Numerical simulation of a dynamic human capital model with demographic delays via the local discrete Galerkin method

Abstract

A strong and dynamic economy depends on various factors, with human capital playing a crucial role in fostering resilience and adaptability in an ever-changing world. Human capital, which depends on the past behavior of the system, requires strategic investments in education, health, and skill development. This study presents a numerical approach for solving the human capital model with age-structured delays, formulated as integro-differential equations with double delays and difference kernels. The proposed method employs a local meshless discrete Galerkin approach based on the moving least squares (MLS) technique, which can work with irregular or non-uniform data. The localized nature of the MLS scheme enhances computational efficiency by focusing on small neighborhoods. Moreover, the stabilized MLS framework, achieved by using shifted and scaled polynomial basis functions, enhances numerical stability and reduces sensitivity to the distribution of nodes, thereby transferring these advantageous properties to the method. The simplicity of the proposed algorithm makes it easy to implement on standard personal computers and extend to a wider class of delay integro-differential equations. To assess its reliability, we analyzed its error and determined the convergence order of the presented method. We have applied it to solve several numerical examples, and the obtained results confirm the method's accuracy, stability, and alignment with theoretical findings.