CONVERGENCE OF THE ITERATIVE EXTENDED KANTOROVICH METHOD FOR VON KARMAN EQUATIONS - Gromov, V. A. (Computer Science and Applied Mathematics, 2017)
Von Karman equations are formulated (in the orthogonal coordinates   1 2 x , x ) as a system of partial differential equations (1), (2), where   const is a function of an external influence; 1 k , 2 k , 1 a , 2 a are positive constants. The problem is defined on a close cylindrical domain. Hereinafter the following notations (2) are used. The problem is completed by boundary conditions 1 u 0   , 2 1 2 1 0 u x     , 2 u 0   , 2 1 0 u x     for parts of the boundary 1 x  const and by periodicity conditions defined on an arbitrary line 2 x  const . The generalized solution of the problem under study is a pair of functions   1 2 U  u ,u ,   1 1 u H  ,   2 0 u H  such that (3), (4) hold true for arbitrary functions   1 1 v H  ,   2 0 v H  , where   1 H  is the Sobolev’s space, the closure of a set of continuous (with 2 derivatives) functions in the norm   2 1 2 , H K   dx dx     ;   0 H  is the Sobolev’s space, the closure (in the same norm) of a set of continuous (with 2 derivatives) functions satisfying homogenous boundary conditions 0, 0 n               . To find solutions of non-linear boundary problem (1), one constructs a sequence of approximations of the generalized solutions (3), (4) with the employment of the representation (7) of the unknowns in the algorithm iterations. For odd iterations, the functions         -1 1 1 i i j j h x  h x are calculated in the previous, the i 1-th iteration, and the unknowns evaluated from these equations are     2 i j g x ; arbitrary functions j v are presented as       1 1 2 i j j j v h x  x   , where   j 2  x are arbitrary functions of a single variable 2 x . Similarly, for even iterations, the functions         -1 2 2 i i j j g x  g x are known, and the unknowns to be found are     1 i j h x ; in that case j v are presented as       1 1 2 i j j j v  x g x   , where   j 1  x are arbitrary functions of a single variable 1 x . Representation (7) allows reducing equations under study (1) to a sequence of boundary problems for ordinary differential equations. On the other hand, it necessitates to consider specific ‘weighted’ functional spaces Hg ,   2 2 g W c;d , determined as a closure of continuous functions     2 C a;b in the specific norm. The spaces introduced make it possible to furnish a means of operator expression of single-dimensional non-linear boundary problems for iterations of the method under discussion (17), (18). What follows is a sequence of statements to analyze convergence of the IGKM for von Karman equations. Lemma 1. If a function p  H can be presented as       1 2 1 2  x , x  h x g x , then boundednes of  in p H is derived from boundednes of   1 h x ,   2 g x in 2 2 W . Lemma 2. The presentation (20), (21) holds true when s P , s G are homogeneous operators of order s . Theorem 1. Each operator P , N , Ps , Ns ( s 1,2 ) is strongly continuous in h1 H , g1 H provided sequences     1 i h ,     1 i g are bounded in     2 min max 2 1 1 W x ; x  ,     2 min max 2 2 2 W x ; x  respectively. Theorem 2. Under hypotheses of Theorem 1 each operator G , H , Gs , Hs ( s  1, 2,3 ), is strongly continuous in h2 H , g2 H respectively. Theorem 3. Provided sequences     1 i g ,     1 i h of IGKM approximations are bounded in 2 2 W .       1 1 i i h g ,       2 2 i i h g are strongly convergent in 1 0 H H .