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Models of multiparameter bifurcation problems for the fourth order ordinary differential equations. (Russian. English summary) Zbl 1413.34080

Summary: We consider the problem of computing the bifurcating solutions of nonlinear eigenvalue problem for an ordinary differential equation of the fourth order, describing the divergence of the elongated plate in a supersonic gas flow, compressing (extending) by external boundary stresses on the example of the boundary conditions (the left edge is rigidly fixed, the right one is free). Calculations are based on the representation of the bifurcation parameter using the roots of the characteristic equation of the corresponding linearized operator. This representation allows one to investigate the problem in a precise statement and to find the critical bifurcation surfaces and curves in the neighborhood of which the asymptotics of branching solutions is being constructed in the form of convergent series in the small parameters. The greatest difficulties arise in the study of the linearized spectral problem. Its Fredholmness is proved by constructing the corresponding Green’s function and for this type of problems it is performed for the first time.

MSC:

34B08 Parameter dependent boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
47J15 Abstract bifurcation theory involving nonlinear operators
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References:

[1] [1] V. V. Bolotin, Non-conservative problems of the theory of elastic stability, Pergamon Press, Oxford, 1963, 320 pp. · Zbl 0121.41305
[2] [2] A. S. Vol’mir, Ustoychivost’ deformiruyemykh sistem [Stability of deformable systems], Nauka, Moscow, 1967, 984 pp. (In Russian)
[3] [3] M. M. Vainberg, V. A. Trenogin, The Theory of Branching of Solutions of Nonlinear Equations, Wolters-Noordhoff, Groningen, 1974 · Zbl 0274.47033
[4] [4] B. V. Loginov, T. E. Badokina, O. V. Makeeva, “Green functions construction for divergence problems in aero-elasticity”, ROMAI Jornal, 4:2 (2008), 33-44 · Zbl 1265.74033
[5] [5] M. A. Naimark, Linear differential operators, v. 1, Elementary theory of linear differential operators, Harrap, London, Toronto, 1968 · Zbl 0198.47502
[6] [6] B. V. Loginov, O. V. Kozhevnikova, “Computation of eigen-bending forms and branching solutions asymptotics for bifurcation problem on rectangular plate divergence”, Izvestiya RAEN,, 2:3 (1998), 112-120 (In Russian)
[7] [7] P. A. Vel’misov, B. V. Loginov, “Group transformtion method and solutions branching in two-point boundary value problems of aeroelasticity“, Materialy Mezhdunar. konf. “Differentsial’nyye uravneniya i ikh prilozheniya” [Materials of the International Conference “Differential Equations and their Applications”], Mordovia Univ., Saransk, 1995, 120-125 (In Russian) · Zbl 1002.58011
[8] [8] P. A. Vel’misov, S. V. Kireyev, A. O. Kuznetsov, “Plate stability in supersonic gas flow”, Vestn. Ul’yanovsk. Gos. Tekh. Univ. Ser. Estestv. Nauki, 1999, no. 1, 44-51 (In Russian) · Zbl 1002.58011
[9] [9] B. V. Loginov, A. V. Tsyganov, O. V. Kozhevnikova, “Strip-plate divergence as bifurcational problem with two spectral parameters”, Proceedings of International Symposium on Trends in Applications of Mathematics to Mechanics (Seeheim, Germany, August 22-28, 2004), Shaker Verlag, Aachen, 2005, 235-246 · Zbl 1084.74021
[10] [10] T. E. Badokina, B. V. Loginov, Yu. B. Rusak, “Construction of the asymptotics of solutions of nonlinear boundary value problems for fourth order differential equation with two bifurcation parameters”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Ser. Matematika, 5:1 (2012), 2-12 (In Russian) · Zbl 1267.34035
[11] [11] E. Kamke, Spravochnik po obyknovennym differentsial’nym uravneniyam [Handbook on Ordinary Differential Equations], Nauka, Moscow, 1971, 576 pp. (In Russian)
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