IMPLICIT MULTI - DERIVATIVE, LINEAR MULTI - STEP METHODS FOR NUMERICAL SOLUTION OF FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

Abstract

This work considered the development, analysis and implementation of a class or implicit multi - derivative linear multi - step methods of the form: ∑_(k=0)^k▒〖a, yₙ₊ᵢ〗^ = ∑_(k=1)^ᶩ▒〖hᶩ ,〗^ ∑_(k=0)^ᶩ▒〖βₙ yͥₙ₊aᵢ=+1〗^ with local truncation error Tn+1; defined as Tₙ^+ₖ=∑_(k=0)^ɭ▒〖aᵢ〗^ Y〖ₙ₊ꙇ〗^ -∑_(Ꙇ=1)^ɭ▒հᶩ ∑_(l-0)^k▒〖βₙ〗 Уᶩ ₙ₊ꙇ for the solution of initial value problems of First Order Ordinary Differential Equations of the form: y=f(x,y ),y(x_0 )=y_(0,) a≤x ≤h The development of the methods adopts the Taylor Series expansion of the functions Уₙ₊ꙇ, Уₙ₊ꙇ and Уₙ₊ꙇ. Accuracy of order P is then imposed on Tₙ^+ₖ. The resulting equations are solved for parameters aᵢˡ ̇⁵ and βˡ ̇⁵to generate the required methods (schemes). The analysis of the basic properties of the methods such as the order of accuracy, consistency, convergence and A-stability were carried out. The results showed that the methods are accurate and absolutely stable (A - stable). The methods are implemented on a digital computer adopting FORTRAN programming language. The programmes are used to solve some sample first order initial value problems. The results showed that the schemes are accurate and convergent. The developed methods are compared with some standard linear multi-step methods like Adams Moulton's and Addison's methods, for which the results showed that the methods are accurate and efficient.