# A New Application of the Sumudu Transform for the Falling Body Problem

Authors: Esra Karatas Akgül; Ali Akgül; Rubayyi T. Alqahtani
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this study, we investigate the falling body problem with three different fractional derivatives. We acquire the solutions of the model by the Sumudu transform. We show the accuracy of the Sumudu transform by some theoretic results and implementations.


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## Body

## 1. Introduction

The Sumudu transform is an important integral transformation. This transformation has been begun with Watugala [1] who has presented a new integral transform to solve differential equations and control engineering problems. Weerakoon [2, 3] has made a big contribution with the application of the Sumudu transform to partial differential equations and complex inversion formula for the Sumudu transform. Then, this transformation has attracted great attention, and lots of work have been done related to this transformation by the authors. Some of them are as follows: Belgacem et al. [4] have searched the analytical investigations of the Sumudu transform. Fundamental properties of the Sumudu transform have been studied by Belgacem and Karaballi [5]. Atangana and Akgül [6] have obtained the transfer function and Bode diagram by the Sumudu transform.Fractional differential equations have taken much interest recently. Arqub and Maayah [7, 8] have studied a fitted fractional reproducing kernel algorithm for the numerical solutions of ABC-fractional Volterra integro-differential equations and solution of the fractional epidemic model by the homotopy analysis method. Jangid et al. [9] have investigated some fractional calculus findings associated with the incomplete functions. Some new fractional-calculus connections between Mittag-Leffler functions have been studied by Srivastava et al. [10]. Singh et al. [11] have investigated the fractional epidemiological model for computer viruses. Ghanbari and Atangana [12, 13] have presented an efficient numerical approach for fractional diffusion partial differential equations and a new application of fractional Atangana-Baleanu derivatives. Abdeljawad et al. [14, 15] have investigated an efficient sustainable algorithm for numerical solutions of systems of fractional-order differential equations by the Haar wavelet collocation method and more general fractional integration by part formulae and applications. For more details, see [16–25].In this study, we examine the falling body problem depending on Newton’s second law that represents that the acceleration of a particle relied on the mass of the particle and the net force action on the particle. Take into consideration an object of massm falling through the air from a height h with velocity v0 in a gravitational field. If we use Newton’s second law, we acquire
(1)mdvdt+mkv=−mg,where k is the positive constant rate and g expresses the gravitational constant. The solution of this equation is presented as [26]
(2)vt=−gk+exp−ktv0+gk,and by integrating for z0=h, we get
(3)zt=h−gtk+1k1−exp−ktv0+gk.Considering all the information presented above, we organize this study as follows. In Section2, some fundamental definitions and lemmas about nonlocal fractional calculus are given. In Section 3, the fractional falling body problem is investigated by means of Caputo, Caputo-Fabrizio sense of Caputo, and ABC. Also, some outstanding consequences are clarified in Section 4.


## 2. Preliminaries

After giving some introduction, we want to present some significant definition and lemma properties of fractional calculus for setting up a mathematically sound theory that will serve the purpose of the current study.Definition 1.
Over(4)B=vt∣∃N,η1,η2>0,∣vt∣<Nexp∣t∣/ηj,ift∈−1j×0,∞,the Sumudu transform is identified by [17]
(5)Vu=Svt=∫0∞futexp−tdt,u∈−η1,η2.Definition 2.
We define the classical Mittag-Leffler functionEαz as [27]
(6)Eαz=∑m=0∞zmΓαm+1z∈ℂ,Reα>0,and the Mittag-Leffler kernel with two parameters is presented by [27]
(7)Eα,βz=∑m=0∞zmΓαm+βz,β∈ℂ,Reα>0.Definition 3.
The generalized Mittag-Leffler function is defined by [28]
(8)Eα,βρz=∑m=0∞zmρmΓαm+βm!z,β,ρ,α∈ℂ,Reα>0,where ρm=ρρ+1⋯ρ+m−1 is the Pochhammer symbol introduced by Prabhakar. As seen clearly, 1m=m! and Eα,β1z=Eα,βz.Definition 4.
Letζ,ψ:0,∞⟶R; then, the convolution of ζ,ψ is given as [27]
(9)ζ∗ψ=∫0tζt−uψudu,and assume that ζ,ψ:0,∞⟶R; then, we have
(10)Sζ∗ψt=uSζtSψt.Definition 5.
Letv:0,∞⟶ℝ be a smooth function. Then, the Caputo fractional derivative is defined as follows [18]:
(11)0CDαvt=1Γn−α∫0tt−zn−α−1vnzdz,where α∈ℂ,Reα>0 and n=Reα+1.Lemma 6.
The Sumudu transform of the Caputo fractional derivative is presented by [6]
(12)S0CDtαvt=Vu−v0uα,
whereVu=Svt.Definition 7.
We identify the Caputo-Fabrizio fractional derivative as [29]
(13)0CFCDαvt=Mα1−α∫0tv′zexp−λt−zdz.Lemma 8.
The Sumudu transform of the CFC fractional derivative is acquired by [6]
(14)S0CFCDtαvt=Mα1−αVu1+α/1−αu−Mα1−αv01+α/1−αu.Definition 9.
We describe the Atangana-Baleanu fractional derivative by [27]
(15)0ABCDαvt=ABα1−α∫0tv′zEα−λt−zαdz.Lemma 10.
The Sumudu transform of the ABC fractional derivative is acquired by [6]
(16)S0ABCDtαvt=ABα1−αVu1+α/1−αuα−ABα1−αv01+α/1−αuα.Definition 11.
The generalized fractional integral is given by [28]
(17)0Iα,βvt=1Γαβα−1∫0ttβ−zβα−1vzzβ−1dz.Definition 12.
The generalized fractional derivatives in the Caputo sense are defined, respectively, by [28]
(18)0CDα,βvt=0In−α,βt1−βddtnvt=1Γn−αβn−α−1∫0ttβ−zβn−α−1t1−βddtnvzzβ−1dz.Lemma 13.
We have [6]
(19)SEα−λtα=11+λuα,(20)S1−Eα−λtα=λuα1+λuα.


## 3. Main Results

The aim of this section is to obtain the solutions for the fractional falling body problem by means of some nonlocal fractional derivative operators such as Caputo, Caputo-Fabrizio, and ABC.


### 3.1. The Falling Body Problem in the Sense of Caputo

The falling body problem in the sense of Caputo depended on Newton’s second law which is given as follows:(21)0CDαvt+kσ1−αvt=−gσ1−α,where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(22)S0CDαvt+kσ1−αSvt=S−gσ1−α.Using the relation equation (9), we can write
(23)Svt−v0uα+kσ1−αSvt=−gσ1−α,(24)Svt=v01+kσ1−αuα−gσ1−αuα1+kσ1−αuα.If we apply the inverse Sumudu transform, we will obtain(25)vt=v0Eα−kσ1−αtα−gk1−Eα−kσ1−αtα.Because ofα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(26)vt=v0Eα−kαα1−αtα−gk1−Eα−kαα1−αtα.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(27)0CDαzt=v0σ1−αEα−kσ1−αtα−gσ1−αk1−Eα−kσ1−αtα.Applying the Sumudu transformation to the above equation, we have(28)S0CDαzt=v0σ1−αSEα−kσ1−αtα−gσ1−αkS1−Eα−kσ1−αtα,Szt−z0uα=v0σ1−α11+kσ1−αuα−gσ1−αk+gσ1−αk11+kσ1−αuα,Szt=z0+v0σ1−αuα1+kσ1−αuα−gσ1−αuαk+gσ1−αkuα1+kσ1−αuα.Using the inverse Sumudu transformation for the last equation and taking thez0=h, we acquire the vertical distance zt as
(29)zt=h+v0k1−Eα−kσ1−αtα−gσ1−αtαkΓα+1+gk21−Eα−kσ1−αtα.Because ofα=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(30)zt=h+v0k1−Eα−kαα1−αtα−gα1−αtαk2−αΓα+1+gk21−Eα−kαα1−αtα.We demonstrate the simulations of the above solution for different values ofα as shown in Figures 1–3. Similar simulations can be shown easily for other solutions.Figure 1
Simulations of the solution forα=0.5 and α=0.9.Figure 2
Simulations of the solution forα=0.8 and α=0.9.Figure 3
Simulations of the solution forα=0.98 and α=0.99.


### 3.2. The Falling Body Problem in the Sense of Caputo-Fabrizio

The falling body problem in the sense of Caputo-Fabrizio depended on Newton’s second law which is given as follows:(31)0CFCDαvt+kσ1−αvt=−gσ1−α.where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(32)S0CFCDαvt+kσ1−αSvt=S−gσ1−α.If we use the relation equation(19), we can write
(33)Mα1−αSvt1+α/1−αu−Mα1−αv01+α/1−αu+kσ1−αSvt=−gσ1−α,(34)Svt=Mαv0/1−α1+α/1−αuMα/1−α1+α/1−αu+kσ1−α−gσ1−αMα/1−α1+α/1−αu+kσ1−α.Applying the inverse Sumudu transform yields(35)vt=S−1Mαv0/1−α1+α/1−αuMα/1−α1+α/1−αu+kσ1−α−S−1gσ1−αMα/1−α1+α/1−αu+kσ1−α.vt=Mαv0Mα+kσ1−α1−αexp−αkσ1−αtMα+kσ1−α1−α−gMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αkMα+kσ1−α1−α.Because ofα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(36)vt=Mαv0Mα+kαα1−α1−αexp−kαα2−αtMα+kαα1−α1−α−gMα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αkMα+kαα1−α1−α.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(37)0CFCDαzt=Mασ1−αv0Mα+kσ1−α1−αexp−αkσ1−αtMα+kσ1−α1−α−gσ1−αMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αkMα+kσ1−α1−α.If we implement the Sumudu transformation to the last equation, we will obtain(38)S0CFCDαzt=Mασ1−αv0Mα+kσ1−α1−αSexp−αkσ1−αtMα+kσ1−α1−α−Sgσ1−αk+gσ1−αMαkMα+kσ1−α1−αSexp−αkσ1−αtMα+kσ1−α1−α,Mα1−αSzt1+α/1−αu−Mα1−αz01+α/1−αu=Mασ1−αv0Mα+kσ1−α1−α11+αkσ1−αu/Mα+kσ1−α1−α−gσ1−αk+gσ1−αMαkMα+kσ1−α1−α11+αkσ1−αu/Mα+kσ1−α1−α,Szt=z0+σ1−αv01−α+αuMα+kσ1−α1−α+αu−gσ1−α1−αMαk−gσ1−ααuMαk+gσ1−αk1−α+αuMα+kσ1−α1−α+αu,If we utilize the inverse Sumudu transformation for equation (13) and take the z0=h, we acquire the vertical distance zt as
(39)zt=h+v0kMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αMα+kσ1−α1−α−gσ1−αMαk−gσ1−ααtMαk+gk2Mα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αMα+kσ1−α1−α.Because ofα=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(40)zt=h+v0kMα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αMα+kαα1−α1−α−gα1−αMαk−gα2−αtMαk+gk2Mα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αMα+kαα1−α1−α.


### 3.3. The Falling Body Problem in the Sense of ABC

The falling body problem in the sense of ABC depended on Newton’s second law which is given as follows:(41)0ABCDαvt+kσ1−αvt=−gσ1−α,where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(42)S0ABCDαvt+kσ1−αSvt=S−gσ1−α.Using the relation equation (15), we can write
(43)ABα1−αSvt1+α/1−αuα−ABα1−αv01+α/1−αuα+kσ1−αSvt=−gσ1−α,Svt=ABαv0−gσ1−α1−α1+α/1−αuαABα+kσ1−α1−α1+α/1−αuα.If we apply the inverse Sumudu transform, we will reach(44)vt=S−1ABαv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−S−1gσ1−α1−α1+α/1−αuαABα+kσ1−α1−α1+α/1−αuα,(45)vt=ABαv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−α1−αABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gk1−Eα−αkσ1−αtαABα+kσ1−α1−α.Becauseα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(46)vt=ABαv0ABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gkα−1α1−α1−αABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gk1−Eα−α2−αktαABα+kαα1−α1−α.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(47)0ABCDαzt=ABασ1−αv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−αk1−Eα−αkσ1−αtαABα+kσ1−α1−α.If we apply the Sumudu transformation to the above equation, we have(48)S0ABCDαzt=ABασ1−αv0ABα+kσ1−α1−αSEα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−αSEα−αkσ1−αtαABα+kσ1−α1−α−Sgσ1−αk+gσ1−αkSEα−αkσ1−αtαABα+kσ1−α1−α,ABα1−αSzt1+α/1−αuα−ABα1−αz01+α/1−αuα=ABασ1−αv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ1−αk+gσ1−αk11+αkσ1−αuα/ABα+kσ1−α1−α,Szt=z0+σ1−α1−αv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α+σ1−αv0αABα+kσ1−α1−αuα1+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−α1−α2ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−αα1−αABαABα+kσ1−α1−αuα1+αkσ1−αuα/ABα+kσ1−α1−α−gσ1−ααuαABαk−gσ1−α1−αABαk+gσ1−ααABαkuα1+αkσ1−αuα/ABα+kσ1−α1−α+gσ1−α1−αABαk11+αkσ1−αuα/ABα+kσ1−α1−α.Using the inverse Sumudu transformation for the last equation and taking thez0=h, we acquire the vertical distance zt as
(49)zt=h+σ1−α1−αv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α+v0k1−Eα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−α2ABαABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−α1−αABαkEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−αABαk1−α+αtαΓ1+α+gσ1−α1−αABαkEα−αkσ1−αtαABα+kσ1−α1−α+gABα+gkσ1−α1−αABαk21−Eα−αkσ1−αtαABα+kσ1−α1−α,where v0=gσ1−α/ABα. Because of α=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(50)zt=h+α1−αkα−11−αv0ABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α+v0k1−Eα−α2−αktαABα+kαα1−α1−α−gα21−αk2α−11−α2ABαABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gα1−αkα−11−αABαkEα−α2−αktαABα+kαα1−α1−α−gα1−αkαABαk21−α+αtαΓ1+α+gσ1−αkα1−αABαkEα−α2−αktαABα+kαα1−α1−α+gABα+gkαα1−α1−αABαk21−Eα−α2−αktαABα+kαα1−α1−α.


## 3.1. The Falling Body Problem in the Sense of Caputo

The falling body problem in the sense of Caputo depended on Newton’s second law which is given as follows:(21)0CDαvt+kσ1−αvt=−gσ1−α,where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(22)S0CDαvt+kσ1−αSvt=S−gσ1−α.Using the relation equation (9), we can write
(23)Svt−v0uα+kσ1−αSvt=−gσ1−α,(24)Svt=v01+kσ1−αuα−gσ1−αuα1+kσ1−αuα.If we apply the inverse Sumudu transform, we will obtain(25)vt=v0Eα−kσ1−αtα−gk1−Eα−kσ1−αtα.Because ofα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(26)vt=v0Eα−kαα1−αtα−gk1−Eα−kαα1−αtα.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(27)0CDαzt=v0σ1−αEα−kσ1−αtα−gσ1−αk1−Eα−kσ1−αtα.Applying the Sumudu transformation to the above equation, we have(28)S0CDαzt=v0σ1−αSEα−kσ1−αtα−gσ1−αkS1−Eα−kσ1−αtα,Szt−z0uα=v0σ1−α11+kσ1−αuα−gσ1−αk+gσ1−αk11+kσ1−αuα,Szt=z0+v0σ1−αuα1+kσ1−αuα−gσ1−αuαk+gσ1−αkuα1+kσ1−αuα.Using the inverse Sumudu transformation for the last equation and taking thez0=h, we acquire the vertical distance zt as
(29)zt=h+v0k1−Eα−kσ1−αtα−gσ1−αtαkΓα+1+gk21−Eα−kσ1−αtα.Because ofα=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(30)zt=h+v0k1−Eα−kαα1−αtα−gα1−αtαk2−αΓα+1+gk21−Eα−kαα1−αtα.We demonstrate the simulations of the above solution for different values ofα as shown in Figures 1–3. Similar simulations can be shown easily for other solutions.Figure 1
Simulations of the solution forα=0.5 and α=0.9.Figure 2
Simulations of the solution forα=0.8 and α=0.9.Figure 3
Simulations of the solution forα=0.98 and α=0.99.


## 3.2. The Falling Body Problem in the Sense of Caputo-Fabrizio

The falling body problem in the sense of Caputo-Fabrizio depended on Newton’s second law which is given as follows:(31)0CFCDαvt+kσ1−αvt=−gσ1−α.where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(32)S0CFCDαvt+kσ1−αSvt=S−gσ1−α.If we use the relation equation(19), we can write
(33)Mα1−αSvt1+α/1−αu−Mα1−αv01+α/1−αu+kσ1−αSvt=−gσ1−α,(34)Svt=Mαv0/1−α1+α/1−αuMα/1−α1+α/1−αu+kσ1−α−gσ1−αMα/1−α1+α/1−αu+kσ1−α.Applying the inverse Sumudu transform yields(35)vt=S−1Mαv0/1−α1+α/1−αuMα/1−α1+α/1−αu+kσ1−α−S−1gσ1−αMα/1−α1+α/1−αu+kσ1−α.vt=Mαv0Mα+kσ1−α1−αexp−αkσ1−αtMα+kσ1−α1−α−gMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αkMα+kσ1−α1−α.Because ofα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(36)vt=Mαv0Mα+kαα1−α1−αexp−kαα2−αtMα+kαα1−α1−α−gMα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αkMα+kαα1−α1−α.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(37)0CFCDαzt=Mασ1−αv0Mα+kσ1−α1−αexp−αkσ1−αtMα+kσ1−α1−α−gσ1−αMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αkMα+kσ1−α1−α.If we implement the Sumudu transformation to the last equation, we will obtain(38)S0CFCDαzt=Mασ1−αv0Mα+kσ1−α1−αSexp−αkσ1−αtMα+kσ1−α1−α−Sgσ1−αk+gσ1−αMαkMα+kσ1−α1−αSexp−αkσ1−αtMα+kσ1−α1−α,Mα1−αSzt1+α/1−αu−Mα1−αz01+α/1−αu=Mασ1−αv0Mα+kσ1−α1−α11+αkσ1−αu/Mα+kσ1−α1−α−gσ1−αk+gσ1−αMαkMα+kσ1−α1−α11+αkσ1−αu/Mα+kσ1−α1−α,Szt=z0+σ1−αv01−α+αuMα+kσ1−α1−α+αu−gσ1−α1−αMαk−gσ1−ααuMαk+gσ1−αk1−α+αuMα+kσ1−α1−α+αu,If we utilize the inverse Sumudu transformation for equation (13) and take the z0=h, we acquire the vertical distance zt as
(39)zt=h+v0kMα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αMα+kσ1−α1−α−gσ1−αMαk−gσ1−ααtMαk+gk2Mα+kσ1−α1−α−Mαexp−αkσ1−αt/Mα+kσ1−α1−αMα+kσ1−α1−α.Because ofα=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(40)zt=h+v0kMα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αMα+kαα1−α1−α−gα1−αMαk−gα2−αtMαk+gk2Mα+kαα1−α1−α−Mαexp−kαα2−αt/Mα+kαα1−α1−αMα+kαα1−α1−α.


## 3.3. The Falling Body Problem in the Sense of ABC

The falling body problem in the sense of ABC depended on Newton’s second law which is given as follows:(41)0ABCDαvt+kσ1−αvt=−gσ1−α,where the initial velocity v0=v0, g is the gravitational constant, and m and k are the positive constant rate which indicates the mass of the body. We have
(42)S0ABCDαvt+kσ1−αSvt=S−gσ1−α.Using the relation equation (15), we can write
(43)ABα1−αSvt1+α/1−αuα−ABα1−αv01+α/1−αuα+kσ1−αSvt=−gσ1−α,Svt=ABαv0−gσ1−α1−α1+α/1−αuαABα+kσ1−α1−α1+α/1−αuα.If we apply the inverse Sumudu transform, we will reach(44)vt=S−1ABαv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−S−1gσ1−α1−α1+α/1−αuαABα+kσ1−α1−α1+α/1−αuα,(45)vt=ABαv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−α1−αABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gk1−Eα−αkσ1−αtαABα+kσ1−α1−α.Becauseα=σk, 0<σ≤1/k, the velocity vt can be put down as follows:
(46)vt=ABαv0ABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gkα−1α1−α1−αABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gk1−Eα−α2−αktαABα+kαα1−α1−α.Note that we put the conditionv0=−g/k in order to satisfy the initial condition v0=v0. By benefiting from the velocity (33), vertical distance zt can be obtained in the following way:
(47)0ABCDαzt=ABασ1−αv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−αk1−Eα−αkσ1−αtαABα+kσ1−α1−α.If we apply the Sumudu transformation to the above equation, we have(48)S0ABCDαzt=ABασ1−αv0ABα+kσ1−α1−αSEα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−αSEα−αkσ1−αtαABα+kσ1−α1−α−Sgσ1−αk+gσ1−αkSEα−αkσ1−αtαABα+kσ1−α1−α,ABα1−αSzt1+α/1−αuα−ABα1−αz01+α/1−αuα=ABασ1−αv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−α1−αABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ1−αk+gσ1−αk11+αkσ1−αuα/ABα+kσ1−α1−α,Szt=z0+σ1−α1−αv0ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α+σ1−αv0αABα+kσ1−α1−αuα1+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−α1−α2ABα+kσ1−α1−α11+αkσ1−αuα/ABα+kσ1−α1−α−gσ21−αα1−αABαABα+kσ1−α1−αuα1+αkσ1−αuα/ABα+kσ1−α1−α−gσ1−ααuαABαk−gσ1−α1−αABαk+gσ1−ααABαkuα1+αkσ1−αuα/ABα+kσ1−α1−α+gσ1−α1−αABαk11+αkσ1−αuα/ABα+kσ1−α1−α.Using the inverse Sumudu transformation for the last equation and taking thez0=h, we acquire the vertical distance zt as
(49)zt=h+σ1−α1−αv0ABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α+v0k1−Eα−αkσ1−αtαABα+kσ1−α1−α−gσ21−α1−α2ABαABα+kσ1−α1−αEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−α1−αABαkEα−αkσ1−αtαABα+kσ1−α1−α−gσ1−αABαk1−α+αtαΓ1+α+gσ1−α1−αABαkEα−αkσ1−αtαABα+kσ1−α1−α+gABα+gkσ1−α1−αABαk21−Eα−αkσ1−αtαABα+kσ1−α1−α,where v0=gσ1−α/ABα. Because of α=σk, 0<σ≤1/k, the vertical distance zt can be put down as
(50)zt=h+α1−αkα−11−αv0ABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α+v0k1−Eα−α2−αktαABα+kαα1−α1−α−gα21−αk2α−11−α2ABαABα+kαα1−α1−αEα−α2−αktαABα+kαα1−α1−α−gα1−αkα−11−αABαkEα−α2−αktαABα+kαα1−α1−α−gα1−αkαABαk21−α+αtαΓ1+α+gσ1−αkα1−αABαkEα−α2−αktαABα+kαα1−α1−α+gABα+gkαα1−α1−αABαk21−Eα−α2−αktαABα+kαα1−α1−α.


## 4. Conclusions

We searched the falling body problem in detail by the Sumudu transform in this study. We obtained the exact solutions of this problem with Caputo, Caputo-Fabrizio, and Atangana-Baleanu derivatives. We demonstrated the effect of the Sumudu transform by these results.



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*Source: 9702569-2021-11-09.xml*