IMPLICIT BLOCK METHODS WITH EXTRA DERIVATIVES FOR SOLVING GENERAL HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH APPLICATIONS

Abstract

Traditionally, higher order ordinary differential equations (ODEs) are solved by reducing them to an equivalent system of first order ODEs. However, it is more cost effective if they can be solved directly by numerical methods. Block methods approximate the solutions of the ODEs at more than one point at one time step, hence faster solutions can be obtained. It is well-known too that a more accurate numerical results can be obtained by incorporating the higher derivatives of the solutions in the method. Based on these arguments, we are focused on developing block methods with extra derivatives for directly solving second, third and fourth order ODEs. In this study, two-point and three-point implicit block methods with extra derivatives are derived using Hermite Interpolating polynomial as the basis function. The technique of integration is used in the derivation as it is more straight forward and can easily be carried out compared to the existing technique of collocation and interpolation in which the points need to be collocated and interpolated resulting in a huge system of linear equations which need to be solved simultaneously. The thesis consists of three parts, the first part of the thesis described the derivation of two-point and three-point implicit block methods which incorporated the second, third and fourth derivatives of the solution for directly solving general second order ODEs. Absolute stability for both block methods is presented. The second part of the thesis is focused on the derivation of two-point and three-point implicit block methods which include the third, fourth and fifth derivatives of the solutions for directly solving general third order ODEs. The last part of the thesis concerned with the construction of two-point and three-point implicit block method which involved the fourth and the fifth derivatives of the solution for directly solving general fourth order ODEs. The basic properties of all the methods, such as algebraic order, zero-stability, and convergence are established. Numerical results clearly show that the new proposed methods are more efficient in terms of accuracy and computational time when compared with well-known existing methods. Applications in several real fields also illustrate the efficiency of the proposed methods. In conclusion, the new block methods with extra derivatives and codes developed based on the methods are suitable for solving second, third and fourth order ODEs respectively and can be applied to solve physical problems.