Generalized Lucas polynomial sequence approach for solving Delay differential equation of fractional order

Abstract

In this paper, the Generalized Lucas Polynomial Sequence method is proposed as an efficient numerical technique for solving fractional-order delay differential equations involving both the modified Riemann–Liouville and Caputo fractional derivatives. The fractionally damped mechanical oscillator is considered as a representative application to demonstrate the effectiveness of the proposed approach. By expanding the unknown solution in terms of generalized Lucas polynomials, the original fractional delay differential equation is transformed into a system of algebraic equations that can be solved efficiently. The proposed method is computationally simple, provides high numerical accuracy, and exhibits good convergence characteristics. The obtained results confirm that the Generalized Lucas Polynomial Sequence approach is a reliable and effective tool for solving a broad class of fractional delay differential equations arising in various scientific and engineering applications