HIGH ORDER ACCURACY SPLITTING FORMULAS FOR COSINE OPERATOR FUNCTION AND THEIR APPLICATIONS J. Rogava, M. Tsiklauri I. Vekua Institute of Applied Mathematics, Iv. Javakhishvili Tbilisi State University 0143 University Street 2, Tbilisi, Georgia (Received: 14.03.10; accepted: 12.10.10) Abstract In the present work the high order accuracy rational splitting for cosine operator function is constructed. On the basis of this formula, the fourth order of accuracy decomposition scheme for homogeneous abstract hyperbolic equation with operator A is constructed. This operator is a self-adjoint, positive definite operator and is represented as a sum of the same type operators. Error of approximate solution is estimated. In the work a method for constructing any order accuracy splitting formula for cosine operator function is also introduced. Key words and phrases: Cosine Operator Function; Decomposition Scheme; Op- erator Split; Abstract Hyperbolic Equation; Rational Approximation. AMS subject classification: 65M12, 65M15, 65M55 1 Introduction Let A is a self-adjoint, positive definite operator and be represented as a finite sum of the same kind of operators. In the present work the high order accuracy rational splitting for cos ( τA 1/2 ) operator function (obviously, ac- curacy of the splitting formula is understand with respect to parameter τ ) is constructed. Our goal is also to introduce the method for constructing 2p + 2 order accuracy splitting formula on the basis of 2p order accuracy formulas. Finally, on the basis of these formulas, we aim to construct high order accuracy decomposition schemes for abstract hyperbolic equation. As is known, the solution of Cauchy problem for an abstract hyperbolic equation can be given by means of sine and cosine operator functions, where square root from the main operator is included in the argument. Using this formula, for the equally distanced values of time variable, the precise three- layer semi-discrete scheme can be constructed whose transition operator is a cosine operator function. On the basis of this relation we can obtain decomposition scheme for an abstract hyperbolic equation. Of course, for this purpose it is necessary to replace cosine operator function by splitting formula. In the present work, using the fourth order of accuracy splitting