An analytical solution for fractional oscillator in a resisting medium

Chaos, Solitons and Fractals 130 (2020) 109395

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

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An analytical solution for fractional oscillator in a resisting medium G.M. Ismail a, H.R. Abdl-Rahim b, A. Abdel-Aty c,d,∗, R. Kharabsheh e, W. Alharbi f, M. Abdel-Aty a,g a

Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt Department of Mathematics, Faculty of Science, South Valley University, Qena, Egypt c Department of Physics, College of Sciences, University of Bisha, Bisha, Saudi Arabia d Physics Department, Faculty of Science, Al-Azhar University, 71524 Assiut, Egypt e College of Administrative Sciences, Applied Science University, Bahrain f Physics Department, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia g Center for Photonics and Smart Materials (CPSM), Zewail City of Science and Technology, October Gardens, 6th of October City, Giza 12578, Egypt b

a r t i c l e

i n f o

Article history: Received 1 July 2019 Revised 16 August 2019 Accepted 20 August 2019

a b s t r a c t In this paper, an analytical exact solution for the fractional differential equation of the oscillator in a resisting medium was obtained successfully via the natural transform method. The fractional derivatives were described in the Caputo sense. The results illustrated the power, efficiency, simplicity, and reliability of the proposed method. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Natural transform Fractional oscillator in a resisting medium Analytical exact solution Fractional calculus

1. Introduction Fractional calculus has various applications in the fields of natural science and technology, including chemistry, biology, physics, engineering, as well as pure and applied mathematics [1–10]. Many physical phenomena can be described more effectively via fractional calculus. Most of them are portrayed via nonlinear fractional differential equations. Thus, scientists in various branches of science have tried to solve them. Additionally, there are numerous methods for calculating the approximate numerical and exact solutions for nonlinear fractional differential equations, and ne f these methods is natural transform method wich used in this paper [11–27]. Also, examles of these methods are, Kudryashov method [2], decomposition method [28], Adomian decomposition method (ADM) is used by Abdou and Elhanbaly [29], extended simple equation method [30], Homotopy Perturbation Method (HPM) [31], Ramswroop presented semi-analytical technique based on the homotopy analysis transform method (HATM) [32]. In this paper, we present a novel technique called the natural transform method, it is a generalization of both Laplace and Sumudu transforms. The main objective for the present paper is gaining an analytical exact solution for the fractional differential equation of the oscillator in

∗

Corresponding author. E-mail address: [email protected] (A. Abdel-Aty).

https://doi.org/10.1016/j.chaos.2019.109395 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

a resisting medium or damping force as in [33,34]. The rest of the paper is organized as follows: Section 2 covers the basic definition and properties of the natural transform method. The fractional differential equation of the oscillator in a resisting medium is offered in Section 3. Section 4 is dedicated to the study’s conclusion. 2. Definition of the natural transform [35] Over the set of functions A = { f (t ) : ∃M, τ1 , τ2 > 0, | f (t )| < Me|t |/τ j , i f t ∈ (−1 ) × [0, ∞ )}. j

(1) The natural transform of f(t) is defined by

N [ f (t )] = R(s, u ) =



0

∞

f (ut )e−st dt, u > 0, s > 0,

(2)

where R(s, u) is the natural transform of the time function f(t) and the variables u and s are the natural transform variables. The properties of the natural transform exist in [36–40]. 3. An application Fractional differential equation of the oscillator in a resisting medium is given by [33]: α

Dα U (t ) + 2 k D 2 U (t ) + w2U (t ) = F f (t ),

1 < α ≤ 2,

(3)

2

G.M. Ismail, H.R. Abdl-Rahim and A. Abdel-Aty et al. / Chaos, Solitons and Fractals 130 (2020) 109395

where k > 0 a constant of proportionality and the right-hand side represents the external driving force. The initial state of the system is

U (0 ) = a, U˙ (0 ) = X, where a, X, F are constants and w is the frequency. The solution to three particular cases is studied, as follows: 3.1. Case 1: k < w (Small damping), in this case n = w2 − k2 > 0 Applying the natural transform of both sides of Eq. (3) and simplifying, we have α −1

R(s, u ) =

u2

α

2

+ k) +

s2 α u2

, Q 2 = w2 − k2 ,

(4)

Q2

i.e . , (− α +1 )−1 α α au 2 u 2α ( s 2 + ku 2 − +1

α ( −α +1 )−1 α (−α +2 )−1 α ) + (X u 2 u −α +2 u 2 + u 2 u 2−α ak )u 2 +1 s

α

R (s, u ) =

2

s

+

α

s 2

 α

α 2 s 2 + ku 2 + Q 2 uα

uα F R (s, f ). α ( s 2 + ku 2 ) + Q 2 uα

(5)

α

Taking the inverse natural transform of Eq. (5), we obtain



α

α

u(− 2 +1)−1

α

U (x, t ) = aN −1 u 2

s



α

+ akN −1 u 2



+1

α

α

2

α

2

α

u(− 2 +1)−1 s

− α2 +1

α

+ F N −1 u 2

α

( s 2 + ku 2 ) + Q 2 ( u 2 )

u(−α +2)−1 u2 s−α +2 (s α2 + ku α2 )2 + Q 2 (u α2 )2

α



−α 2



α

( s 2 + ku 2 ) α

+ X N −1 u 2

α

u2 α

α

α

2



α

2

α

α

U (x, t ) = aN −1

Q=



This means that when the resistance is small, the modified frequency is obviously smaller than the natural frequency w It should also be noted that the amplitude decays exponentially to zero as time t → ∞.The phase of the motion is also changed by the small resistance. Thus, the motion is called the damped oscillatory motion. In Fig. 1. the analytical approximate solution U(t) of Eq. (3), at α → 2 and F = 0, the fractional solution arrives at the exact solution obtained via [40].

From Eq. (4), we obtain

2

α

 α

2

( s 2 + ku 2 ) + Q ( u 2 ) ( s 2 + ku 2 ) + ( u 2 )

. (6)

R(s, u ) =

+ F N −1 u

(s + ku )2 + Q u2

F Q

t

0

U (x, t ) = aN ⎣

X + ak U (x, t ) = e−kt (a cos(Qt ) + sin(Qt )), Q U (x, t ) = Ae

(a cos(Qt − ϕ )),



(7)



A=

a2 +

(X + ak )2 Q2

,

ϕ = tan−1 (

X + ak ). aQ

α

α



s2

α

u 2 −1 s

− α +1 2

+ F N −1 uα −1

2



u α

α

.

u α

α

U (x, t ) = aN −1 (8)

(9)

 + F N −1 u

α

2



( s 2 + ku 2 )







α

( s 2 + ku 2 )

R( f, u ) . 2

1 u + X N −1 s + ku (s + ku )2

+ akN −1



u

2

( s 2 + ku 2 )

In the case α → 2 from (12), we have

(10)

where

α

 ⎦ + X N−1 sα−2

s 2 + k u α2

+ akN −1

X + ak −kt e sin(Qt ) Q

f (t − τ )e−kt sin(Qt )dτ .

α

⎤

α

s 2 −1

−1

This is the most general solution to the problem for an arbitrary form of the external driving force obtained via [40]. In absence of an external driving force (i.e. F = 0), we have

−kt

α

⎡

Thus,

+

α

Taking the inverse natural transform of Eq. (11), we obtain



u R(s, f ) . (s + ku ) + Q u2 (s + ku )2 + u2

U (x, t ) = ae−kt cos(Qt ) +

α

( s 2 + ku 2 )

u





α

as 2 −1 (s 2 + ku 2 ) + X s2u−α + s 2 −1 aku 2 + uα F R(s, f )

(11)

   (s + ku ) u −1 + X N (s + ku )2 + Q 2 u2 (s + ku )2 + Q 2 u2  

+ akN −1

1 k2 + ... ), 0 < k < w. 2 w2

w2 − k2 = ( 1 −

2

( s 2 + ku 2 ) + Q ( u 2 ) α u2 R(s, f ) α

Like the harmonic oscillator in a vacuum, the motion is oscillatory with the time-dependent amplitude Ae−kt and the modified frequency

3.2. Case 2: k = w (Critical damping), in this case Q = 0



In the case of α → 2 from (6), we have



) and the exact solution

α −1

α −2

a suα + X us α−1 + 2ak s 2 α + F R(s, f )

(

Fig. 1. Comparison between the analytical solution ( ). (





u

(s + ku )2 u

(12)



R( f, u ) . 2

(13)

(s + ku )

Then,

U (x, t ) = ae−kt + (X + ak )t e−kt + F



t 0

f (t − τ )τ e−kτ dτ .

(14)

This is the exact solution obtained by [33]. In Fig. 2. the analytical approximate solution U(t) of Eq. (4), at α → 2 and F = 0, the fractional solution arrives at the exact solution obtained via [40].

G.M. Ismail, H.R. Abdl-Rahim and A. Abdel-Aty et al. / Chaos, Solitons and Fractals 130 (2020) 109395

3

4. Conclusion In this work a new technique based on natural transform method for obtaining new fractional solutions for the oscillator in a resisting medium is proposed. From the obtained results we can observe that, at α → 2 we have the exact solution. Thus, it can be concluded that the natural transform is a very powerful and efficient tool in finding analytical as well as numerical solutions for wide classes of fractional differential equations. Fig. 2. Comparison between the analytical solution ( ). (

) and the exact solution

Declaration of Competing Interest I declare that I have no significant competing financial, professional, or personal interests that might have influenced the performance or presentation of the work.

References

Fig. 3. Comparison between analytical solution (

) and exact solution (

).

Case 3: k > w (Large damping), in this case Q 2 = w2 − k2 = −(k2 − w2 ) = −m2 So that k2 − w2 > 0 Taking the inverse natural transform of Eq. (5) and substituting Q 2 = −m2 , we obtain



α

U (x, t ) = aN −1 u 2

α

s

  + akN

 + F N −1

u

−α 2

+1



α

( s 2 + ku 2 ) α

α

α

2

2

( s 2 + ku 2 ) − m2 ( u 2 ) α

u(−α +2)−1 u2 s−α +2 (s α2 + ku α2 )2 − m2 Q 2 (u α2 )2

α

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(15)

In the case α → 2 from (15), we have



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+ akN −1

 + F N −1 u

u

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(s + ku )2 + u2



R( f, s ) .

(16)

Thus,

X + ak −kt U (x, t ) = ae−kt cosh(mt ) + ( )e sinh(mt ) m  t F + f (t − τ )e−kt sinh(mt )dτ . m 0

(17)

This is the standard form obtained by [33]. In Fig. 3. the analytical approximate solution U(t) of Eq. (5). at α → 2 and F = 0, the fractional solution arrives at the exact solution obtained via [40].

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