Asymptotic Solution to the Time-Periodic Boundary Control Problem for a Reaction-Advection Equation with a Linear Source

Abstract

We consider a boundary control problem for a singularly perturbed parabolic equation with a linear time-periodic source having a solution in the form of a moving front. Our goal is to obtain a boundary condition that provides the required law of front motion. Based on the asymptotic analysis of the direct problem, we reduce the boundary control problem to solving algebraic equations connecting the asymptotic approximation of the front motion law with the coefficients in the equation and boundary conditions. We also provide a method for estimating the error that occurs when replacing the exact solution to a control problem with an asymptotic one.

Keywords

Internal transition layer / Front motion / Reaction-diffusion-advection / Boundary control problem / Asymptotic solution / Asymptotic method of differential inequalities / 35B25 / 35B30

References

[1] Lukyanenko, D.V., Borzunov, A.A., Shishlenin, M.A.: Solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection type with data on the position of a reaction front. Commun. Nonlinear Sci. Numer. Simul. 99, 105824 (2021)

[2] Lukyanenko, D.V., Shishlenin, M.A., Volkov, V.T.: Asymptotic analysis of solving an inverse boundary value problem for a nonlinear singularly perturbed time-periodic reaction-diffusion-advection equation. J. Inverse Ill-Posed Probl. 27(5), 745–758 (2019)

[3] Nefedov, N.N.: Development of methods of asymptotic analysis of transition layers in reaction-diffusion-advection equations: theory and applications. Comput. Math. Math. Phys. 61(12), 2068–2087 (2021)

[4] Nefedov, N.N, Recke, L., Schneider, K.: Existence and asymptotic stability of periodic solutions with an interior layer of reaction-advection-diffusion equations. J. Math. Anal. Appl. 405(1), 90–103 (2013)

[5] Rozenman, G.G., Schleich, W.P., Shemer, L., Arie, A.: Periodic wave trains in nonlinear media: Talbot revivals, Akhmediev breathers, and asymmetry breaking. Phys. Rev. Lett. 128(21), 214101.1–1214101.6 (2022)

[6] Rudenko OV. Three nonlinearities in physics of acoustic flows. Dokl. Phys., 2020, 65: 317-322.

[7] Shishkin GI. Mesh approximation of singularly perturbed boundary-value-problems for systems of elliptic and parabolic equations. Comput. Math. Math. Phys., 1995, 35(4): 429-446. DOI:

[8] Vasil’eva AB, Butuzov VF, Kalachev LV. The Boundary Function Method for Singular Perturbation Problems, 1995, Philadelphia. SIAM.

[9] Volkov, V.T., Nefedov, N.N.: Asymptotic solution of coefficient inverse problems for Burgers-type equations. Comput. Math. Math. Phys. 60(6), 950–959 (2020)

[10] Volkov, V.T., Nefedov, N.N.: Asymptotic solution of the boundary control problem for a Burgers-type equation with modular advection and linear gain. Comput. Math. Math. Phys. 62(11), 1849–1858 (2022)

[11] Wang J. Monotone method for diffusion equations with nonlinear diffusion coefficients. Nonlinear Anal., 1998, 34: 113-142.

[12] Weber, R.O.: Toward a comprehensive wildfire spread model. Int. J. Wildland Fire. 1(4), 245–248 (1991)

[13] Zakharova SA, Davydova MA, Lukyanenko DV. Use of asymptotic analysis for solving the inverse problem of source parameters determination of nitrogen oxide emission in the atmosphere. Inverse Probl. Sci. Eng., 2021, 29(3): 365-377.

[14] Zayko JS, Gareev LR, Chicherina AD, Trifonov VV, Vedeneev VV, Reshmin AI. Experimental validation of linear-stability theory applied to a submerged jet. Dokl. Phys., 2021, 66(4): 106-109.

Funding

Russian Science Foundation(23-11-00069)

RIGHTS & PERMISSIONS

Shanghai University

Just Accepted

This article has successfully passed peer review and final editorial review, and will soon enter typesetting, proofreading and other publishing processes. The currently displayed version is the accepted final manuscript. The officially published version will be updated with format, DOI and citation information upon launch. We recommend that you pay attention to subsequent journal notifications and preferentially cite the officially published version. Thank you for your support and cooperation.