Iterative solution of a nonlinear static beam equation

The paper deals with a boundary value problem for the nonlinear integro-differential equation $u^{\prime\prime\prime\prime}-m\left(\int_0^l {u^\prime}^2dx\right)u^{\prime\prime}=f(x,u,u^\prime), \; m(z)\geq \alpha>0, \; 0\leq z<\infty$, modelling the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation which is solved using the Picard iteration method. The convergence of the iteration process is established and the error estimate is obtained.


Statement of the Problem
Let us consider the nonlinear beam equation with the conditions u(0) = u(l) = 0, u (0) = u (l) = 0.
(2) Here u = u(x) is the displacement function of length l of the beam subjected to the action of a force given by the function f (x, u, u ), the function m(z), describes the type of a relation between stress and strain. Namely, if the function m(z) is linear, this means that this relation is consistent with Hooke's linear law, while otherwise we deal with material nonlinearities. Equation (1) is the stationary problem associated with the equation m(z) ≥ const > 0, which for the case where m(z) = m 0 + m 1 z, m 0 , m 1 > 0, and f (x, t, u, u x ) = 0, was proposed by Woinowsky-Krieger [11] as a model of deflection of an extensible dynamic beam with hinged ends. The nonlinear term l 0 u 2 x dx was for first time used by Kirchhoff [3] who generalized D'Alembert's classical linear model. Therefore (1) is frequently called a Kirchhoff type equation for a static beam.
The problem of construction of numerical algorithms and estimation of their accuracy for equations of type (1) is investigated in [1], [5], [8] and [9]. In [4], the existence of a solution of problem (1), (2) is proved when the right-hand part of equation is written in the form q(x)f (x, u, u ), where f ∈ C([0, l] × [0, ∞) × R) is a nonnegative function and q ∈ C[0, l] is a positive function.
In the present paper, in order to obtain an approximate solution of the problem (1),(2), an approach is used, which differs from those applied in the above-mentioned references. It consists in reducing the problem (1),(2) by means of Green's function to a nonlinear integral equation, to solve which we use the iterative process. The condition for the convergence of the method is established and its accuracy is estimated.
The Green's function method with a further iteration procedure has been applied by us previously also to a nonlinear problem for the axially symmetric Timoshenko plate [6].

Assumptions
Let us assume that besides (3) the function m(z) also satisfies the Lipschitz condition Suppose that f (x, u, v) ∈ L 2 ((0, l), R, R) and, additionally, that the inequalities We impose one more restriction on the beam length l and the parameters α and σ 2 (x), σ 3 (x) from the conditions (3) and (4), (5) in the form Let us assume that there exists a solution of the problem (1),(2) and u ∈ W 2,2 0 (0, l) [2].

The Method
We will need the Green function for the problem In order to obtain this function, we split problem (7) into two problems Substituting the first of these formulas into the second and performing integration by parts, we obtain The application of (7) to problem (1), (2) makes it possible to replace the latter problem by the integral equation where The equation (8) is solved by the method of the Picard iterations. After choosing a function u 0 (x), 0 ≤ x ≤ l, which together with its second derivative vanish for x = 0 and x = l, we find subsequent approximations by the formula and u k (x) is the k th approximation of the solution of equation (8).

The Equation for the Method Error
Our aim is to estimate the error of the method, by which we understand the difference between the approximate and exact solutions For this, it is advisable to use not formula (9), but the system of equalities which follows from (9). If we subtract the respective equalities in (1) and (2) from (11) and (12), then we get δu k (0) = δu k (l) = 0, δu (0) = δu (l) = 0, k = 1, 2, · · · .
We will come back to (13),(14) to estimate the error of method (9). In meantime we have to derive several a priori estimates.

Auxiliary Inequalities
Let The symbol (·, ·) is understood as a scalar product in L 2 (0, l).

From this relation and (6) follows (18).
Lemma 4. Suppose where given some numbers v k ≥ 0, k = 0, 1, · · · , for which the in- where 0 ≤ a < 1, b > 0, holds. Then we have the following uniform estimate with respect to the index k Proof. By virtue of (20), by the method of mathematical induction we have v k ≤ a k v 0 + (a k−1 + a k−2 + · · · + 1)b, k = 1, 2, · · · , which implies In the first case ν k ≤ 0 and by virtue of (22) v k ≤ b 1 − a , k = 1, 2, · · · . In the second case From this conclusions the validity of estimate (21) follows.
By Lemma 3 and Lemma 5 it will be natural to require that the initial approximatio u 0 (x) in (9) satisfy the condition Then, by virtue of (24) and (23), we have u k (x) 1 ≤ c 1 , which, with (19) taken into account, implies 6. Convergence of the Method Multiplying (13) by δu k (x), integrating the resulting equality with respect to x from 0 to l and using (14), we come to the relation Theorem 1. Let assumptions (3)- (6) and (25) are fulfilled. Suppose besides Then the approximations of the iteration method (9) converge to exact solution of problem (1), (2) and for the error the folloving estimate is true.

Numerical Experiment
The theoretical results about the convergence of approximations of iteration method (9) to exact solution u(x) of problem (1), (2) is confirmed. For illustration, the results of numerical computations of one of the test problems are given below.
We consider a special case, where m (z) = m 0 +m 1 ·z, m 0 , m 1 > 0, m 0 = 1, We carried out five, seven and nine iterations. To compute the integrals on [0, 1] we divided the interval into n = 10, 20 parts (h = 0.1, 0.05, respectively) and used the square formula of trapezoid. The error in the k-iteration is defined as Numerical values for the errors are calculated (see Table 1).
The function u 0 (x) = 0 is taken as the initial approximation. In case of five, seven and nine iterations for n = 10, 20 the exact and approximate solutions are graphically illustrated (Figs. 1-6).
Remark. In the figures the green line color denotes the exact solution graph, yellow is the first approximation, red -the second, blue -the third, pink -the fourth, golden -the fifth, brown -the sixth, purple -the seventh, orange -the eighth and black -the ninth.  The numerical experiments clearly show the convergence of iteration approximate solutions to the exact solution of the problem. The error decreases with the growth of the parameters n and k.