In this paper, we consider a boundary value problem (BVP) for a fourth-order nonlinear integro-differential equation. By reducing the problem to an operator equation, we establish the existence and uniqueness of the solution and construct a numerical method for solving it. We prove that the method is of second-order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

中文翻译：

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求解四阶非线性积分微分方程的存在性结果及数值方法

在本文中，我们考虑四阶非线性积分微分方程的边值问题 (BVP)。通过将问题简化为算子方程，我们建立了解的存在唯一性，并构造了求解它的数值方法。我们证明该方法具有二阶精度并获得总误差的估计。一些算例证明了所得理论结果的有效性和数值方法的有效性。