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## Archive of Issues

Uzbekistan Tashkent
Year
2013
Issue
1
Pages
3-10
 Section Mathematics Title A boundary value problem for a fourth order partial differential equation with the lowest term Author(-s) Amanov D.a, Murzambetova M.B.a Affiliations Institute of Mathematics and Information Technologies, National Academy of Sciences of Uzbekistana Abstract In this paper we study a boundary value problem for the fourth order partial differential equation with the lowest term in a rectangular domain. For the solution of the problem a priori estimate is obtained. From a priori estimate the uniqueness of the solution of the problem follows. For the proof of the solvability of this problem we use the method of separation of variables. The solvability of this problem is reduced to the Fredholm integral equation of the second kind with respect to unknown function. Integral equation is solved by the method of successive approximations. We find the sufficient conditions for the absolute and uniform convergence of series representing the solution of the problem and the series obtained by differentiation four times with respect $x$ and two times with respect to $t$. Keywords boundary value problem, a priori estimate, regular solvability, Fredholm integral equation of the second kind, resolvent, method of successive approximations UDC 517.95 MSC 35G15, 35D05 DOI 10.20537/vm130101 Received 30 November 2012 Language Russian Citation Amanov D., Murzambetova M.B. A boundary value problem for a fourth order partial differential equation with the lowest term, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 3-10. References Dzhurayev T.D., Sopuev.A.K. K teorii differentsial’nykh uravnenii v chastnykh proizvodnykh chetvertogo poryadka (Theory of partial differential equations of fourth order), Tashkent: Fan, 2000, 144 p. Otarova Zh.A. Solvability and spectral properties of the boundary value problems for fourth order equations of mixed type, Abstract of Cand. Sci. (Phys.–Math.) Dissertation, Tashkent, 2009, 16 p. Salimova G. On a boundary value problem for a fourth order partial differential equation, Proc. Inst. Math. Mech., Azerb. Acad. Sci., 2006, vol. 25, pp. 95–104. Baikuziev K.B., Kasimova M. Initial boundary value problem for nonlinear degenerating fourth order equation, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1968, no. 5, pp. 7–12. Karimov D.Kh., Kasimova M. Initial boundary value problems for linear degenerating fourth order equation, Izv. Akad. Nauk UzSSR, Ser. Fiz.-Mat. Nauk, 1968, no. 2, pp. 27–31. Meredov M.M. About uniqueness of solutions of boundary value problems for fourth order mixed type equation, Izv. Akad. Nauk Turkm. SSR, Ser. Fiz.-Tekh., Khim. Geol. Nauk, 1967, no. 4, pp. 11–16. Salakhitdinov M.S., Amanov D. Solvability and spectral properties of the self-adjoint problems for fourth order equation, Modern Problems of Mathematical Physics and Information Technologies: Transaction of the International Conference, Ulugbek National University of Uzbekistan, Tashkent, 2005, pp. 151–155. Salakhitdinov M.S., Amanov D. Solvability and spectral properties of a self-adjoint problem for a equation of fourth order, Uzb. Mat. Zh., 2005, no. 3, pp. 72–77. Amanov D., Yuldasheva A.V. Solvability and spectral properties of a self-adjoint problem for a equation of fourth order, Uzb. Mat. Zh., 2007, no. 4. pp. 3–8. Hudaverdiev K.I., Alieva A.G. About existence in small of the classical solution of one-dimensional mixed problem for one class of the semilinear Sobolev type equations of the fourth order, Vestn. Bakin. Univ., Ser. Fiz.-Mat. Nauk, 2009, no. 1, pp. 5–17. Yang Yang, Zhang Jihui. Existence of infinitely many mountain pass solutions for some fourth-order boundary value problems with a parameter, Nonlinear Analysis: Theory, Methods and Applications, 2009, vol. 71, issue 12, pp. 6135–6143. Karaca Ilkay Yaslan. On positive solutions for fourth-order boundary value problem with impulse, Journal of Computational and Applied Mathematics, 2009, vol. 225, no. 2, pp. 356–364. Khankhasaev V.N. First boundary value problem for nonlinear degenerating equation of fourth order, Vestn. Buryat. Gos. Univ., 2010, no. 9, pp. 183–186. Zhao Junfang, Wang Libo, Ge Weigao. Necessary and sufficient conditions for the existence of positive solutions of fourth order multi-point boundary value problems, Nonlinear Analysis: Theory, Methods and Applications, 2010, vol. 72, issue 2, pp. 822–835. Mustafaev A.P. Representation of partial solutions of fourth order composite type equation, Est. Tekhn. Nauki, 2010, no. 2, pp. 51–53. Runzhang Xu, Yacheng Liu. Global existence and blow-up of solutions for generalized Pochhammer-Chree equations, Acta Mathematica Scientia, 2010, vol. 30, issue 5, pp. 1793–1807. Full text