On the accuracy of an iteration method for one nonlinear oscillation equation

Abstract

Let us consider the initial boundary value problemFor the homogeneous case, the equation (1) was obtained by Kirchhoff [1] who investigated the dynamic state of a string. A great number of works is devoted to the solvability of this equation as well as to the construction and investigation of numerical algorithms for their solution (see, for example, [2] and the references therein).Suppose a solution of the problem (1), (2) is sought for in the form of a series w n = n i=1 w ni (t) sin ix, where the coefficients w ni (t) are found according to the Galerkin method from a system of ordinary differential equationswhere wπ π 0 f (x, t) sin ix dx, p = 0, 1, i = 1, 2, . . . , n. On a time interval [0, T ], we introduce a grid with step τ = T M and nodes t m = mτ , m = 0, 1, . . . , M . Setting w 0 ni = w (0) i and denoting, when m ≥ 1, by w m ni an approximate value of w ni (t m ), we solve system (3) by the difference scheme)) , m = 2, 3, . . . , M.The system of cubic equations (4) will be solved layer-by-layer. Assuming that, for m = 1, w ni and, for m ≥ 2, w m−2 ni and w m−1 ni , i = 1, 2, . . . , n, have already been obtained, to find w m ni , i = 1, 2, . . . , n, we apply the nonlinear Jacobi