We consider a boundary-value problem for the nonlinear integrodifferential equation u ′ ′ ′ ′ − m ∫ 0 l u ′ 2 dx u ″ = f x u u ′ , m z ≥ α > 0 , 0 ≤ z < ∞ , $$ {u}^{\prime \prime \prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, $$ simulating the static state of the Kirchhoff beam. The problem is reduced to a nonlinear integral equation, which is solved by using the Picard iterative method. The convergence of the iterative process is established and the error is estimated.

中文翻译：

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非线性静力梁方程的迭代求解

我们考虑非线性积分微分方程的边值问题 u ′ ′ ′ ′ − m ∫ 0 lu ′ 2 dx u ″ = fxuu ′ , mz ≥ α > 0 , 0 ≤ z < ∞ , $$ {u}^{ \prime \prime \prime \prime }-m\left(\underset{0}{\overset{l}{\int }}{u}^{\prime 2} dx\right){u}^{{\ prime\prime} }=f\left(x,u,{u}^{\prime}\right),\kern1em m(z)\ge \upalpha >0,\kern1em 0\le z<\infty, $ $ 模拟基尔霍夫光束的静态状态。将问题简化为非线性积分方程，采用Picard迭代法求解。建立迭代过程的收敛性并估计误差。