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We make use of the so-called Sumudu transform method (STM), a type of ordinary differential equations with both integer and noninteger order derivative. Firstly, we give the properties of STM, and then we directly apply it to fractional type ordinary differential equations, both homogeneous and inhomogeneous ones. We obtain exact solutions of fractional type ordinary differential equations, both homogeneous and inhomogeneous, by using STM. We present some numerical simulations of the obtained solutions and exhibit two-dimensional graphics by means of Mathematica tools. The method used here is highly efficient, powerful, and confidential tool in terms of finding exact solutions.

Computation and analysis of solutions for nonlinear fractional differential equations now span a half-century or more and play a crucial role in several theoretical and applied sciences such as, but certainly not limited to, theoretical biology and ecology, solid state physics, viscoelasticity, fiber optics, signal processing and electric control theory, stochastic based finance, and hemo-, hydro-, and thermodynamics [

The Sumudu transform method (STM) was initiated in 1993, by Watugala who used it to solve engineering control problems [

In this work, our aim is to exhibit exact solutions of some homogeneous and nonhomogeneous fractional ordinary differential equations by using the STM. For paper layout, we first recall basic features of fractional calculus, in Section

In this section, we primarily introduce main features of fractional calculus following notations [

The Sumudu transform is obtained over the set of functions [

If

The Sumudu transform of the first derivative of

If

At the moment, we harness the properties developments of the STM and utilize them for finding exact solutions of fractional ordinary differential equations.

We consider the general linear fractional ordinary differential equation (FODE) as follows:

When we get Sumudu transform of (

When we get inverse Sumudu transform of (

In this Section, we implement the STM to homogeneous and inhomogeneous fractional ordinary differential equations (HFODEs and IHFODEs) in the following three examples.

Initially, we consider the inhomogeneous fractional ordinary differential equation [

Upon Sumudu inverting (

The exact solution of (

Secondly, we consider homogeneous fractional ordinary differential equation as follows [

When we take inverse Sumudu transform of (

To our knowledge, the solution of (

Thirdly, we investigate the inhomogeneous fractional ordinary differential equation [

Again, the exact solution of (

We plot solution (

Two-dimensional graphic for exact solution of (

Two-dimensional graphic for exact solution of (

Two-dimensional graphic for exact solution of (

In this paper, we considered homogeneous and inhomogeneous fractional ordinary differential equations (HFODEs and IHFODEs) and treated them by applying the Sumudu transform method and obtained their exact solutions. To date and to our knowledge, the exact solutions obtained in the last two examples are new. We also exhibited two-dimensional graphics for the obtained solutions by means of programming language Mathematica. Accordingly, we can conclude that not only does the Sumudu transform method play an important role in treating homogeneous and inhomogeneous fractional ordinary differential equations, but also it is highly effective with regard to yielding exact solutions. We expect the STM to be equally successful when treating even more complex applications than presented involving FODEs.

The authors declare that there is no conflict of interests regarding the publication of this paper.