Newton Polynomial Collocation Method for Linear Differential Equations

Newton polynomials are used to develop a collocation method for the first time. The fundamental idea of our technique is to estimate the solution by employing Newton polynomials and collocating at Legendre-type points. Newton polynomials have been used to approximate the solution as a series. The roots of the Legendre orthogonal polynomial have been used as collocation points to ensure a closer estimation to the analytical solution of the given problem. The presented method brings down the ordinary differential equation into a set of algebraic equations. We have numerically computed several examples to demonstrate the validity and applicability of the proposed method. The numerical outcomes are collated with the solutions of already presented techniques to highlight the effectiveness of the proposed technique. We have tabulated the absolute point-wise errors, maximum absolute errors, relative errors, and reference errors at different points in the domain for different numbers of collocations to exhibit the convergence of our method.