Collocation method for Volterra integro-differential equations with piecewise delays

This paper provides a rigorous framework for collocation methods applied to Volterra integro-differential equations (VIDEs) with piecewise linear delays, addressing important gaps in managing discontinuous delay transitions and solution regularity. The proposed method uniquely combines quasi-$\theta$-invariant meshes with polynomial spline spaces to handle discontinuities induced by delays while preserving optimal convergence. First, the collocation method is extended to VIDEs with piecewise linear delays. Then, the existence and uniqueness of the solution is proved. Convergence analysis is also investigated. At the end, four examples, including two practical examples, are given to illustrate the accuracy and efficiency of the proposed method and confirmation the theoretical results.