A Review of Polynomial Matrix Collocation Methods in Engineering and Scientific Applications

Abstract Ordinary, partial, and integral differential equations are indispensable tools across diverse scientific domains, enabling precise modeling of natural and engineered phenomena. The polynomial collocation method, a powerful numerical technique, has emerged as a robust approach for solving these equations efficiently. This review explores the evolution and applications of the collocation method, emphasizing its matrix-based formulation and utilization of polynomial sequences such as Chebyshev, Legendre, and Taylor series. Beginning with its inception in the late 20th century, the method has evolved to encompass a wide array of differential equation types, including integro-differential and fractional equations. Applications span mechanical vibrations, heat transfer, diffusion processes, wave propagation, environmental pollution modeling, medical uses, biomedical dynamics, and population ecology. The method’s efficacy lies in its ability to transform differential equations into algebraic systems using orthogonal polynomials at chosen collocation points, facilitating accurate numerical solutions across complex systems and diverse engineering and scientific disciplines. This approach circumvents the need for mesh generation and simplifies the computational complexity associated with traditional numerical methods. This comprehensive review consolidates theoretical foundations, methodological advancements, and practical applications, highlighting the method’s pivotal role in modern computational mathematics and its continued relevance in addressing complex scientific challenges.