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13  Numerical Stability Study of the Explicit-Implicit Range-Kuta Synthesis Methods (IMEX-ARK) Thair Y. Thanoon Younis Abd Hussain thairthanoon@uomosul.edu.iq College of Computer Sciences and Mathematics University of Mosul Received on:9/11/2004 Accepted on:5/4/2005 ABSTRAC  In this research we studied the numerical stability for ImplicitExplicit Additive Runge-Kutta (IMEX-ARK) methods which have the A[ ]-stability .This stability is equivalent A-stability of (B( )-ARK) methods where B( )= B1+(1- )B2 of second and third order for different intervals of values .This methods are suitable for solving stiff non-linear differential equations which has the form :