Growth, decay and bifurcation of shock amplitudes under the type-II flux law

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 463, Issue 2087, 2007, Pages 2783-2798

  Jordan, P M ^a  

^a NAVAL RESEARCH LABORATORY   (United States)

Author keywords

Fisher KPP equation; Green and Nagdhi's type II flux law; Shock and acceleration waves; Transcritical bifurcation; Travelling wave solution

Indexed keywords

EID: 36348970153     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2007.1895     Document Type: Article

References (45)

1. Aronson, D. G. & Weinberger, H. F. 1975 Nonlinear diffusion in population genetics, combustion, and nerve propagation. In Partial differential equations and related topics, vol. 446 (ed. J. A. Goldstein) Lecture Notes in Mathematics, pp. 5-49. Berlin, Germany: Springer.
2. Bland, D. R. 1988 Wave theory and applications. Oxford, UK: Oxford University Press.
3. Chandrasekharaiah, D. S. 1998 Hyperbolic thermoelasticity: a review of recent literature. Appl. Mech. Rev. 51, 705-729.
4. Chen, P. J. 1973 Growth and decay of waves in solids. In Handbuch der physik, vol. VIa/3 (eds S. Flügge & C. Truesdell), pp. 303-402. Berlin, Germany: Springer.
5. Christov, C. I. & Jordan, P. M. 2005 Heat conduction paradox involving second-sound propagation in moving media. Phys. Rev. Lett. 94, 154301. (doi:10.1103/PhysRevLett.94.154301)
6. Christov, I., Christov, C. I. & Jordan, P. M. 2007 Cumulative nonlinear effects in acoustic wave propagation. CMES: Comp. Model. Eng. Sci. 17, 47-54.
7. Debnath, L. 2004 Nonlinear partial differential equations for scientists and engineers, 2nd edn. Boston, MA: Birkhäuser.
8. Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. & Morris, H. C. 1982 Solitons and nonlinear wave equations, pp. 30-31. London, UK: Academic Press.
9. Dolak, Y. & Hillen, T. 2003 [Erratum: 'Cattaneo models for chemosensitive movement: numerical solution and pattern formation' J. Math. Biol. 46, 153-170 (2003)]. J. Math. Biol. 46, 461-478. (doi:10.1007/s00285-003-0221-y)
10. Dreyer, W. & Struchtrup, H. 1993 Heat pulse experiments revisited. Cont. Mech. Thermodyn. 5, 3-50. (doi:10.1007/BF01135371)
11. Ebert, U. & van Saarloos, W. 2000 Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. Physica D 146, 1-99. (doi:10.1016/S0167-2789(00)00068-3)
12. Fisher, R. A. 1937 The wave of advance of advantageous genes. Ann. Eugen. 7, 355-369.
13. Franchi, F. & Straughan, B. 1994 Continuous dependence on the relaxation time and modelling, and unbounded growth, in theories of heat conduction with finite propagation speeds. J. Math. Anal. Appl. 185, 726-746. (doi:10.1006/jmaa.1994.1279)
14. Fu, Y. B. & Scott, N. H. 1990 One-dimensional shock waves in simple materials with memory. Proc. R. Soc. A 428, 547-571. (doi:10.1098/rspa.1990.0047)
15. Fu, Y. B. & Scott, N. H. 1991 The transition from acceleration wave to shock wave. Int. J. Eng. Sci. 29, 617-624. (doi:10.1016/0020- 7225(91)90066-C)
16. Green, A. E. & Naghdi, P. M. 1991 A re-examination of the basic postulates of thermomechanics. Proc. R. Soc. A 432, 171-194. (doi:10.1098/rspa.1991.0012)
17. Green, A. E. & Naghdi, P. M. 1993 Thermoelasticity without energy dissipation. J. Elasticity 31, 189-208.
18. Gurtin, M. E. & Pipkin, A. C. 1968 A general theory of heat conduction with finite wave speeds. Arch. Rat. Mech. Anal. 31, 113-126.
19. Hale, J. & Koçak, H. 1991 Dynamics and bifurcations, ch. 1,2. New York, NY: Springer.
20. Harris, S. 2005 Diffusive logistic population growth with immigration. Appl. Math. Lett. 18, 261-265. (doi:10.1016/j.aml.2003.03.009)
21. Jaisaardsuetrong, J. & Straughan, B. 2007 Thermal waves in a rigid heat conductor. Phys. Lett. A 366, 433-436. (doi:10.1016/j. physleta.2007.02.058)
22. Jordan, P. M. 2005a Growth and decay of shock and acceleration waves in a traffic flow model with relaxation. Physica D 207, 220-229. (doi:10.1016/j.physd.2005.06.002)
23. Jordan, P. M. 2005b Growth and decay of acoustic acceleration waves in Darcy-type porous media. Proc. R. Soc. A 461, 2749-2766. (doi:10.1098/rspa.2005.1477)
24. Jordan, P. M. & Puri, A. 2002 Qualitative results for solutions of the steady Fisher-KPP equation. Appl. Math. Lett. 15, 239-250. (doi:10.1016/S0893-9659(01)00124-0)
25. Kay, A. L., Sherratt, J. A. & McLeod, J. B. 2001 Comparison theorems and variable speed waves in a scalar reaction-diffusion equation. Proc. R. Soc. Edin. A 131, 1133-1161.
26. Kolmogoroff, A., Petrovsky, I. & Piscounoff, N. 1937 Étude de l'équations de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. Bull. Univ. Moskou, Ser. Int., Sec. A 1, 1-25.
27. Lighthill, M. J. & Whitham, G. B. 1955 On kinematic waves II. A theory of traffic flow on long crowded roads. Proc. R. Soc. A 229, 317-345. (doi:10.1098/rspa.1955.0089)
28. Logan, J. D. 1994 An introduction to nonlinear partial differential equations, pp. 196-199. New York, NY: Wiley.
29. Méndez, V. & Camacho, J. 1997 Dynamics and thermodynamics of delayed population growth. Phys. Rev. E 55, 6476-6482. (doi:10.1103/PhysRevE.55.6476)
30. Mickens, R. E. & Jordan, P. M. 2004 A positivity-preserving nonstandard finite difference scheme for the DWE. Num. Meth. Partial Diff. Eqn 20, 639-649. (doi:10.1002/num.20003)
31. Mickens, R. E. & Jordan, P. M. 2005 A new positivity-preserving nonstandard finite difference scheme for the DWE. Num. Meth. Partial Diff. Eqn 21, 976-985. (doi:10.1002/num.20073)
32. Morro, A. 2006 Jump relations and discontinuity waves in conductors with memory. Math. Comp. Model. 43, 138-149. (doi:10.1016/j.mcm.2005. 04.016)
33. Müller, I. & Ruggeri, T. 1993 Extended thermodynamics. In Springer tracts in natural philosophy, vol. 37 (ed. C. Truesdell), New York, NY: Springer.
34. Murray, J. D. 1993 Mathematical biology, 2nd edn. Berlin, Germany: Springer.
35. Needham, D. J. & King, A. C. 2002 The evolution of travelling waves in the weakly hyperbolic generalized Fisher model. Proc. R. Soc. A 458, 1055-1088. (doi:10.1098/rspa.2001.0902)
36. Newman, W. I. 1980 Some exact solutions to a non-linear diffusion problem in population genetics and combustion. J. Theor. Biol. 85, 325-334. (doi:10.1016/0022-5193(80)90024-7)
37. Niwa, H. S. 1998 Migration dynamics of fish schools in heterothermal environments. J. Theor. Biol. 193, 215-231. (doi:10.1006/jtbi. 1998.0675)
38. Ockendon, H., Ockendon, J. R. & Falle, S. A. E. G. 2001 The Fanno model for turbulent compressible flow. J. Fluid Mech. 445, 187-206.
39. Ostoja-Starzewski, M. & Trȩbicki, J. 1999 On the growth and decay of acceleration waves in random media. Proc. R. Soc. A 455, 2577-2614. (doi:10.1098/rspa.1999.0446)
40. Pierce, A. D. 1989 Acoustics: an introduction to its physical principles and applications, pp. 589-591. New York, NY: Acoustical Society of America.
41. Quintanilla, R. & Straughan, B. 2004 A note on discontinuity waves in type III thermoelasticity. Proc. R. Soc. A 460, 1169-1175. (doi:10.1098/rspa.2003.1131)
42. Rosen, G. 1966 Solutions of a certain nonlinear wave equation. J. Math. Phys. 45, 235-265.
43. Sherratt, J. A. & Marchant, B. P. 1996 Algebraic decay and variable speeds in wavefront solutions of a scalar reaction-diffusion equation. IMA J. Appl. Math. 56, 289-302.
44. Whitham, G. B. 1974 Linear and nonlinear waves, pp. 26-30. New York, NY: Wiley.
45. Xin, J. 2000 Front propagation in heterogeneous media. SIAM Rev. 42, 161-230. (doi:10.1137/S0036144599364296)