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

On Convergence of a Three-Layer Semi-Discrete Scheme for the Nonlinear Dynamic String Equation of Kirchhoff-Type with Time-Dependent Coefficients