×

Waves interaction in the Fisher-Kolmogorov equation with arguments deviation. (English) Zbl 1486.65145

Summary: We considered the process of density wave propagation in the logistic equation with diffusion, such as Fisher-Kolmogorov equation, and arguments deviation. Firstly, we studied local properties of solutions corresponding to the considered equation with periodic boundary conditions using asymptotic methods. It was shown that increasing of period makes the spatial structure of stable solutions more complicated. Secondly, we performed numerical analysis. In particular, we considered the problem of propagating density waves interaction in infinite interval. Numerical analysis of the propagating waves interaction process, described by this equation, was performed at the computing cluster of YarSU with the usage of the parallel computing technology – OpenMP. Computations showed that a complex spatially inhomogeneous structure occurring in the interaction of waves can be explained by properties of the corresponding periodic boundary value problem solutions by increasing the spatial variable changes interval. Thus, the complication of the wave structure in this problem is associated with its space extension.

MSC:

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65Y05 Parallel numerical computation
35A18 Wave front sets in context of PDEs
35B10 Periodic solutions to PDEs
35K57 Reaction-diffusion equations
35K91 Semilinear parabolic equations with Laplacian, bi-Laplacian or poly-Laplacian
35R10 Partial functional-differential equations
35R07 PDEs on time scales
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kolmogorov, A. N.; Petrovsky, I. G.; Piskunov, N. S., No article title, Bull. Univ. d’EtatMoscou, Ser. A, 1, 1-26 (1937)
[2] Fisher, R. A., No article title, Ann. Eugen., 7, 355-369 (1937) · doi:10.1111/j.1469-1809.1937.tb02153.x
[3] Aleshin, S. V.; Glyzin, S. D.; Kaschenko, S. A., No article title, Model. Anal. Inform. Sist., 22, 304-321 (2015) · doi:10.18255/1818-1015-2015-2-304-321
[4] Aleshin, S. V.; Glyzin, S. D.; Kaschenko, S. A., No article title, Model. Anal. Inform. Sist., 22, 609-628 (2015) · doi:10.18255/1818-1015-2015-5-609-628
[5] Kakutani, S.; Markus, L., No article title, Ann. Math. Stud., 4, 1-18 (1958)
[6] Kaschenko, S. A., No article title, in Studies in Stability and the Theory of Oscillation, 1, 64-85 (1981)
[7] Kolesov, Yu. S., No article title, in Mathematical Models in Biology and Medicine, 1, 93-103 (1985)
[8] Y. Kuang, Delay Differential Equations. With Applications in Population Dynamics (Academic, Boston, 1993). · Zbl 0777.34002
[9] Kashchenko, S. A., No article title, Autom. Control Comput. Sci., 47, 470-494 (2013) · doi:10.3103/S0146411613070079
[10] J. Wu, Theory and Applications of Partial Functional Differential Equations Theory and Applications of Partial Functional Differential Equations (Springer, New York, 1996). · Zbl 0870.35116 · doi:10.1007/978-1-4612-4050-1
[11] Glyzin, S. D., No article title, Model. Anal. Inform. Sist., 16, 96-116 (2009)
[12] Kashchenko, A. A., No article title, Model. Anal. Inform. Sist., 18, 58-62 (2015)
[13] Glyzin, S. D., No article title, Autom. Control Comput. Sci., 47, 452-469 (2013) · doi:10.3103/S0146411613070031
[14] Glyzin, S. D.; Kolesov, A. Yu.; Rozov, N. Kh., No article title, Comput.Math. Math. Phys., 50, 816-830 (2010) · Zbl 1224.35198 · doi:10.1134/S0965542510050076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.