Application of Green's operator to quadratic variational problems

• Autorzy: Azbelev N. V. , Tsalyuk V. Z.
• Języki publikacji: EN

Abstrakty

EN

We use Green's function of a suitable boundary value problem to convert the variational problem with quadratic functional and linear constraints to the equivalent unconstrained extremal problem in some subspace of the space L2 of quadratically summable functions. We get the neccessary and sufficient criterion for unique solvability of the variational problem in terms of the spectrum of some integral Hilbert-Schmidt operator in L2 with symmetric kernel. The numerical technique is proposed to estimate this criterion. The results are demonstrated on examples: 1) a variational problem with deviating argument, and 2) the problem of the critical force for the vertical pillar with additional support point (the qualities of the pillar may vary discontinuously along the pillar's axis).

Słowa kluczowe

EN

quadratic variational problem; Sobolev space; boundary value problem; Hilbert space; Green's operator; Fredholm integral operator; spectrum

• Wydawca: AGH University of Science and Technology Press
• Czasopismo: Opuscula Mathematica
• Rocznik: 2006
• Tom: Vol. 26, no. 2
• Strony: 243--256
• Opis fizyczny: Bibliogr. 8 poz., tab., rys.

Twórcy

autor

Azbelev N. V.

autor

Tsalyuk V. Z.

• Perm State Technical University, Perm, Russia

Bibliografia

• [1] Azbelev N.V., Kultyshev S.Yu., Tsalyuk V.Z., Functional Differential Equations and Variational Problems, R&C Dynamics, 2006 (in Russian; http//shop.rcd.ru).
• [2] Azbelev N. V., Makagonova M. A., Plaksina V.P., Variational method of solving Euler's problem on stability of elastic pillar, Izvestiya RAEN. Differentsial'nye uravneniya, 8 (2004), 10-13 (in Russian).
• [3] Azbelev N. V., Maksimov V. P., Rakhmatullina L.F., The Elements of the Contemporary Theory of the Functional Differential Equations, Methods and Applications. - Moscow, Institute of Computer Research, 2002 (in Russian; http://shop.rcd.ru).
• [4] Azbelev N.V., Rakhmatullina L.F., Theory of linear abstract functional differential equations, Memoirs on Differential Equations and Mathematical Physics, (1996) 8, 1-102.
• [5] Riesz F. , Nagy B. Sz., Functional Analysis, Dover Publications, 1990.
• [6] Trenogin V. A., Functional Analysis. Nauka, Moscow 1993 (in Russian).
• [7] Tsalyuk V. Z., Essays on shape of a beam (a method for solving variational problems), Izvestiya RAEN, Differentsial'nye uravneniya, 8 (2004), 98-112 (in Russian).
• [8] Tsalyuk V. Z., Solving some variational problems by W-method, Vestnik Permskogo universiteta, Matematika. Informatika. Mechanika, 2 (2005) (in Russian).