Scientific Research Reports

Duvar Publications, İzmir, 2024

Various polynomial methods employed in solving differential equations are examined. These methods,

founded on Taylor polynomials, offer a wide range of applications, from high-order linear differential

equations to integro-differential and delay differential equations. The Taylor polynomial approach has

been effectively applied in numerous areas, as demonstrated by the work of Sezer and colleagues. In

addition to the Taylor polynomial approach, several other polynomial methods have been developed,

including the Bernoulli matrix method, Lucas polynomials, and Euler polynomials. These methods offer

alternative strategies for solving both linear and nonlinear differential equations, particularly those

involving variable delays or nonlinear terms. The Lucas polynomial approach is particularly effective

in solving nonlinear differential equations with variable delay and functional integro-differential

equations. In the field of mechanical vibrations, polynomial methods have also been applied to solve

single degree of freedom systems. Contributions from several researchers have demonstrated the

applicability of Euler and Taylor polynomial methods in this field. Moreover, the hybrid use of

polynomial methods with collocation techniques has improved computational efficiency and accuracy.

For instance, the hybrid Taylor-Lucas collocation method offers a robust solution for high-order

pantograph-type delay differential equations. Notably, advancements in polynomial methods play a

critical role in the numerical solution of differential equations. These methods not only provide accurate

and efficient solutions but also have broad applications across various scientific and engineering

disciplines. This study provides a detailed examination of the theoretical foundations and practical

implementations of different polynomial methods.