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

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.
Citation: Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of Fisher-Kolmogorov type equations. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3567-3582. doi: 10.3934/dcds.2013.33.3567
References:
[1]

Bull. Math. Biol., 41 (1979), 835-840. doi: 10.1016/S0092-8240(79)80020-8.  Google Scholar

[2]

Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975.  Google Scholar

[3]

Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[4]

Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.  Google Scholar

[5]

Phys. Rev. Lett., 75 (1995), 2047-2050. Google Scholar

[6]

Clarendon Press, 1991.  Google Scholar

[7]

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. Google Scholar

[8]

J. Fluid Mech., 38 (1969), 279-303. Google Scholar

[9]

Anal. Appl. (Singap.), 9 (2011), 187-199. doi: 10.1142/S0219530511001807.  Google Scholar

[10]

J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.  Google Scholar

[11]

J. Math. Biol., 35 (1997), 713-728. doi: 10.1007/s002850050073.  Google Scholar

[12]

J. Fluid Mech., 38 (1969), 203-224. Google Scholar

[13]

SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[14]

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. Google Scholar

show all references

References:
[1]

Bull. Math. Biol., 41 (1979), 835-840. doi: 10.1016/S0092-8240(79)80020-8.  Google Scholar

[2]

Partial Differential Equations and Related Topics (Program, Tulane Univ., New Orleans, La., 1974), 5-49. Lecture Notes in Math., 446, Springer, Berlin, 1975.  Google Scholar

[3]

Ann. Eugenics, 7 (1937), 355-369. Google Scholar

[4]

Nonlinearity, 23 (2010), 2977-3001. doi: 10.1088/0951-7715/23/12/001.  Google Scholar

[5]

Phys. Rev. Lett., 75 (1995), 2047-2050. Google Scholar

[6]

Clarendon Press, 1991.  Google Scholar

[7]

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. Google Scholar

[8]

J. Fluid Mech., 38 (1969), 279-303. Google Scholar

[9]

Anal. Appl. (Singap.), 9 (2011), 187-199. doi: 10.1142/S0219530511001807.  Google Scholar

[10]

J. Differential Equations, 117 (1995), 281-319. doi: 10.1006/jdeq.1995.1055.  Google Scholar

[11]

J. Math. Biol., 35 (1997), 713-728. doi: 10.1007/s002850050073.  Google Scholar

[12]

J. Fluid Mech., 38 (1969), 203-224. Google Scholar

[13]

SIAM Rev., 42 (2000), 161-230. doi: 10.1137/S0036144599364296.  Google Scholar

[14]

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. Google Scholar

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