On a Periodic Inner Layer in the Reaction–Diffusion Problem with a Modular Cubic Source

Analytical Aspects of the Theory of Tikhonov Systems

“…If ϕ 1 (t), ϕ 2 (t) are regularizing functions, and V [1] (t, y, z), V [2] (t, y, z) are functionally independent solutions of the equation L g V = 1, turning to zero at the point (0, y 0 , z 0 ), then the four independent first integrals of system (19), determining its solution with the initial conditions (21), has the following form:…”

“…). From the form V [1] and V [2] it follows that we can take the following functions as regularizing functions…”

“…Let us return to the series of Equation (38). You can directly verify that with the previously selected U [1] 0 and U [2] 0 , the functions U We have four functionally independent first integrals of system (35):…”

“…ln(e t z 1 + z 2 ) − εU Here, as usual, R(t) is a transition operator from variables ( x, s) to variables (z 1 , z 2 ), R −1 (t, s) is a transition operator from variables (z 1 , z 2 ), to variables ( x, s), and for U [1] 2 and U [2] 2 the following formulas hold:…”

“…A.N. Tikhonov's theorem on the limit transition [1] served as an impetus for the development of one of the most practically used methods for solving nonlinear singularly perturbed problems-the Vasilyeva-Butuzov-Nefedov boundary function method [1][2][3]. On the other hand, Poincaré's theorems on the decomposition of [4], which are the basis of the analytical theory of differential equations, and the regularization method of S.A. Lomov [5,6] allowed us to formulate conditions for the usual (and not asymptotic) convergence of series by degrees of a small parameter representing solutions of singularly perturbed equations [7].…”