Spectral-block hybrid method for high-order numerical solution of Volterra integro-differential equations

摘要

This paper focuses on the numerical solution of first-order Volterra integro-differential equations (VIDEs), which frequently arise in science and engineering to describe systems with memory and hereditary effects. Conventional numerical approaches often exhibit limited accuracy or increased computational cost when dealing with such problems. To address these challenges, a
Spectral-Block Hybrid Method (SHBM) is proposed for the high-order approximation of VIDEs. The method employs shifted Legendre polynomials as basis functions for the spectral representation of the approximate solution, while Simpson’s 3/8 quadrature rule is utilized to evaluate the integral term within each block efficiently. The proposed SHBM combines the global accuracy of spectral methods with the parallel efficiency of block formulations, enabling multiple solution points to be computed simultaneously. This integration enhances accuracy, stability, and computational efficiency. Numerical experiments show that the SHBM yields significantly smaller absolute errors and reduced CPU times compared with existing methods in literature. The results confirm that SHBM maintains strong convergence and numerical stability as the step-size decreases. Given its precision and efficiency, the SHBM is a promising tool for solving practical integro-differential problems encountered in fields such as population dynamics, viscoelastic material modeling, heat transfer, and biological systems with memory.