AN EXPONENTIAL METHOD OF VARIABLE ORDER FOR GENERAL NONLINEAR (STIFF AND NONSTIFF) ODE SYSTEMS

C. C. Jibunoh

Abstract


In this paper, an explicit method, hereby called an Exponential Method of variable order, is derived from the earlier published Exponential Method of orders 2 and 3. The present method of variable order commands higher accuracy since it obtains numerical solutions which coincide with the exact theoretical solutions, to eight or more decimal places, in virtually all stiff and nonstiff, (linear and nonlinear) ODE systems. Numerical applications show that it has faster convergence and much higher accuracy than many existing methods. New formats are now introduced to make it easy to integrate any $K \times K$ systems. Other remarkable features include the use of the exact Jacobians of nonlinear systems; implementation of a phase to phase integration of stiff systems, with exact formulas for determining the terminal points of phases; avoidance of matrix inversions, LU decompositions and the cumbersome Newton iterations, since the method is explicit; solving oscillatory systems without additional refinements and a straight forward application of the method without starters. Implementations show that any program of the Exponential Method of variable order (e.g the QBASIC program) produces a very fast or instant output in automatic computation.

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