Time-Varying Initialization and Corrected Laplace Transform of the Caputo Derivative

This paper derives the time-varying initialization function for the Caputo derivative of any order u>0 where m–1 < u < m, m = 1,2,…. The derivative is redefined to include this initialization function. Then, the Laplace transform for the redefined Caputo derivative is determined which corrects (supplants) that given for the derivative in the literature and properly accounts for time-varying initialization effects. The paper generalizes previous results for order 0 < u < 1.