A Chebyshev Derived Numerical Block Method for Solving Fourth-Order ODEs

Abstract

This study presents a numerical block method by using Chebyshev polynomials of the first kind as the basis function for the direct solution of initial-value problems for fourth-order ordinary differential equations, without the need to reduce the problems to a first-order system of differential equations. The method was derived by applying interpolation and collocation procedures to a Chebyshev approximating polynomial. The unknown parameters were obtained using the Gaussian elimination method and then substituted into the approximate solution to obtain the continuous scheme, and the resulting scheme was evaluated at the selected point to yield the discrete scheme. The method derived had an order of seven and was convergent, consistent, and zero-stable, and, as shown by the region of absolute stability, was p-stable. Four numerical examples were solved to test both the accuracy and usability of the derived method. The computational results show that the derived method was highly efficient because it produced low errors. The numerical results of the derived method, when compared with those of the reported works in the literature, are found to be better, as they produce lower errors.