Advancements in Polynomial Matrix Collocation Methods for Engineering and Scientific Applications

Baykuş Savaşaneril N.

10TH INTERNATIONAL "AZERBAIJAN" CONGRESS ON LIFE, ENGINEERING, MATHEMATICAL, AND APPLIED SCIENCES, Baku, Azerbaijan, 13 - 15 March 2025, pp.12-24, (Full Text)

  • Publication Type: Conference Paper / Full Text
  • Doi Number: 10.5281/zenodo.1261529
  • City: Baku
  • Country: Azerbaijan
  • Page Numbers: pp.12-24
  • Dokuz Eylül University Affiliated: Yes

Abstract

Ordinary, partial, and integral differential equations serve as indispensable tools across various scientific disciplines, enabling precise modeling of natural and engineering systems. The polynomial collocation method, a powerful numerical technique, has emerged as a robust approach for efficiently solving these equations.

The application areas of this method span multiple scientific and engineering fields, including mechanical vibrations, heat transfer, diffusion processes, wave propagation, environmental pollution modeling, medical applications, biomedical dynamics, and population ecology.

The effectiveness of this method lies in its ability to transform differential equations into algebraic systems by utilizing orthogonal polynomials at selected collocation points. This approach ensures accurate numerical solutions for complex systems while simplifying computational complexity by eliminating the need for mesh generation, which is typically required in traditional numerical methods.

This study consolidates theoretical foundations, methodological advancements, and practical implementations, highlighting the critical role of polynomial collocation methods in modern computational mathematics and their ongoing significance in addressing scientific problems.