids incorporating microscopic thermal vibrations, Continuum Mech. Thermodyn., 16 (2004), 185-198. doi: 10.1007/s00161-003-0150-4.

[56]

C. E. Walters, B. Straughan and J. Kendal, Modelling alcohol problems: Total recovery, Ric. Mat., 62 (2013), 1-18. doi: 10.1007/s11587-012-0138-0.

[57]

W. Wang, P. Fergola, S. Lombardo and G. Mulone, Mathematical models of innovation diffusion with stage structure, Appl. Math. Model., 30 (2006), 129-146. doi: 10.1016/j.apm.2005.03.011.

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