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 temporal 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 concerning 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：本文研究含时变系数的基尔霍夫型非线性动力学弦方程的柯西问题。本文旨在开发一种时间离散算法，以近似求解该初边值问题。为此，针对时间变量采用对称三层半离散格式，其中非线性项的值在中点处进行评估。该方法通过求解线性算子逆运算，得到每个时间步的数值解，进而形成一个二阶线性常微分方程组。证明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。我们针对多个测试问题运用该算法进行了数值实验，以验证其性能。所得数值结果与理论结论一致。