To solution of a boundary value problem with parameter for ordinary differential equations
Keywords:
principle of immersion, optimization problem, minimizing sequences, integral equation, Sturm-Liouville problemAbstract
A method for solving of a boundary value problem with parameter at presence of phase and integral constraints is supposed. Necessary and sufficient conditions for the existence of solution of the boundary value problem with parameter for ordinary differential equations are obtained. A method for constructing solution of the boundary value problem with parameter and constraints is developed by constructing minimizing sequences. The basis of the proposed method for solving the boundary value problem is the principle of immersion. The immersion principle is established by building the general solution of a class of Fredholm integral equations of the first kind. Solution of the Sturm-Liouville problem for the parameter value on the interval is presented as example. The principal difference between the proposed method lies in the fact that the solubility and the construction of the solution of the boundary value problem with a parameter and constraints are solved together, by constructing minimizing sequences focused on the use of computer technology. Solvability and construction of the solution of the boundary value problem are determined by solving the optimization problem. The creation of the general theory of boundary value problems with parameters for ordinary differential equations of any order with complicated boundary conditions in the presence of phase and integral constraints are an important issue.
References
[2] Tihonov A.N., Vasileva A.B., Svetnikov A.G. Differentsialnyie uravneniya. - M.: Nauka, 1985, 231 s.
[3] Klokov Yu.A. O nekotoryih kraevyih zadachah dlya sistem dvuh uravneniy vtorogo poryadka. // Differentsialnyie uravneniya, 2012, t. 48, No 10, s. 1368-1373.
[4] Kolmogorov D.P., Sheyka B.A. Zadacha o kratnyih sobstvennyih i polozhitelnyih sobstvennyih funktsiyah dlya odnorodnogo kvazilineynogo uravneniya vtorogo poryadka. // Differentsialnyie uravneniya, t. 48, No 9, s. 1475-1486.
[5] Makin A.S., Tompson G.V. O razlozheniyah po sobstvennyim funktsiyam nelineynogo operatora Shturma-Liuvillya s kraevyimi uloviyami zavisyaschimi ot spektralnogo parametra. // Differentsialnyie uravneniya, t. 27, No 8, s. 1096-1104.
[6] Aisagaliev S.A. Upravlyaemost nekotoroy sistemyi differentsialnyih uravneniy // Differentsialnyie uravnniya. 1991, t. 27, No 9. s. 1475-1486.
[7] Aisagaliev S.A. Obschee reshenie odnogo klassa integralnyih uravneniy // Matematicheskiy zhurnal. - 2005. - t. 5, No 14(18).
[8] Aisagaliev S.A., Belogurov A.P. Upralyaemost i byistrodeystvie protsessa, opisyivaemogo parabolicheskim uravneniem s ogranichennyim upravleniem. Sibirskiy matematicheskiy zhurnal, yanvar-fevral, 2011, t. 53, No 1, s. 20-37.
[9] Aisagaliev S.A., Kabidoldanova A.A. Ob optimalnom upravlenii lineynyimi sistemami s lineynyim kriteriem kachestva i ogranicheniyami // Differentsialnyie uravneniya. 2012. t. 48, No 6, s. 826-838.
[10] Aisagaliev S.A., Kabidoldanova A.A. optimalnoe upravlenie dinamicheskih sistem. Palmarium Academic Publishing (Verlag, Germaniya). -2012. -288 s.
[11] Aisagaliev S.A., Kalimoldaev M.N., Zhunusova Zh.H. Printsip pogruzheniya dlya kraevyih zadach obyiknovennyih differentsialnyih uravneniy // Matematicheskiy zhurnal. 2012. T. 12. No 2(44). s. 5-22.
[12] S.A. Aisagaliev, M.N. Kalimoldaev, E.M. Pozdeeva K kraevoy zadache obyiknovennyih differentsialnyih uravneniy. - ISSN 1563-0285. Vestnik KazNU, ser. mat., meh., inf. 2012, No 2(76), s. 5-24.
