Precise integration method for duhamel terms arising from non-homogenous dynamic systems

With the precise integration method (PIM) proposed for lineartime-invariant systems, one can obtain precise numerical results approaching theexact solution at the integration points. However, it is more or lessdifficult to use the algorithm in the Duhamel's integration arisingfrom the non-homogenous dynamic systems due to the inverse matrixcalculations. So the precise integration method for Duhamel terms withoutthe inverse matrix calculations is proposed. By applying the techniques ofaddition theorem and increment storage, which are the key ideas of PIM,directly to the Duhamel integration terms, it can also give precisenumerical results can be obtained close to the computer precision when the non-homogenous termsare polynomial, sinusoidal, exponential or theircombinations. In particular, this method is not affected by the qualityof the system matrix (or the relative matrix). If the system matrix issingular or nearly singular, the advantages of the method will be more remarkable.Numerical examples are given to demonstrate the validity and efficiency ofthe method.