Engineering Applications of Polynomial Matrix Method: A Review

The modeling of natural phenomena is based on ordinary and partial differential equations, which appear in all branches of science and engineering. For this reason, applied mathematicians and engineers have tried to develop new methods of solutions to these equations. One of the widely used numerical methods for the solution of ordinary and partial differential, integro-differential, and integral equations is the Polynomial Matrix Method (PMM). In this study, these methods are introduced first and then a brief history of the development of the method is given. Almost 30 polynomials used in this collocation approach are mentioned. Fundamental principle of the PMM is explained. Engineering applications such as in single and multi-degree of freedom systems, mechanical vibrations, heat equations, diffusion equation and others are reviewed.