IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 4 Ser. III (Jul.–Aug. 2020), PP 01-13 www.iosrjournals.org DOI: 10.9790/5728-1604030113 www.iosrjournals.org 1 | Page Construction of a Family of Stable One-Block Methods Using Linear Multi-Step Quadruple 1 Ajie, I.J.; 2 Durojaye M.O.; 1 Utalor, K and 1 Onumanyi, P 1 (Mathematics Programme, National Mathematical Centre, Abuja, Nigeria) 2 (Department of Mathematics, University of Abuja, Abuja, Nigeria) Abstract Background: This paper deals with the construction of a family of implicit one-block methods for the solution of stiff problems using four different linear multistep methods. Method: This is done by applying shift operator on the quadruple: Reversed Generalized Adams Moulton (RGAM), Generalized Backward Differentiation Formula (GBDF), Top Order Method (TOM) and Backward Differentiation Formula (BDF). Results: The application of the shift operator on the quadruples is done in such a manner that the resultant one- block methods are self-starting and forms a family. Orders four and seven are L-stable. Conclusion: Numerical experiments carried out using orders four, seven and ten of the family show that the methods are good for solving stiff initial value problems. Keywords: Stiff initial value problem; One-block methods; Self-starting; quadruple and shift operator. --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 17-07-2020 Date of Acceptance: 01-08-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction This paper deals with the construction of methods for finding the numerical solution ) (t y to the stiff initial value problems (sivp) in ode m m m y f b a t y t y t y t f t y             : ; : ; ] , [ ; ) ( ; )) ( , ( ) ( 0 0 (1.1) The problem in (1.1) can only be handled adequately by high order A-Stable methods. These high order A- Stable methods are difficult to come by due to the severe restrictions imposed by Dalquist order barrier theorem 7 . To circumvent this barrier, unconventional means were adopted by many researchers to achieve high order numerical integrators to handle (1.1). These include but not limited to: boundary value methods 3, 4 ; second derivative methods 6, 8 ; implicit two points numerical integration formula 9 , general linear methods 5 , second derivative general linear methods 14 and rational one-step numerical integrators 16 . The use of collocation and interpolation in the construction of some linear multistep formulas for solving ordinary differential equations has been mostly with two-point boundary value problems until 6 showed the connection to the backward differentiation formula (BDF). Current trend following Onumanyi et al 15 have extended this connections to many families of traditional linear multistep methods, including boundary value methods (BVMs) 4 possessing good stability properties suitable for efficient solutions of (1.1). Notwithstanding these desirable developments, the cumbersomeness in the construction process is a drawback and need to be eliminated. This paper will approach the construction of continuous linear multistep formulas from the perspective of the order definition. The already known families that will be used in this paper are: Reversed Adams Moulton (RAM) methods, Generalized Backward Differentiation Formulas (GBDF), Top Order Methods (TOM) and Backward Differentiation Formulas (BDF). These four families will be used to demonstrate both the construction of the continuous linear multistep formulas and the new family of methods which this paper is proposing.