Spectral Tau Method with Legendre Polynomials to Approximate Second Order Boundary Value Problem
Nur Sariza Mat Salleh · International Journal of Academic Research in Accounting Finance and Management Sciences · 2024
Spectral methods also known as the Method of Weighted Residuals (MWR) are commonly used in many fields such as Mathematics, Engineering, Physics and others.This method is global smooth functions, usually by high order polynomials which differ from the finite element and finite difference which are local smooth functions, usually by low order polynomials.The most popular spectral methods that are commonly used by researchers are Tau, Collocation and Galerkin methods.Since not all the differential equations can be solved analytically, therefore, the numerical solution of the Legendre Tau method is presented.In this study, the Legendre Tau method is proposed and the comparison with the Chebyshev Tau method has been presented.The objectives of this study are to approximate the second order Boundary Value Problem (BVP) using Spectral Tau method by using the Legendre polynomials as the basis function and to make a comparison between the Legendre Tau method with Chebyshev Tau method.The accuracy of the Legendre Tau method is also presented by calculating the absolute error.Besides, the efficiency of both methods was proposed in this study by calculating their CPU times.Previous literature shows that many researchers approximated differential equations using Chebyshev Tau method while the Legendre Tau method has never been used before.The numerical structures established in this study are in line with solutions attained with renowned and standard spectral methods.To validate the results and claim, several test problems were presented in this study.