Kings University OER

This work describes the development of a class of implicit one-step multi-derivative methods of the form; n j i j ij j l i i j n j y h β y + = = = ? ? + = ? 1 1 0 1 α ’ 1 α1 = + with local truncation error T α y h β y j n j i ij l i i n j j n ? j ? ?= + = + = + = − 1 1 0 1 1 Also, it investigates the pattern of the order of accuracy of the methods with varying order of derivative. The development of the methods adopts the Taylor Series expansion of the functions n j y + , y n+ j ' , y n+ j '' , ---- . Accuracy of order P is imposed on Tn+1 and the resulting equations are solved for parameters αj’ s and βij’ s to generate the required methods (schemes). The order of accuracy and error constants of the methods are determined by substituting the results of parameters αj’ s and βij’ s into the original equations. The methods were implemented by using them to solve some initial value problems of first order ordinary differential equations in order to establish the pattern of the accuracy. The result showed that the order of accuracy of the methods increased as the order of the derivative increased with optimum accuracy obtained at l =3.