J. Peradze

I. Javakhishvili Tbilisi State University Tbilisi, Georgia

THE SOLVABILITY OF A NONLINEAR INITIAL BOUNDARY VALUE PROBLEM FOR A BEAM AND THE ALGORITHM OF CONSTRUCTION OF ITS APPROXIMATE

SOLUTION

The problem

\begin{matrix}\frac{∂^{2}w}{∂t^{2}} = (cd − a + b ∫_{0}^{1} (\frac{∂w}{∂x} )^{2} dx) \frac{∂^{2}w}{∂x^{2}} − cd \frac{∂ψ}{∂x} , \\ \frac{∂^{2}ψ}{∂t^{2}} = c \frac{∂^{2}ψ}{∂x^{2}} − c^{2}d(ψ − \frac{∂w}{∂x} ), \\ 0 < x < 1, 0 < t ≤ T, \end{matrix}  (1)

\frac{∂^{s}w}{∂t^{s}} (x, 0) = w^{s}(x), \frac{∂^{s}ψ}{∂t^{s}} (x, 0) = ψ^{s}(x), s = 0, 1,  (2)
w(0, t) = w(1, t) = 0, \frac{∂ψ}{∂x} (0, t) = \frac{∂ψ}{∂x} (1, t) = 0,  (3)

describing the vibration of a beam with hinged ends, is considered in the Timoshenko model. Here a, b, c, d, T are the given positive constants, cd − a > 0, and w^{s}(x), ψ^{s}(x) are the given functions, s = 0, 1.

Assuming that w^{s}(x) and ψ^{s}(x) are analytic functions of the forms

w^{s}(x) = ∑_{i=1}^{∞} a_{i}^{s} sin iπx, ψ^{s}(x) = \frac{b_{0}^{s}}{\sqrt{2}} + ∑_{j=1}^{∞} b_{j}^{s} cos jπx, s = 0, 1,

and applying S. Bernstein’s approach, it is proved that there exist functions w(x, t) and ψ(x, t) which are a solution of problem (1)–(3) and which are analytic functions with respect to x for all 0 < t ≤ T.

Further, a numerical algorithm of finding a solution of problem (1)–(3) is proposed. This algorithm consists in representing the solution as

w_{n}(x, t) = ∑_{i=1}^{n} w_{ni}(t) sin iπx, ψ_{n}(x, t) = \frac{ψ_{n0}(t)}{\sqrt{2}} + ∑_{j=1}^{n} ψ_{nj}(t) cos jπx

and using Galerkin’s method. As a result, we obtain the Cauchy problem for a system of ordinary differential equations with respect to functions w_{ni}(t) and ψ_{nj}(t), i = 1, 2, …, n, j = 0, 1, …, n, which is solved by means of the Crank–Nicholson implicit difference scheme. Therefore on the time grid layers we have to solve a system of nonlinear algebraic equations. For this we use the Picard iteration method. The error of each of three constituent parts of the algorithm is estimated.