Convergence Analysis and Error Propagation of the Laplace Residual Power Series Method for Linear Delay Matrix Differential Equations

Abstract

Matrix differential equations with time delay are crucial to the modeling of complex multivariate systems. However, the existing semi-analytic Laplace residual power series method (LRPSM) literature mainly focuses on scalar or no-delay problems, and lacks rigorous theoretical guarantees for matrix-valued time delay systems. This study systematically generalizes LRPSM to the linear time-delay matrix differential equation X′(t)=AX(t)+BX(t−τ)+F(t) , where X(t)∈Rn×n, A and B are constant matrices, τ>0 is a constant delay, and the forcing term F(t) and the history function Φ(t) are analytic. The method of steps is employed to construct the solution piecewise: on each local interval, the solution is expanded asymptotically in the Laplace domain, and the coefficients are determined recursively via the Laplace residual function. We establish local error bounds on the initial interval and derive a global error propagation bound across successive delay interfaces using a variation-of-constants framework. Numerical experiments, including non-diagonal matrices, non-zero history functions, and multi-interval tests, illustrate the effectiveness of the approach. The proposed method reduces exactly to the standard LRPSM for scalar cases, demonstrating its validity as a natural and rigorous generalization of the existing semi-analytical framework.