INSIGHT INTO DIRECT APPROXIMATION OF GENERAL SECOND ORDER ORDINARY DIFFERENTIAL EQUATIONS: 5-STEP COLLOCATION METHOD

INSIGHT INTO DIRECT APPROXIMATION

Abstract

One of the key advantages of the five-step linear multistep approach is its ability to numerically solve stiff differential equations with remarkable stability, making it an invaluable tool across various scientific and engineering disciplines. Notably, there has been limited exploration of the direct application of the five-step method for approximating general second-order ordinary differential equations. This study addresses that gap by employing collocation and interpolation techniques to develop a five-step method for directly approximating the initial value problems (IVPs) of such equations. The power series serves as the approximate solution framework, with the differential function collocated at all points and the basis function interpolated at points near the evaluation point to derive the required linear multistep method. The proposed method exhibits symmetry, consistency, convergence, and zero stability. Its application to several examples demonstrates superior efficiency, achieving highly accurate results with reduced computational effort.