Multidimensional exact solutions to the reaction-diffusion system with power-law nonlinear terms

Kosov, A. A.; Семенов, Е. И.

doi:10.1134/s0037446617040085

Cited by 6 publications

(3 citation statements)

References 11 publications

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“…Note that the case c = 0 leads to a trivial solution. Functions a(t) = c 1 e c 2 t (for γ = 1) and a(t) = (c 1 t + c 2 ) (γ−2)/(2γ−2) (for γ ≥ 3) are solutions to Equation (27). Therefore, for all natural γ = 2, the system in Equation ( 26) can be rewritten as…”

Section: Exact Solutionsmentioning

confidence: 99%

“…Note that in the literature, you can find systems of a slightly different form, which are also based on second-order parabolic equations with power nonlinearity. In particular, reaction-diffusion systems where the functions f and p also depend on independent variables are considered in [27,28]; systems describing heat and mass transfer are studied in [29]. The objective of this research is the problem of constructing HDW-type solutions of the system in Equation ( 2) that has a known law of front motion (a problem with a given diffusion front).…”

Section: Introductionmentioning

confidence: 99%

“…In article [28], the solutions of the two-component reaction-diffusion system, which differs slightly from Equation (2), are constructed by the method of linear defining equations. Multicomponent systems are considered in [27], where solutions have the form of special matrix constructions. However, we could not find any publications deal with exact solutions of Equation ( 2) that have HDW-type.…”

Section: Introductionmentioning

confidence: 99%

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Analytical Solutions to the Singular Problem for a System of Nonlinear Parabolic Equations of the Reaction-Diffusion Type

2020

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The paper deals with a system of two nonlinear second-order parabolic equations. Similar systems, also known as reaction-diffusion systems, describe different chemical processes. In particular, two unknown functions can represent concentrations of effectors (the activator and the inhibitor respectively), which participate in the reaction. Diffusion waves propagating over zero background with finite velocity form an essential class of solutions of these systems. The existence of such solutions is possible because the parabolic type of equations degenerates if unknown functions are equal to zero. We study the analytic solvability of a boundary value problem with the degeneration for the reaction-diffusion system. The diffusion wave front is known. We prove the theorem of existence of the analytic solution in the general case. We construct a solution in the form of power series and suggest recurrent formulas for coefficients. Since, generally speaking, the solution is not unique, we consider some cases not covered by the proved theorem and present the example similar to the classic example of S.V. Kovalevskaya.

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Section: Exact Solutionsmentioning

confidence: 99%