A New Method for Laplace Transforms of Multiterm Fractional Differential Equations of the Caputo Type

Abstract

The Laplace transform method is one of the powerful tools in studying the fractional differential equations (FDEs). In this paper, it is shown that the Heaviside expansion method for integer order differential equations is also applicable to the Laplace transforms of multiterm Caputo FDEs of zero initial conditions if the orders of Caputo derivatives are integer multiples of a common real number. The particular solution of a linear multiterm Caputo FDE is obtained by its Laplace transform and the Heaviside expansion method. A Caputo FDE of nonzero initial conditions is transformed to an Caputo FDE of zero initial conditions by an appropriate change of variables. In the latter, the terms originated from the initial conditions appear as nonhomogeneous terms. Thus, the solution to the Caputo FDE of nonzero initial conditions is obtained as the particular solutions to the equivalent Caputo FDE of zero initial conditions. The solutions of a linear multiterm Caputo FDEs of nonzero initial conditions are expressed through the two parameter Mittag–Leffler functions.