GENERALIZED LAPLACE TRANSFORM AND TEMPERED Ψ-CAPUTO FRACTIONAL DERIVATIVE

In this paper, images of the tempered Ψ-Hilfer fractional integral and the tempered Ψ-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 Ψ-Caputo fractional derivative of an order α for n− 1 <α<n N. An illustrative example is given for 0 <α<1 comparing solutions to the same initial value problem but with different tempering and Ψ.