On Shehu Transform with Application of Solutions of Fractional Differential Equations

This review article explains the Shehu transform as a tool used for solving linear differential equations of fractional order, where the definition of the Caputo differential operator of order α>0, is taken into consideration. The transformation is used to convert Initial Value Problems (IVPs) of the fractional order of Caputo sense into simple algebraic equations. Then the inverse of the transform is used to obtain the analytical solution of the problem. We solved some illustrative examples


Introduction
ehT Integral transformations are successfully used as effective tools for solving differential and integral equations [1]-, [2]-.Fractional Differential Equations (FDEs) are an important topic due to their application in modelling problems in many fields including mechanics, electroanalytical chemistry, electrical circuits, and other physical, chemical, biological, and economic aspects [3]-, [4]-, [5]-, [6]-, [7]-.Due to their importance, researchers investigated many ideas searching for solutions to FDEs.Some of these ideas involve integral transformations such as Laplace and Sumudu transforms [8]-, [9]-that are used to solve differential equations of fractional order in the form of initial value problems (IVPs).Recently, the Shehu transform has been derived from the classical Fourier integral transform as a generalization of Laplace and Sumudu transforms [10]-.It is successfully used to deal with fractional derivatives in the Caputo sense [11]-that can be used to model various natural phenomena involving fractional derivatives.This work is considered as a review study of what was stated in the literature.We focus on the definition and some properties of the Shehu transform for derivatives in the Caputo sense.Shehu transform is applied to IVPs of Caputo derivatives of fractional order to convert them into an algebraic equation.We used the transformation to obtain analytical solutions of fractional models as IVPs of fractional order by applying the inverse fractional Shehu transform.The study supplied illustrative examples.

Preliminaries
In this section, we introduce some basics that need to be well known for proceeding in solving fractional differential equations.Definition 1.The function is defined as,  It converges if the limit of the integral exists, and diverges otherwise.

Discussion
In this section, we apply the Shehu transformation and its properties to obtain the analytic solution of linear differential equations as IVPs.We consider differentiations of order  > 0 that are defined in the Caputo sense, within the use of the Shehu integral transform technique and its inverse, we solve two examples of IVPs of fractional order.This method is considered an additional tool added to other transformations that are used to find solutions to differential equations of fractional order.Applying the Shehu transform on both sides of Eq.( 10) yields Applying the Shehu inverse on both sides of Eq. ( 11) we obtain () =   (−  ).Some solution plots for several values  are shown in Figure 1.Example 2. Consider the initial value problem (Bagley-Torvik equation) Fig. 1: Solutions () of the linear fractional initial value problem given by Eq. ( 10), where  is in the interval (0, 4], for different values, α = . From the relation (7)  By applying the Shehu inverse transform we get which yields () = 1 + .

Conclusion
This review article introduced the Shehu integral transform and some of its properties.We found the transform is a useful tool to solve initial value problems involving fractional order differentiations using the Caputo definition.The transformation is applied to convert initial value problems of fractional order into a simpler algebraic equation that can be easily solved then an analytic solution is obtained by using the inverse of the Shehu transform.With the use of the Shehu integral transform technique, the obtained results in terms of elementary mathematical functions agree well with those achieved using other integral transforms such as Laplace Transform.Hence, the Shehu transform is considered as an additional tool for solving continuous dynamical systems of fractional order of Caputo type and it is closely connected with the Sumudu transform.

Example 1 .
Consider the linear fractional initial value problem given as