This paper considers the Cauchy problem for the nonlinear dynamic string equation of Kirchhoff-type with time-varying coefficients. The objective of this work is to develop a time domain discretization algorithm capable of approximating a solution to this initial-boundary value problem. To this end, a symmetric three-layer semi-discrete scheme is employed with respect to the temporal variable, wherein the value of a nonlinear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second-order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings.

翻译：本文研究了具有时变系数的Kirchhoff型非线性动力弦方程的柯西问题。本文旨在开发一种能够逼近该初边值问题解的时域离散算法。为此，对时间变量采用对称的三层半离散格式，其中非线性项的值在中节点处进行求值。该方法使得每个时间步的数值解可通过求逆线性算子得到，从而生成一组二阶线性常微分方程。建立了所提格式的局部收敛性，并在局部时间区间上实现了关于时间离散步长的二次收敛。我们使用所提算法针对多个测试问题进行了数值实验以验证其性能。可以认为，所获得的数值结果与理论结果一致。