Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1635-1646 © Research India Publications http://www.ripublication.com Implementing an Order Six Implicit Block Multistep Method for Third Order ODEs Using Variable Step Size Approach Jimevwo G. Oghonyon 1* , Nicholas A. Omoregbe 2 , Sheila A. Bishop 1 1 Department of Mathematics, College of Science and Technology, Covenant University, Ota, Ogun State, Nigeria. 2 Department of Computer and Information Sciences, College of Science and Technology, Covenant University, Ota, Ogun State, Nigeria. Abstract In this paper, an order six implicit block multistep method is implemented for third order ordinary differential equations using variable step size approach. The idea which originated from Milne’s possess a number of computational vantages when equated with existing methods. They include; designing a suitable step size/changing the step size, convergence criteria (tolerance level) and error control/minimization. The approach employs the estimates of the principal local truncation error on a pair of explicit and implicit of Adams type formulas which are implemented in P(CE) m mode. Gauss Seidel method is adopted for the execution of the suggested method. Numerical examples are given to examine the efficiency of the method and will be compared with subsisting methods. Keywords and phrase: Variable step size approach • Implicit block multistep method • convergence criteria (tolerance level) • Gauss Seidel Method • Principal Local Truncation Error • Adams type formulas 1. Introduction Consider the initial value problem of the form ( ) y y y x f x y ' ' , ' , , = ) ( ' ' ' , α a y = ) ( , β a y = ) ( ' , ψ a y = ) ( ' ' , [ ] b a x , ∈ and R m R m R f → × : . (1) The solution to (1) is generally, written as