On the Iterative Methods of Linearization, Decrease of Order and Dimension of the Karman-Type PDEs

Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 792829, 15 pages http://dx.doi.org/10.1155/2014/792829 Research Article On the Iterative Methods of Linearization, Decrease of Order and Dimension of the Karman-Type PDEs A. V. Krysko,1 J. Awrejcewicz,2 S. P. Pavlov,3 M. V. Zhigalov,3 and V. A. Krysko3 1 Engels Institute of Technology (Branch), Saratov State Technical University, Department of Higher Mathematics and Mechanics, Russian Federation, Ploschad Svobodi 17, Engels, Saratov 413100, Russia 2 Department of Automation and Biomechanics, Lodz University of Technology, and Department of Vehicles, Warsaw University of Technology, 84 Narbutta Street, 02-524 Warszawa, Poland 3 Saratov State Technical University, Department of Mathematics and Modeling, Russian Federation, Polytehnicheskaya 77, Saratov 410054, Russia Correspondence should be addressed to J. Awrejcewicz; Received 8 November 2013; Accepted 30 December 2013; Published 16 March 2014 Academic Editors: D. Baleanu and H. Jafari Copyright © 2014 A. V. Krysko et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Iterative methods to achieve a suitable linearization as well as a decrease of the order and dimension of nonlinear partial differential equations of the eighth order into the biharmonic and Poisson-type differential equations with their simultaneous linearization are proposed in this work. Validity and reliability of the obtained results are discussed using computer programs developed by the authors. 1. Introduction Mathematical models of continuous mechanical structures are described by nonlinear partial differential equations which may be solved analytically only in a few rare cases. However, a direct application of the numerical methods is associated also with big difficulties regarding a high order of both dimension and differential operator, as well as nonlinearity of the PDEs studied. This is why it is tempting to develop approaches that offer a reduction of the input differential equations. The mentioned methods can be divided into three groups: (1) linearization; (2) order decrease of the PDEs; (3) order decrease of a differential operator. The so far existing methods of solutions of nonlinear problems, depending on the introduced linearization level, can be divided into two groups. The first one deals with the linearization of PDEs, whereas the second one is dedicated to the linearization of algebraic equations obtained through the discretization procedures applied to the input PDEs. Below, we consider the methods associated with the first group. This group contains the Newton and Newton-Kantorovich methods [1]. One of the linearization methods is the method of quasilinearization, widely illustrated in monograph [2]. It presents a further development of Newton’s method, and it generalizes the method proposed by Kantorovich. On the other hand, there is a seminal approach known as the Agmon-Douglis-Nirenberg (ADN) theory for elliptic PDEs still attracting a large number of imitators [3, 4]. In particular, the abstract least squares theory is developed satisfying the ADN elliptic theory assumptions [5–7]. Furthermore, in the case of corners in plane domains the ADN system exhibits singularities, which imply a need for construction of singular exponents and angular functions [8]. Our approach does not have this disadvantage and it is simple in direct applications to the real world systems. The so far briefly addressed approaches linearize the input problem; that is, they reduce it to the solution of linear problems. However, there is one more important question to be solved, that is, a reduction of the space dimension of the initial problem. One of the methods to solve the stated problem is focused on averaging (integration) along such a coordinate on which the object dimension is lesser in comparison to the two remaining coordinates. On the other hand, it 2 The Scientific World Journal is well known that mathematical problems related to the theory of material strength can be formulated as variational problems, that is, the problems of finding extrema of a certain functional. Variational statements create a foundation for the construction of direct difference and variational methods, as it is widely described in monograph [9]. We mention only a few works [10–12] devoted to the third group, that is, aiming at a decrease of the PDE order. Note that the so far presented state of the art of the proposed and applied methods allows us to solve each of the mentioned problems separately: either a decrease of the system order or its linearization. However, we show how all these problems can be solved simultaneously. Our paper is focused mainly on the method of dimension decrease and linearization of the Karman-type PDEs. However, the presented approach can be successfully applied to other nonlinear PDEs. In particular, in the modified version two variants of the proposed method are presented: (i) the first iterative method consists of the reduction of the eighth order linear PDEs to a successive solution of linear PDEs of the fourth order biharmonic equations; that is, the system dimension is reduced twice with simultaneous linearization of the problem; (ii) the applied second iterative procedure includes a further order decrease of the earlier obtained (first iterative method) linear system of biharmonic PDEs of the fourth order to the successive solution to the system of the second order Poisson-type equations. In other words, the application of these two iterative procedures implies a fourfold reduction of the PDEs order with the linearization procedure carried out simultaneously. The proposed iterative procedures regarding the nonlinear PDEs order decrease and linearization can also be applied to PDEs with a curvilinear boundary. The application of FDM (finite difference method) to solve biharmonic equations and PDEs of the Poisson-type requires a solution to the so-called Sapondzhyan-Babuška problem. The paradox of Sapondzhyan-Babuška (see [13–15]) was discovered when studying the asymptotic behavior of solutions to an elasticity system in a thin polygonal plate (inscribed in the plate with smooth boundary) as the length of the side of the polygon tends to zero and the number of sides goes to infinity. In Section 2 of our work we prove the proposed iterative procedure to remove this paradox (this problem concerns smoothness of the curvilinear boundary). In Section 3 of our work the reliability and validity of the method of variational iterative procedure to solve PDEs described by positively defined operators are illustrated and discussed. Namely, the convergence of the method of variational iterations generalizes the Kantorovich-Vlasov method [16] aimed at the reduction of PDEs to ODEs. 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