An exponential matrix function and its application

A new exponential matrix function is introduced to obtain the special solutions, based on Duhamel convolution integration for the non-homogenous polynomial terms of the first order constant coefficient ordinary equation, exponential functions or their products., Based on the new exponential matrix, an efficient recursive algorithm is proposed in this paper with a precise integration method (PTM) applied to the special solution. The processes of PTM for general solution and the proposed method for special solution are closely integrated. This combination is confirmed to have a high computational efficiency.. The proposed method is also validated to have more convenient and more applied range. A general formula is given to greatly save computation time. Numerical examples are also demonstrated the validity and efficiency of the method.