Generalized Laplace Transform and Tempered ?-Caputo Fractional Derivative

摘要

In this paper, images of the tempered W-Hilfer fractional integral and the tempered W-Caputo fractional derivative under the generalized Laplace transform are derived. The results are applied to find a solution to an initial value problem for a nonhomogeneous linear fractional differential equation with the tempered W-Caputo fractional derivative of an order alpha for n - 1 < alpha < n is an element of N. An illustrative example is given for 0 < alpha < 1 comparing solutions to the same initial value problem but with different tempering and W.