Direct solution of initial and boundary value problems of third order ODEs using maximal-order fourth-derivative block method

Abstract

Improved accuracy has been observed in block methods with the presence of higher derivatives when implemented to solve first order and higher order ordinary differential equations. This improvement in accuracy is as a result of the increased order possessed by the higher derivative block method. In this article, a fourth-derivative block method of maximal-order is introduced to solve third order initial and boundary value problems. The block method possesses convergent properties required for any good numerical method and it is suitable for solving third order ODE models. This is evident in its improved performance over other methods in terms of comparison to the exact solution of the numerical problems considered. INTRODUCTION Numerous real life scenarios can be described as differential equation models, and third order ordinary differential equations (ODEs) of the form y′′′ = f (x, y, y′, y′′) . (1) are not an exception. Equation (1) can be presented with initial conditions defined (initial value problems) or having the presence of boundary conditions (boundary value problems) [1, 2]. In common real-life situations, the differential equations are quite complicated to solve exactly, thus the need to adopt numerical approaches such as finite element methods, finite difference methods, splines methods, collocation methods, differential transform methods, amongst others [3, 4, 5]. However, since the aim of any approximate method is to obtain more accurate solutions, block methods have shown to compete favourably with these previously mentioned numerical approaches [6, 7, 8]. Therefore, the solution of problems in the form of Equation (1) will be presented in this article, using block method. The block method is developed to be of maximal-order 2k + 2 as defined by [9] to ensure improved accuracy. [10] introduced a linear multistep method with the presence of a second-derivative term defined as