An Efficient Block Multistep Method for the Numerical Approximation of General Fourth-Order Ordinary Differential Equations

Abstract

Fourth-order ordinary differential equations (ODEs) are applied in real-life situations such as analyzing the vibration and stability of structural elements, including beams, plates, and airplane wings. Other applications include modeling fluid flow, such as in the lungs, and the development of surface profiles in material science. In engineering, they are used in areas such as beam theory to predict beam failure and in analyzing the stress and strain in materials like reinforced concrete shells. The purpose of this study is to develop a class of continuous hybrid numerical methods for the direct solution of general fourth-order initial value problems of ordinary differential equations. The technique adopted in this work involves interpolation and collocation of a basis function and its corresponding differential system, respectively. The differential systems and the basis functions are collocated and interpolated, respectively, at selected grid and intra-step grid points. The unknown parameters in the system of linear equations arising from the collocation and interpolation procedures were determined, and the values were substituted into the approximate solution. The required continuous methods were obtained for different step numbers after the necessary simplifications. The derived methods were tested and found to be consistent, convergent, and to possess low error constants. The discrete schemes obtained from the continuous methods were implemented in block mode. The methods were applied to solve linear and nonlinear fourth-order initial value problems directly. The errors in the results obtained were compared with those of existing methods of the same and even higher order of accuracy to establish the superiority of the newly proposed method.