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August  2013, 33(8): 3567-3582. doi: 10.3934/dcds.2013.33.3567

Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations

Armengol Gasull 1, , Hector Giacomini 2, and  Joan Torregrosa 3,

1. 

Dept. de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona
2. 

Laboratoire de Mathématique et Physique Théorique, C.N.R.S. UMR 7350, Faculté des Sciences et Techniques, Université de Tours, Parc de Grandmont,37200 Tours
3. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Edifici C, 08193 Bellaterra, Barcelona, Spain

Received  July 2012 Revised  November 2012 Published  January 2013

It is well-known that the existence of traveling wave solutions for reaction-diffusion partial differential equations can be proved by showing the existence of certain heteroclinic orbits for related autonomous planar differential equations. We introduce a method for finding explicit upper and lower bounds of these heteroclinic orbits. In particular, for the classical Fisher-Kolmogorov equation we give rational upper and lower bounds which allow to locate these solutions analytically and with very high accuracy. These results allow one to construct analytical approximate expressions for the traveling wave solutions with a rigorous control of the errors for arbitrary values of the independent variables. These explicit expressions are very simple and tractable for practical purposes. They are constructed with exponential and rational functions.
Keywords: reaction-diffusion partial differential equation, invariant manifold., heteroclinic orbit, Fisher-Kolmogorov equation, Traveling wave.
Mathematics Subject Classification: 34C37, 35C07, 37C2.
Citation: Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
References:
[1]

M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), 835-840. doi: 10.1016/S0092-8240(79)80020-8.

[2]

D. G. Aronson and H. F. Weinberger, "Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation," Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975.

[3]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.

[4]

A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.

[5]

A. Goriely, A simple solution to the nonlinear front problem, Phys. Rev. Lett., 75 (1995), 2047-2050.

[6]

P. Grindrod, Patterns and Waves, "The Theory and Applications of Reaction-Diffusion Equations," Clarendon Press, 1991.

[7]

A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, in "Selected Works of A. N. Kolmogorov I" (editor, V. M. Tikhomirov), 248-270. Kluwer 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25.

[8]

A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303.

[9]

M. R. Rodrigo and R. M. Miura, Exact and approximate traveling waves of reaction-diffusion systems via a variational approach, Anal. Appl. (Singap.), 9 (2011), 187-199. doi: 10.1142/S0219530511001807.

[10]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.

[11]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol., 35 (1997), 713-728. doi: 10.1007/s002850050073.

[12]

L. A. Segel, Distant sidewalls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224.

[13]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[14]

Y. B. Zeldovich and D. A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physicochimica URSS 9 (1938), 341-350. English Translation: Dynamics of Curved Fronts, editor P. Pelcé, Perspectives in Physics, Academic Press, New York, (1988), 131-140.

show all references

References:
[1]

M. J. Ablowitz and A. Zeppetella, Explicit solutions of Fisher's equation for a special wave speed, Bull. Math. Biol., 41 (1979), 835-840. doi: 10.1016/S0092-8240(79)80020-8.

[2]

D. G. Aronson and H. F. Weinberger, "Nonlinear Diffusion in Population Genetics, Combustion, and Nerve Pulse Propagation," Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975.

[3]

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 355-369.

[4]

A. Gasull, H. Giacomini and J. Torregrosa, Some results on homoclinic and heteroclinic connections in planar systems, Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.

[5]

A. Goriely, A simple solution to the nonlinear front problem, Phys. Rev. Lett., 75 (1995), 2047-2050.

[6]

P. Grindrod, Patterns and Waves, "The Theory and Applications of Reaction-Diffusion Equations," Clarendon Press, 1991.

[7]

A. Kolmogorov, I. Petrovskii and N. Piskunov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, in "Selected Works of A. N. Kolmogorov I" (editor, V. M. Tikhomirov), 248-270. Kluwer 1991. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25.

[8]

A. C. Newell and J. A. Whitehead, Finite bandwidth, finite amplitude convection, J. Fluid Mech., 38 (1969), 279-303.

[9]

M. R. Rodrigo and R. M. Miura, Exact and approximate traveling waves of reaction-diffusion systems via a variational approach, Anal. Appl. (Singap.), 9 (2011), 187-199. doi: 10.1142/S0219530511001807.

[10]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in some degenerate reaction-diffusion equations, J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.

[11]

F. Sánchez-Garduño and P. K. Maini, Travelling wave phenomena in non-linear diffusion degenerate Nagumo equations, J. Math. Biol., 35 (1997), 713-728. doi: 10.1007/s002850050073.

[12]

L. A. Segel, Distant sidewalls cause slow amplitude modulation of cellular convection, J. Fluid Mech., 38 (1969), 203-224.

[13]

J. Xin, Front propagation in heterogeneous media, SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.

[14]

Y. B. Zeldovich and D. A. Frank-Kamenetskii, A theory of thermal propagation of flame, Acta Physicochimica URSS 9 (1938), 341-350. English Translation: Dynamics of Curved Fronts, editor P. Pelcé, Perspectives in Physics, Academic Press, New York, (1988), 131-140.

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