Research Information System
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.