Chebyshev Polynomials and Their Applications in Numerical Analysis

Abstract

Differential equations play a crucial role in modeling diverse phenomena in science and technology, including growth, decay, epidemics, and oscillatory systems. Number-based methods, especially for solving first-order ordinary differential equations (ODEs), have grown because people want quick and correct answers to these equations. The main goal of this study is to come up with and evaluate a numerical method for using Chebyshev polynomials of the first kind to solve general first-order ODEs. Chebyshev polynomials are famous for being orthogonal and having uses in approximation theory. They are also used to build continuous linear multistep methods (LMMs). These methods utilize interpolation and collocation techniques to achieve high accuracy. The Chebyshev method is systematically derived and analyzed for order, consistency, zero-stability and convergence. Employing boundary locus methods, the stability region is determined, affirming the method's robustness. The numerical applications include solving problems such as the Malthusian growth model, Prothero-Robinson oscillatory equations and highly stiff differential equations. Comparisons of absolute errors with existing methods highlight the Chebyshev method's superior accuracy and efficiency. Results show minimal errors across various test cases, confirming its applicability to real-world problems involving oscillatory and stiff systems. This study contributes to advancing numerical analysis by providing a reliable, efficient tool for solving initial value problems. The findings emphasize the significance of Chebyshev polynomials in enhancing the accuracy and stability of numerical schemes, offering potential for broader applications in science and engineering disciplines.