Geodesic
A boundary value problem for a~fourth order partial differential equation with the lowest term
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2013), pp. 3-10

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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. 
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     author = {D. Amanov and M. B. Murzambetova},
     title = {A boundary value problem for a~fourth order partial differential equation with the lowest term},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
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D. Amanov; M. B. Murzambetova. A boundary value problem for a~fourth order partial differential equation with the lowest term. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 1 (2013), pp. 3-10. http://geodesic.mathdoc.fr/item/VUU_2013_1_a0/