Investigation of the uniqueness solution of the Showalter-Sidorov problem for the mathematical Hoff model. Phase space morphology
Автор: Nikolaeva N.G., Gavrilova O.V., Manakova N.A.
Рубрика: Математическое моделирование
Статья в выпуске: 1 т.17, 2024 года.
Бесплатный доступ
The study of the phase space morphology of the mathematical model deformation of an I-beam, which lies on smooth Banach manifolds with singularities (k-Whitney assembly) depending on the parameters of the problem, is devoted to the paper. The mathematical model is studied in the case when the operator at time derivative is degenerate. The study of the question of non-uniqueness of the solution of the Showalter-Sidorov problem for the Hoff model in the two-dimensional domain is carried out on the basis of the phase space method, which was developed by G.A. Sviridyuk. The conditions of non-uniqueness of the solution in the case when the dimension of the operator kernel at time derivative is equal to 1 or 2 are found. Two approaches for revealing the number of solutions of the Showalter-Sidorov problem in the case when the dimension of the operator kernel at time derivative is equal to 2 are presented. Examples illustrating the non-uniqueness of the solution of the problem on a rectangle are given.
Sobolev type equations, showalter-sidorov problem, phase space method, whitney assemblies, the hoff equation, non-uniqueness of solutions
Короткий адрес: https://sciup.org/147243951
IDR: 147243951 | DOI: 10.14529/mmp240105
Список литературы Investigation of the uniqueness solution of the Showalter-Sidorov problem for the mathematical Hoff model. Phase space morphology
- Sviridyuk G.A. The Manifold of Solutions of an Operator Singular Pseudoparabolic Equation. Doklady Akademii Nauk SSSR, 1986, vol. 289, no. 6, pp. 1-31. (in Russian)
- Pyatkov S.G. Boundary Value and Inverse Problems for Some Classes of Nonclassical Operator-Differential Equations. Siberian Mathematical Journal, 2021, vol. 62, no. 3, pp. 489-502. DOI: 10.1134/S0037446621030125
- Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-Up in Nonlinear Sobolev Type Equations. Berlin, Walter de Gruyter, 2011. DOI: 10.1515/9783110255294
- Keller A.V., Zagrebina S.A. Some Generalizations of the Showalter-Sidorov Problem for Sobolev-Type Models. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 5-23. DOI: 10.14529/mmp150201
- Zamyshlyaeva A.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Linear Sobolev Type Equations of Higher Order. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 4, pp. 5-16. DOI: 10.14529/mmph160401
- Hoff N.J. Creep Buckling. Journal of the Aeronautical Science, 1956, no. 7, pp. 1-20.
- Bayazitova A.A. The Showalter-Sidorov Problem for the Hoff Model on a Geometric Graph. The Bulletin of Irkutsk State University. Series: Mathematics, 2011, vol. 4, no. 1, pp. 2-8. (in Russian)
- Sviridyuk G.A., Shemetova V.V. Hoff Equations on Graphs. Differential Equations, 2006, vol. 42, no. 1, pp. 139-145. DOI: 10.1134/S0012266106010125
- Shafranov D.E., Shvedchikova A.I. The Hoff Equation as a Model of Elastic Shell. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2012, no. 18(227), iss. 12, pp. 77-81.
- Sidorov N.A., Romanova O.A. On Application of Some Results of the Branching Theory in the Process of Solving Differential Equations with Degeneracy. Differential Equations, 1983, vol. 19, no. 9, pp. 1516-1526.
- Sviridyuk G.A., Zagrebina S.A. The Showalter-Sidorov Problem as a Phenomena of the Sobolev Type Equations. The Bulletin of Irkutsk State University. Series: Mathematics, 2010, vol. 3, no. 1, pp. 104-125. (in Russian)
- Manakova N.A., Gavrilova O.V., Perevozchikova K.V. Semilinear Models of Sobolev Type. Non-Uniqueness of Solution to the Showalter-Sidorov Problem. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2022, vol. 15, no. 1, pp. 84-100. DOI: 10.14529/mmp220105
- Gil'mutdinova A.F. On the Non-Uniqueness of Solutions of Showalter-Sidorov Problem for One Plotnikov Model. Vestnik of Samara State University, 2007, no. 9/1, pp. 85-90. (in Russian)
- Bokareva T.A., Sviridyuk G.A. Whitney Folds in Phase Spaces of Some Semilinear Sobolev-Type Equations. Mathematical Notes, 1994, vol. 55, no. 3, pp. 237-242. DOI: 10.1007/BF02110776
- Sviridyuk G.A., Sukacheva T.G. The Phase Space of a Class of Operator Equations of Sobolev Type. Differential Equations, 1990, vol. 26, no 2, pp. 250-258. (in Russian)
- Sviridyuk G.A., Kazak V.O. The Phase Space of an Initial-Boundary Value Problem for the Hoff Equation. Mathematical Notes, 2002, vol. 71, no. 1-2, pp. 262-266. DOI: 10.4213/mzm347
- Sviridyuk G.A., Trineeva I.K. A Whitney Fold in the Phase Space of the Hoff Equation. Russian Mathematics (Izvestiya VUZ. Matematika), 2005, vol. 49, no. 10, pp. 49-55.
- Gavrilova O.V., Nikolaeva N.G. Numerical Study of the Non-Uniqueness of Solutions to the Showalter-Sidorov Problem for a Mathematical Model of I-beam Deformation. Journal of Computational and Engineering Mathematics, 2022, vol. 9, no. 1, pp. 10-23. DOI: 10.14529/jcem220102
- Sviridyuk G.A., Sukacheva T.G. Galerkin Approximations of Singular Nonlinear Equations of Sobolev Type. Izvestiya VUZ. Matematika, 1989, vol. 33, no. 10, pp. 56-59.
- Bogatyreva E.A., Manakova N.A. Numerical Simulation of the Process of Nonequilibrium Counterflow Capillary Imbibition. Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 1, pp. 132-139. DOI: 10.1134/S0965542516010085
- Bychkov E.V. Analytical Study of the Mathematical Model of Wave Propagation in Shallow Water by the Galerkin Method. Bulletin of the South Ural State University. Series: Mathematical Modeling, Programming and Computer Software, 2021, vol. 14, no. 1, pp. 26-38. DOI: 10.14529/mmp210102
- Manakova N.A., Sviridyuk G.A. Nonclassical Equations of Mathematical Physics. Phase Space of Semilinear Sobolev Type Equations. Bulletin of the South Ural State University. Series: Mathematics. Mechanics. Physics, 2016, vol. 8, no. 3, pp. 31-51. DOI: 10.14529/mmph160304
