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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:
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