Application of Backward Differentiation Formula on Fourth-Order Differential Equations

Abstract

Higher order ordinary differential equations are typically encountered in engineering, physical science, biological sciences, and numerous other fields. The analytical solution of the majority of engineering problems involving higher-order ordinary differential equations is not a simple task. Various numerical techniques have been proposed for higher-order initial value problems (IVP), but a higher degree of precision is still required. In this paper, we propose a novel two-step backward differentiation formula in the class of linear multistep schemes with a higher order of accuracy for solving ordinary differential equations of the fourth order. The proposed method was created by combining interpolation and collocation techniques with the use of power series as the basis function at some grid and off-grid locations to generate a hybrid continuous two-step technique. The method's fundamental properties, such as order, zero stability, error constant, consistency, and convergence, were explored, and the analysis showed that it is zero stable, consistent and convergent. The developed method is suitable for numerically integrating linear and nonlinear differential equations of the fourth order. Four Numerical tests are presented to demonstrate the efficiency and accuracy of the proposed scheme in comparison to some existing block methods. Based on what has been observed, the numerical results indicate that the proposed scheme is a superior method for estimating fourth-order problems than the method previously employed, confirming its convergence.