ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.12(2011) No.2,pp.196-204 Hybrid Extended Backward Differentiation Formulas for Stiff Systems Ali K. Ezzeddine 1,2 , Gholamreza Hojjati 2 ∗ 1 Faculty of Arts and science, Lebanese International University, Beirut, Lebanon 2 Faculty of Mathematical sciences, University of Tabriz , Tabriz, Iran (Received 27 April 2011, accepted 29 July 2011) Abstract: In this paper we present details of a new class of hybrid methods which are based on backward differentiation formula (BDF) for the numerical solution of ordinary differential equations. In these methods, the first derivative of the solution in one super future point as well as in one off-step point is used to improve the absolute stability regions. The constructed methods are A(α)–stable up to order 9 so that, as it is shown in the numerical experiments, they are superior for stiff systems. Keywords: stiff ODEs; multistep methods; hybrid methods; stability aspects 1 Introduction The numerical integration of ordinary differential equations has been one of the principal concerns of numerical analysis. Many applications modeled by system of ordinary differential equations exhibit a behavior known as stiffness. Although there has been much controversy about the mathematical definition, simply we can say that the problem y ′ (x)= f (x, y(x)), y(x 0 )= y 0 , (1) on the finite interval I =[x 0 ,x N ] where y :[x 0 ,x N ] → R m and f :[x 0 ,x N ] × R m → R m , is stiff if its Jacobian (in the neighborhood of the solution) has eigenvalues that verify max|Reλi| min|Reλ i | >> 1 (usually it is considered that Reλ i < 0). A potentially good numerical method for the solution of stiff systems of ODEs must has good accuracy and some reasonably wide region of absolute stability [3]. The search for higher order A-stable multistep methods is carried out in the two main directions: • use higher derivatives of the solutions, • throw in additional stages, off-step points, super-future points and like. This leads into the large field general linear methods [6]. Backward differentiation formulas (BDFs) y n+k + k−1 ∑ j=0 α j y n+j = hβ k f n+k , of order k are A-stable up to order 2. Adaptive BDFs [5], blended methods of implicit and explicit BDF, k ∑ j=0 (α j - t ¯ α j )y n+j = hβ k f n+k - ht ¯ β k f n+k−1 , of order k, are A-stable up to order 3. Extended backward differentiation formulas (EBDFs) [1] y n+k + k−1 ∑ j=0 ˆ α j y n+j = h ˆ β k f n+k + h ˆ β k+1 ¯ f n+k+1 , ∗ Corresponding author. E-mail address: ghojjati@tabrizu.ac.ir, ghojjati@yahoo.com. Tel. : +98-411-339 2899; Fax: +98-411- 3342102. Copyright c ⃝World Academic Press, World Academic Union IJNS.2011.10.15/530