An innovative shifted generalized Chebyshev collocation scheme for pantograph-type delay equations

Delay differential equations (DDEs) are recognized as a significant class of functional differential equations, arising naturally in various scientific and engineering disciplines such as biology, physics, and control systems. In this study, a novel collocation framework is proposed for solving generalized pantograph-type differential equations (PTDEs). Furthermore, the methodology is extended to handle systems of second-order DDEs effectively. The approach is constructed upon a set of carefully chosen basis functions derived from shifted generalized Chebyshev polynomials (SGCPs), whose mathematical properties underpin the theoretical foundation of the proposed algorithms. A rigorous error analysis is provided to assess the accuracy of the SGCP-based expansion. To demonstrate the robustness and practicality of the method, a set of illustrative examples is included, highlighting the effectiveness and reliability of the developed numerical schemes.