Nonlinear singularly perturbed parabolic equations with boundary conditions of the first kind

I. V. Denisov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, geometry, differential equations, Tome 217 (2022), pp. 29-36.

Résumé

This paper is a review of applications of the method of angular boundary functions to nonlinear equations. We consider the first boundary-value problem for the following singularly perturbed parabolic equation in a rectangle: \begin{equation*} \epsilon^2\left(a^2\frac{\partial^2 u}{\partial x^2} -\frac{\partial u}{\partial t}\right)=F(u,x,t,\epsilon), \end{equation*} where the function $F$ is nonlinear with respect to the variable $u$. We consider the case where the function $F$ is quadratic or cubic in the variable $u$ at the corner points of the rectangle and examine the possibility of constructing a complete asymptotic expansion of the solution of the problem as $\epsilon\rightarrow 0$.

Détail

Keywords: boundary layer, asymptotic approximation, singularly perturbed equation.

Bibliographie

[1] Butuzov V. F., “Asimptotika resheniya raznostnogo uravneniya s malymi shagami v pryamougolnoi oblasti”, Zh. vychisl. mat. mat. fiz., 12:3 (1972), 582–597

[2] Butuzov V. F., Nesterov A. V., “Ob odnom singulyarno vozmuschennom uravnenii parabolicheskogo tipa”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. mat. kibern., 1978, no. 2, 49–56

[3] Vasileva A. B., Butuzov V. F., Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shkola, M., 1990 | MR

[4] Denisov A. I., Denisov I. V., “Uglovoi pogranichnyi sloi v kraevykh zadachakh dlya singulyarno vozmuschennykh parabolicheskikh uravnenii s nelineinostyami”, Zh. vychisl. mat. mat. fiz., 59:1 (2019), 102–117

[5] Denisov A. I., Denisov I. V., “Uglovoi pogranichnyi sloi v kraevykh zadachakh dlya singulyarno vozmuschennykh parabolicheskikh uravnenii s nemonotonnymi nelineinostyami”, Zh. vychisl. mat. mat. fiz., 59:9 (2019), 1581–1590

[6] Denisov A. I., Denisov I. V., “Matematicheskie modeli protsessov goreniya”, Itogi nauki tekhn. Sovr. mat. prilozh. Temat. obz., 185 (2020), 82–88

[7] Denisov I. V., “Pervaya kraevaya zadacha dlya kvazilineinogo singulyarno vozmuschennogo parabolicheskogo uravneniya v pryamougolnike”, Zh. vychisl. mat. mat. fiz., 36:10 (1996), 56–72 | MR

[8] Denisov I. V., “Uglovoi pogranichnyi sloi v kraevykh zadachakh dlya singulyarno vozmuschennykh parabolicheskikh uravnenii s kvadratichnoi nelineinostyu”, Zh. vychisl. mat. mat. fiz., 57:2 (2017), 255–274

[9] Denisov I. V., “Uglovoi pogranichnyi sloi v kraevykh zadachakh dlya singulyarno vozmuschennykh parabolicheskikh uravnenii s monotonnoi nelineinostyu”, Zh. vychisl. mat. mat. fiz., 58:4 (2018), 1–11

[10] Denisov I. V., “Uglovoi pogranichnyi sloi v kraevykh zadachakh dlya singulyarno vozmuschennykh parabolicheskikh uravnenii s kubicheskimi nelineinostyami”, Zh. vychisl. mat. mat. fiz., 61:2 (2021), 256–267

[11] Nefedov N. N., “Metod differentsialnykh neravenstv dlya nekotorykh klassov nelineinykh singulyarno vozmuschennykh zadach s vnutrennimi sloyami”, Differ. uravn., 31:7 (1995), 1132–1139 | MR

[12] Chaplygin S. A., Osnovaniya novogo sposoba priblizhennogo integrirovaniya differentsialnykh uravnenii, 1919, M.

[13] Amann H., “On the existence of positive solutions of nonlinear elliptic boundary-value problems”, Indiana Univ. Math. J., 21:2 (1971), 125–146 | DOI | MR

[14] Amann H., “Periodic solutions of semilinear parabolic equations”, Nonlinear Analysis. A Collection of Papers in Honor of E. H. Rothe, eds. Cesari L., Kannan R., Weinberger H., Academic Press, New York, 1978, 1–29 | MR

[15] Sattinger D. H., “Monotone methods in nonlinear elliptic and parabolic boundary-value problems”, Indiana Univ. Math. J., 21:11 (1972), 979–1000 | DOI | MR