Nonclassical Mathematical Physics Models

G. A. Sviridyuk ; S. A. Zagrebina

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 14 (2012), pp. 7-18

Résumé

Nonclassical called the models of mathematical physics, whose representation in the form of equations or systems of partial differential equations do not fit into one of the classical types — elliptic, parabolic or hyperbolic. In particular, the non-classical model are described by the equations of mixed type (eg, Tricomi equation), the degenerate equation (for example, the Keldysh equation) or the equations of Sobolev type (eg, Barenblatt–Zheltov–Kochina equation). The article provides an overview of some, in our opinion — the main A. I. Kozhanov achievements in the field of non-classical models of mathematical physics. His major achievements in the field of non-classical linear models belong to the theory of composite type equations, where he developed almost to perfection the method of a priori estimates and did the maximum possible generalization. Furthermore, the method of a priori estimates, along with the principle of comparing A. I. Kozhanov very effectively applied to the study of non-linear non-classical models such as the generalized Boussinesq filtration equation and classical nonlinear models, including models of the Josephson junction. Special place in activity of A. I. Kozhanov take the inverse problem, which, along with the decision and want to find another unknown factor. Here he received outstanding results in both linear and nonlinear cases.

Détail

Mots-clés : composite type equations, Sobolev type equations
Keywords: weakened Showalter–Sidorov problem, generalized filtration Boussinesq equation, inverse coefficient problems.