Nonlinear Diffusion–Reaction Phenomena

  • Lokenath Debnath


Many physical phenomena are described by the interaction of convection and diffusion and also by the interaction of diffusion and reaction. From a physical point of view, the convection–diffusion process and the diffusion–reaction process are quite fundamental in describing a wide variety of problems in physical, chemical, biological, and engineering sciences. Some nonlinear partial differential equations that model these processes provide many new insights into the question of interaction of nonlinearity and diffusion. It is well known that the Burgers equation is a simple nonlinear model equation representing phenomena described by a balance between convection and diffusion. The Fisher equation is another simple nonlinear model equation which arises in a wide variety of problems involving diffusion and reaction.


Shock Wave Diffusion Equation Travel Wave Solution Burger Equation Wave Profile 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Abdelkader, M.A. (1982). Travelling wave solutions for a generalized Fisher equation, J. Math. Anal. Appl. 85, 287–290. MathSciNetCrossRefGoogle Scholar
  2. Abramowitz, M. and Stegun, I.A. (1972). Handbook of Mathematical Functions, Dover, New York. MATHGoogle Scholar
  3. Ablowitz, M.J. and Zeppetella, A. (1979). Explicit solutions of Fisher’s equation for a special wave speed, Bull. Math. Biol. 41, 835–840. MathSciNetMATHGoogle Scholar
  4. Ammerman, A.J. and Cavalli-Sforva, L.L. (1971). Measuring the rate of spread of early farming, Man 6, 674–688. CrossRefGoogle Scholar
  5. Ammerman, A.J. and Cavalli-Sforva, L.L. (1983). The Neolithic Transition and the Genetics of Populations in Europe, Princeton University Press, Princeton. Google Scholar
  6. Aoki, K. (1987). Gene-culture waves of advance, J. Math. Biol. 25, 453–464. MathSciNetMATHCrossRefGoogle Scholar
  7. Arnold, R., Showalter, K., and Tyson, J.J. (1987). Propagation of chemical reactions in space, J. Chem. Educ. 64, 740–742. Google Scholar
  8. Aronson, D.G. (1980). Density-dependent interaction–diffusion systems, in Dynamics and Modelling of Reactive Systems (eds. W.E. Stewart, W.H. Ray, and C.C. Conley). Academic Press, Boston, 161–176. Google Scholar
  9. Barenblatt, G.I. (1979). Similarity, Self-similarity and Intermediate Asymptotics, Consultants Bureau, New York. MATHGoogle Scholar
  10. Barenblatt, G.I. and Zel’dovich, Y.B. (1972). Self-similar solutions as intermediate asymptotics, Annu. Rev. Fluid Mech. 4, 285–312. CrossRefGoogle Scholar
  11. Benton, E.R. and Platzman, G.W. (1972). A table of solutions of one-dimensional Burgers equation, Q. Appl. Math. 30, 195–212. MathSciNetMATHGoogle Scholar
  12. Birkhoff, G. (1950). Hydrodynamics, Princeton University Press,Princeton. MATHGoogle Scholar
  13. Blackstock, D.T. (1964). Thermoviscous attenuation of plane, periodic finite-amplitude sound waves, J. Acoust. Soc. Am. 36, 534–542. CrossRefGoogle Scholar
  14. Bramson, M. (1983). Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Am. Math. Soc. 285, 1–190. MathSciNetGoogle Scholar
  15. Britton, N.F. (1986). Reaction-Diffusion Equations and Their Applications to Biology, Academic Press, New York. MATHGoogle Scholar
  16. Burgers, J.M. (1948). A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1, 171–199. MathSciNetCrossRefGoogle Scholar
  17. Burgers, J.M. (1974). The Nonlinear Diffusion Equation, Reidel, Dordrecht. MATHGoogle Scholar
  18. Canosa, J. (1969). Diffusion in nonlinear multiplicative media, J. Math. Phys. 10, 1862–1868. CrossRefGoogle Scholar
  19. Canosa, J. (1973). On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17, 307–313. MathSciNetMATHCrossRefGoogle Scholar
  20. Case, K.M. and Chiu, C.S. (1969). Burgers’ turbulence models, Phys. Fluids 12, 1799–1808. MathSciNetMATHCrossRefGoogle Scholar
  21. Cole, J.D. (1951). On a quasilinear parabolic equation occurring in aerodynamics, Q. Appl. Math. 9, 225–236. MATHGoogle Scholar
  22. Crank, J. (1975). The Mathematics of Diffusion, 2nd edition, Oxford University Press, Oxford. Google Scholar
  23. Crighton, D.G. (1979). Model equations of nonlinear acoustics, Annu. Rev. Fluid Mech. 11, 11–23. CrossRefGoogle Scholar
  24. Crighton, D.G. and Scott, J.F. (1979). Asymptotic solution of model equations in nonlinear acoustics, Philos. Trans. R. Soc. Lond. A292, 101–134. MathSciNetGoogle Scholar
  25. Dunbar, S.R. (1983). Travelling wave solutions of diffusive Lotka–Volterra equations, J. Math. Biol. 17, 11–32. MathSciNetMATHCrossRefGoogle Scholar
  26. Fife, P.C. (1979). Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomathematics, Vol. 28, Springer, Berlin. MATHGoogle Scholar
  27. Fisher, R.A. (1936). The wave of advance of advantageous genes, Ann. Eugen. 7, 335–369. Google Scholar
  28. Gazdag, J. and Canosa, J. (1974). Numerical solution of Fisher’s equation, J. Appl. Probab. 11, 445–457. MathSciNetMATHCrossRefGoogle Scholar
  29. Ghez, R. (1988). A Primer of Diffusion Problems, Wiley, New York. CrossRefGoogle Scholar
  30. Grindrod, P. (1991). Patterns and Waves, Oxford University Press, Oxford. MATHGoogle Scholar
  31. Gurney, W.S.C. and Nisbet, R.M. (1975). The regulation of inhomogeneous populations, J. Theor. Biol. 52, 441–457. CrossRefGoogle Scholar
  32. Gurtin, M.E. and MacCamy, R.C. (1977). On the diffusion of biological populations, Math. Biosci. 33, 35–49. MathSciNetMATHCrossRefGoogle Scholar
  33. Hagstrom, T. and Keller, H.B. (1986). The numerical calculations of traveling wave solutions of nonlinear parabolic equations (Preprint) Google Scholar
  34. Hopf, E. (1950). The partial differential equation u t+uu x=μu xx, Commun. Pure Appl. Math. 3, 201–230. MathSciNetMATHCrossRefGoogle Scholar
  35. Hoppensteadt, F.C. (1975). Mathematical Theories of Populations: Demographics, Genetics, and Epidemic, CBMS Lectures, Vol. 20, SIAM, Philadelphia. Google Scholar
  36. Johnson, R.S. (1970). A nonlinear equation incorporating damping and dispersion, J. Fluid Mech. 42, 49–60. MathSciNetMATHCrossRefGoogle Scholar
  37. Jones, D.S. and Sleeman, B.D. (1983). Differential Equations and Mathematical Biology, Allen and Unwin, London. MATHGoogle Scholar
  38. Kaliappan, P. (1984). An exact solution for travelling waves, Physica D11, 368–374. MathSciNetGoogle Scholar
  39. Kametaka, Y. (1976). On the nonlinear diffusion equation of Kolmogorov–Petrovskii–Piskunov type, Osaka J. Math. 13, 11–66. MathSciNetMATHGoogle Scholar
  40. Kolmogorov, A., Petrovsky, I., and Piscunov, N. (1937). A study of the equation of diffusion with increase in the quantity of matter and its application to a biological problem, Bull. Univ. Moscow, Ser. Int. Sec. A1, 1–25. Google Scholar
  41. Kopell, N. and Howard, L.N. (1973). Plane wave solutions to reaction–diffusion equations, Stud. Appl. Math. 42, 291–328. MathSciNetGoogle Scholar
  42. Kriess, H.O. and Lorenz, J. (1989). Initial-Boundary Value Problems and the Navier–Stokes Equations, Academic Press, New York. Google Scholar
  43. Lardner, R.W. (1986). Third order solutions of Burgers equation, Q. Appl. Math. 44, 293–302. MathSciNetMATHGoogle Scholar
  44. Larson, D.A. (1978). Transient bounds and time asymptotic behaviour of solutions, SIAM J. Appl. Math. 34, 93–103. MathSciNetMATHCrossRefGoogle Scholar
  45. Lighthill, M.J. (1956). Viscosity effects in sound waves of finite amplitude, in Surveys in Mechanics (eds. G.K. Batchelor and R.M. Davies), Cambridge University Press, Cambridge, 250–351. Google Scholar
  46. Logan, J.D. and Dunbar, S.R. (1992). Travelling waves in model reacting flows with reversible kinetics, IMA J. Appl. Math. 49, 103–121. MathSciNetMATHCrossRefGoogle Scholar
  47. Logan, J.D. and Shores, T.S. (1993a). Steady state solutions in a model reacting flow problem, Appl. Anal. 48, 273–286. MathSciNetMATHCrossRefGoogle Scholar
  48. Logan, J.D. and Shores, T.S. (1993b). On a system of nonlinear hyperbolic conservation laws with sources, Math. Models Methods Appl. Sci. 3, 341–358. MathSciNetMATHCrossRefGoogle Scholar
  49. Manoranjan, V.S. and Mitchell, A.R. (1983). A numerical study of the Belousov–Zhabotinskii reaction using Galerkin finite element methods, J. Math. Biol. 16, 251–260. MathSciNetMATHCrossRefGoogle Scholar
  50. McKean, H.P. (1975). Application of Brownian motion to the equations of Kolmogorov–Pertovskii–Piskunov, Commun. Pure Appl. Math. 28, 323–331. MathSciNetMATHCrossRefGoogle Scholar
  51. Montroll, E.W. and West, B.J. (1973). Models of population growth, diffusion, competition and rearrangement, in Synergetic (ed. H. Haken), B.G. Teubner, Stuttgart, 143–156. Google Scholar
  52. Munier, A., Burgen, J.R., Gutierrez, J., Fijalkow, E., and Feix, M.R. (1981). Group transformations and the nonlinear heat diffusion equation, SIAM J. Appl. Math. 40, 191–207. MathSciNetMATHCrossRefGoogle Scholar
  53. Murray, J.D. (1970a). Perturbation effects on the decay of discontinuous solutions of nonlinear first order wave equations, SIAM J. Appl. Math. 19, 273–298. MathSciNetMATHCrossRefGoogle Scholar
  54. Murray, J.D. (1970b). On the Gunn-effect and other physical examples of perturbed conservation equations, J. Fluid Mech. 44, 315–346. CrossRefGoogle Scholar
  55. Murray, J.D. (1973). On Burgers’ model equation for turbulence, J. Fluid Mech. 59, 263–279. MATHCrossRefGoogle Scholar
  56. Murray, J.D. (1993). Mathematical Biology, 2nd corrected edition, Springer, Berlin. MATHCrossRefGoogle Scholar
  57. Naumkin, P.I. and Shishmarev, I.A. (1994). Nonlinear Nonlocal Equations in the Theory of Waves, Vol. 133, Am. Math. Soc., Providence. MATHGoogle Scholar
  58. Newman, W.I. (1980). Some exact solutions to a nonlinear diffusion problem in population genetics and combustion, J. Theor. Biol. 85, 325–334. CrossRefGoogle Scholar
  59. Newman, W.I. (1983). Nonlinear diffusion: Self-similarity and travelling waves, Pure Appl. Geophys. 121, 417–441. CrossRefGoogle Scholar
  60. Okubo, A. (1980). Diffusion and Ecological Problems: Mathematical Models, Springer, Berlin. MATHGoogle Scholar
  61. Parker, D.F. (1980). The decay of saw-tooth solutions to the Burgers equation, Proc. R. Soc. Lond. A369, 409–424. Google Scholar
  62. Pattle, R.E. (1959). Diffusion from an instantaneous point source with a concentration-dependent coefficient, Q. J. Mech. Appl. Math. 12, 407–409. MathSciNetMATHCrossRefGoogle Scholar
  63. Penel, P. and Brauner, C.M. (1974). Identification of parameters in a nonlinear selfconsistent system including a Burgers equation, J. Math. Anal. Appl. 45, 654–681. MathSciNetMATHCrossRefGoogle Scholar
  64. Riemann, B., (1858). Uber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite, Göttingen Abhandlunger, Vol. viii, p. 43 (Werke, 2te Aufl., Leipzig, 1892, p. 157). Google Scholar
  65. Rodin, E.Y. (1970). On some approximate and exact solutions of boundary value problems for Burgers equations, J. Math. Anal. Appl. 30, 401–414. MathSciNetMATHCrossRefGoogle Scholar
  66. Rosenau, P. (1982). A nonlinear thermal wave in a reacting medium, Physica 5D, 136–144. Google Scholar
  67. Shigesada, N. (1980). Spatial distribution of dispersing animals, J. Math. Biol. 9, 85–96. MathSciNetMATHCrossRefGoogle Scholar
  68. Smoller, J. (1994). Shock Waves and Reaction–Diffusion Equations, 2nd edition, Springer, New York. MATHGoogle Scholar
  69. Sparrow, E.M., Quack, H., and Boerner, C.J. (1970). Local nonsimilarity boundary-layer solutions, AIAA J. 8, 1342–1350. CrossRefGoogle Scholar
  70. Tang, S. and Webber, R.O. (1991). Numerical study of Fisher’s equation by a Petrov–Galerkin finite element method, J. Aust. Math. Soc. B33, 27–38. Google Scholar
  71. Walsh, R.A. (1969). Initial-value problems associated with u t(x,t)=δu xx(x,t)−u(x,t)u x(x,t), J. Math. Anal. Appl. 26, 235–247. MathSciNetMATHCrossRefGoogle Scholar
  72. Whitham, G.B. (1974). Linear and Nonlinear Waves, Wiley, New York. MATHGoogle Scholar
  73. Zel’dovich, Ya.B. and Raizer, Yu.P. (1966). Physics of Shock Waves and High Temperature Hydrodynamic Phenomena, Academic Press, New York. Google Scholar
  74. Zel’dovich, Ya.B. and Raizer, Yu.P. (1968). Elements of Gas Dynamics and the Classical Theory of Shock Waves, Academic Press, New York. Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas, Pan AmericanEdinburgUSA

Personalised recommendations