Abstract
We study a system of the reaction–diffusion type, where diffusion coefficients depend in an arbitrary way on spatial variables and concentrations, while reactions are expressed as homogeneous functions whose coefficients depend in a special way on spatial variables. We prove that the system has a family of exact solutions that are expressed through solutions to a system of ordinary differential equations (ODE) with homogeneous functions in right-hand sides. For a special case of theODE systemwe construct a general solution represented by Jacobi higher transcendental functions. We also prove that these periodic solutions are analytic functions that can be expressed near each point on the period by convergent power series.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Cherniha, R., King, J. R. “Nonlinear Reaction–Diffusion Systems With Variable Diffusivities: Lie Symmetries, Ansätze and Exact Solutions”, J.Math. Anal. Appl. 308, No. 1, 11–35 (2005).
Polyanin, A. D. and Zaitsev, V. F. Handbook of Nonlinear Equations of Mathematical Physics: Exact Solutions (Fizmatlit,Moscow, 2002) [in Russian].
Slin’ko, M. G., Zelenyak, T. I., Akramov, T. A., Lavrent’ev, M. M. and Shcheplev, V. S. “Nonlinear Dynamics of Catalytic Reactions and Processes (review)”, Mat. Model. 9, No. 12, 87–109 (1997) [in Russian].
Nefedov, N. N., Nikulin, E. I. “Existence and Stability of Periodic Solutions for Reaction–Diffusion Equations in the Two-Dimensional Case”, Model. Anal. Information Syst. 23, No. 3, 342–348 (2016).
Erugin, N. P. Book for Reading on General Course of Differential Equations (Nauka i Tekhnika, Minsk, 1972) [in Russian].
Cherepennikov, V. B. “Analytic Solutions of the Cauchy Problem for Some Linear Systems of Functional-Differential Equations of Neutral Type”, RussianMathematics 38, No. 6, 88–96 (1994).
Cherepennikov, V. B. “On the Solvability in the Class of Analytic Functions of Some Linear Systems of Functional-Differential Equations in a Neighborhood of a Regular Singular Point”, Russian Mathematics 40, No. 5, 71–76 (1996).
Cherepennikov, V. B. and Ermolaeva, P. G. “Numerical Experiment in the Investigation of Polynomial Quasisolutions of Linear Differential-Difference Equations”, RussianMathematics 52, No. 7, 48–60 (2008).
van Brunt, B., Kim, Hong Oh, Derfel, G. “Holomorphic Solutions to Functional Differential Equations”, J. Math. Anal. Appl. 365, No. 1, 350–357 (2010).
Akhiezer, N. I. Elements of the Theory of Elliptic Functions (OGIZ, Gostekhizdat, 1948) [in Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Kosov, E.I. Semenov, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 10, pp. 34–42.
About this article
Cite this article
Kosov, A.A., Semenov, E.I. On Analytic Periodic Solutions to Nonlinear Differential Equations With Delay (Advance). Russ Math. 62, 30–36 (2018). https://doi.org/10.3103/S1066369X18100043
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X18100043