On the Simulation of Higher Order Linear Block Algorithm for Modelling Fourth Order Initial Value Problems

Abstract

The introduction of new linear block method for the direct simulation of fourth order IVPs has been developed in this article. The reason for adopting direct simulation of fourth order initial value problems is to address some setbacks in reduction method. When developing the method, we adopted the linear block approach through a one step method. We have validated the accuracy of the method on some fourth order initial value problems without reduction process, and the results are better than the conventional method. The numerical experiments were given and the results obtained were found to be better in accuracy than the existing methods in literature.

On the Numerical Approximation of Higher Order Differential Equation

“…The newly derived scheme is consistent according to [3] if all the following conditions are fulfilled i.…”

“…The most common mathematical designed tools by some researchers includes numerical methods such as Euler method, Runge-Kutta (RK) method, Trapezoidal rule and Taylor series method, which are used to solve the first order IVPs. These numerical methods are also used to solve the higher order IVPs indirectly by reducing it to the first order system of equations [2][3][4][5][6]. However, this process is easy to implement but it will increase the number of equations as well as increase the cost for the process [7][8][9].…”

“…[2,10] discourse the computational burden and zero-stability barrier in hybrid block method. The use of block hybrid method to approximate IVPs (1) directly is considered by some researcher in literature such as [3,7,[10][11][12][13][14]. [10] developed the four step hybrid block method for the direct solution of fourth order ordinary differential equations, where he adopt power series as a basic function and directly applied it on fourth order IVPs.…”

Numerical Application of Higher Order Linear Block Method for Solving Some Fourth Order Initial Value Problems

“…The system is arranged from first order to higher order problems [2]. Some examiners in literature have treat (1), without following the reduction method because of its setbacks, among others are [2][3][4][5][6][7].…”

“…Solving fourth order initial value problems of ordinary differential equations without reduction process was developed by [3][4][5][6][7], where [4] proposed a six-step zero-stable block method for solving fourth order initial value problems using power series as a basic function and overcome for the setbacks in reduction methods. [5] developed the block algorithm using the linear block method and his method improved on reduction method.…”