The solution of Klein–Gordon equation by using modified Adomian decomposition method

Journal Pre-proof The solution of Klein–Gordon equation by using modified Adomian decomposition method Jeerawan Saelao, Natsuda Yokchoo PII: DOI: Reference:

S0378-4754(19)30313-1 https://doi.org/10.1016/j.matcom.2019.10.010 MATCOM 4881

To appear in:

Mathematics and Computers in Simulation

Received date : 31 January 2019 Revised date : 11 October 2019 Accepted date : 16 October 2019 Please cite this article as: J. Saelao and N. Yokchoo, The solution of Klein–Gordon equation by using modified Adomian decomposition method, Mathematics and Computers in Simulation (2019), doi: https://doi.org/10.1016/j.matcom.2019.10.010. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. c 2019 Published by Elsevier B.V. on behalf of International Association for Mathematics and ⃝ Computers in Simulation (IMACS).

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Journal Pre-proof

The Solution of Klein-Gordon Equation by Using Modified Adomian Decomposition Method Jeerawan Saelaoa,∗, Natsuda Yokchoob b Department

a Division of Mathematics, Faculty of Science, Maejo University, Chiang Mai, 50290, Thailand. of Mathematics, Faculty of Applied Science, King’s Mongkut University of Technology North Bangkok, Bangkok, 10800, Thailand.

Abstract In this paper, a modification of Adomian decomposition method is used to find a solution of linear and nonlinear Klein-Gordon equation. The modified Adomian decomposition method is based on the application of conventional Adomian decomposition method that only requires calculation of the first Adomian polynomial. This method is very effective, easy to calculation, solution faster convergence traditional Adomian decomposition method and can be applied to other nonlinear problems. The demonstration of the efficiency from this method with Klein-Gordon equation are illustrated four examples such as homogeneous linear, inhomogeneous linear and nonlinear.

f

Keywords: Klein-Gordon equation, Adomian decomposition method, Modified Adomian decomposition method.

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1. Introduction

Jo u

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lP

rep

The mathematical solving method is one of the essential tool for finding the solution of nonlinear equation. The obtain solution from nonlinear equation is used to analyze and control the system follow as stability analysis [1,2], robustness analysis [3] and controllability analytical [4]. The one of interesting nonlinear equation is Klein-Gordon equation [5] which proposed by Oscar Klein and Walter Gordon in 1927. The Klein-Gordon equations is interesting equation of physician and mathematician because of this equation has capability to study the quantum movement, the distribution of physic plasma and the movement of other wave. In recent years, many methods for solving KleinGordon equation such as Sumudu Decomposition Method (SDM) [6], Variational Iteration Method (VIM) [7], Laplace Decomposition Method (LDM) [8], Adomian Decomposition Method (ADM) [9], Natural Decomposition Method (NDM) [10], Sumudu Transform Method (STM) and Homotopy Perturbation Method (HPM) [11], KAM Theory [12], Asymptotic iteration method [13], and fourth-order compact and conservative scheme [14]. In nine methods above, there are still some disadvantages about the nonlinear, especially nonlinear term has a hard for computation. In 2016 Biazar, J. and Hosseini, K. [15] propose the modified Adomain decomposition method in order to solve singular initial value problem instead the Adomian decomposition Method conventional that consider nonlinear with Adomian polynomial which difficult to calculation and complexity procedure. The modified Adomian decomposition method has pass the test to solve Emden-Fowler type equations in linear and nonlinear and this method can find the solution in term of series convergence because the calculation only fist term of Adomian polynomial then the procedure of computation has decreased. At the present, many research which related about the solution of KleinGordon equation finding are studied, however there are no research found that the modified Adomian decomposition method has been applied for this equation. Consequently, the modified Adomian decomposition method is proposed to solve the problem Klein-Gordon equation in term of linear and nonlinear. In this paper, introduction is initiated in first section, then in the second section, the principle of traditional, the modified Adomain decomposition method is described and then application of Klein-Gordon in three forms of equation are illustrated follow as homogeneous linear equation, inhomogeneous linear equation and nonlinear equation respectively. Finally, conclusions of this paper are summarized. ∗ Corresponding

Author. Email address: [email protected]

Submitted to xxx

October 11, 2019

Journal Pre-proof New Submission 2. Modified Adomian Decomposition Method (MADM) We consider a general nonlinear inhomogeneous Klein-Gordon Equation with the initial conditions as follows utt (x, t) − uxx (x, t) + cu(x, t) + Nu(x, t) = h(x, t) With the initial conditions

(1)

u(x, 0) = f (x) , ut (x, 0) = g(x)

Where the complex valued functions u(x, t) is the solutions of the equation, t is the time variable, x is the space variables, c is a constant, N represents the general nonlinear differential operator and h(x, t), f (x), g(x) is source term. In an operator form Eq. (1) can be rewritten as Lu(x, t) = uxx (x, t) − cu(x, t) − Nu(x, t) + h(x, t) (2) ! ∂2 Where L is second order differential operator and the inverse operator L−1 is a two fold integral defined by ∂t2 Z tZ t −1 L = (·) dtdt (3) 0

−1

Applying L

0

to both sides of Eq. (2), we have L−1 Lu(x, t) = L−1 uxx (x, t) − cL−1 u(x, t) − L−1 Nu(x, t) + L−1 h(x, t)

Such that

Z tZ 0

t

[utt (x, t)] dtdt = u(x, t) − ut (x, 0)t − u(x, 0)

roo

L−1 Lu(x, t) =

f

Consider L−1 Lu(x, t) gives

0

u(x, t) = u(x, 0) + ut (x, 0)t + L−1 uxx (x, t) − cL−1 u(x, t) − L−1 Nu(x, t) + L−1 h(x, t)

rep

From the initial condition, the Eq. (4) is given

u(x, t) = f (x) + g(x)t + L−1 uxx (x, t) − cL−1 u(x, t) − L−1 Nu(x, t) + L−1 h(x, t)

(4)

The solution as an infinite series is given below

u(x, t) =

∞ X

un (x, t)

(5)

lP

n=0

The nonlinear term can be decomposed as

Nu(x, t) =

∞ X

An

(6)

n=0

rna

Where the Adomian polynomials An can be determined as follow  ∞  1 dn  X n  N  λ un  An = n! dλn   n=0

λ=0

Substituting the decomposition series Eq. (5) and Eq. (6) into both sides of Eq.(4) gives ∞  ∞ ∞ X X X  −1   un (x, t) = f (x) + g(x)t + L  unxx (x, t) − c un (x, t) n=0 n=0 n=0 ∞  X  − L−1  An  + L−1 [h(x, t)]

Jo u

(7)

n=0

2

(8)

Journal Pre-proof New Submission Adding the term L−1

P∞

n n=0 an t



− pL−1

P∞

n=0 an t

n



into the right hand side of Eq(8), we obtain   ∞  ∞ ∞ X X  X  −1  n −1 n    un (x, t) = f (x) + g(x)t + L  an t  − pL  an t  n=0 n=0 n=0 ∞  ∞  ∞ X X  X  −1  −1       + L  unxx (x, t) − c un (x, t) − L  An  + L−1 [h(x, t)] n=0

n=0

(9)

n=0

Where p is an artificial parameter and an , n = 0, 1, 2, ... are unknown coefficients. The Eq. (10) is defined  ∞ X  n −1   u0 = f (x) + g(x)t + L  an t  n=0 ∞  X    −1 −1  n u1 = L [h(x, t)] − pL  an t  + L−1 u0xx (x, t) − cu0 (x, t) − L−1 [A0 ] n=0  −1  u2 = L u1xx (x, t) − cu1 (x, t) − L−1 [A1 ]   u3 = L−1 u2xx (x, t) − cu2 (x, t) − L−1 [A2 ] .. .

(10)

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f

To avoid calculation of An , n = 0, 1, 2, ... let determine an , n = 0, 1, ... such that u1 = 0 . This implies that u2 = u3 = ... = 0 Setting p = 1 , yields the solution of Eq. (1) with the initial conditions as follows in Eq. (11) ∞  X  −1  n  u(x, t) = f (x) + g(x)t + L  an t  (11) n=0

rep

3. Application

In this section, the various applications of linear and nonlinear Klein-Gordon equation are shown and then the solution from the MADM are compared with the traditional ADM. Example 1. Consider the following homogeneous linear Klein-Gordon equation [9].

lP

utt (x, t) − uxx (x, t) − u(x, t) = 0

With the initial conditions

(12)

u(x, 0) = 0 , ut (x, 0) = sin(x)

The traditional ADM is expressed as follow in Eq. (13)

rna

∞ X

−1

un (x, t) = f (x) + g(x)t + L

n=0

∞  ∞ X X   unxx (x, t) − un (x, t)  n=0

n=0

Jo u

To solve the problem by the MADM, let us rewrite Eq.(13) as follows ∞  ∞  ∞ X X  X  −1  n −1 n    un (x, t) = f (x) + g(x)t + L  an t  − pL  an t  n=0 n=0 n=0  ∞ ∞ X X  + L−1  unxx (x, t) − un (x, t) n=0

n=0

3

(13)

Journal Pre-proof New Submission We have following recursive relation ∞  X  u0 = f (x) + g(x)t + L−1  an tn  n=0 ∞  X    un+1 = −pL−1  an tn  + L−1 unxx (x, t) − un (x, t) ; n ≥ 0 n=0

The the equation is obtained h i u0 (x, t) = t sin(x) + L−1 a0 + a1 t + a2 t2 + a3 t3 + ... Z tZ th i = t sin(x) + a0 + a1 t + a2 t2 + a3 t3 + ... dtdt 0

0

1 1 1 = t sin(x) + a0 t2 + a1 t3 + a2 t4 + ... 2 6 12 ∞  X    n  u1 (x, t) = −pL  an t  + L−1 u0xx (x, t) + u0 (x, t) n=0 h i −1 = −pL a0 + a1 t + a2 t2 + a3 t3 + ...   1 1 1 + L−1 −t sin(x) + t sin(x) + a0 t2 + a1 t3 + a2 t4 + ... 2 6 12 Z tZ th i = −p a0 + a1 t + a2 t2 + a3 t3 + ... dtdt 0

roo

f

−1

0

Z tZ t

 1 2 1 3 1 a0 t + a1 t + a2 t4 + ... dtdt 2 6 12 0 0     1 1 1 1 1 1 = − pa0 t2 − pa1 t3 + a0 − pa2 t4 + a1 − pa3 t5 + ... 2 6 24 12 120 20

rep

+

Setting u1 (x, t) = 0, we find

    1 1 1 1 1 1 − pa0 t2 − pa1 t3 + a0 − pa2 t4 + a1 − pa3 t5 + ... = 0 2 6 24 12 120 20

lP

It can be shown that a0 = 0, a1 = 0, a2 = 0, a3 = 0, ... Substituting the initial condition and the value of an , n = 0, 1, 2, ... in Eq. (11), we get u(x, t) = t sin(x) + L−1 [0] = t sin(x)

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It found that the calculated solution has accuracy and precision as same as the traditional Adomian decomposition method [9]. However, the calculation method has less complexity and fast convergence. Example 2. Consider the following inhomogeneous linear Klein-Gordon equation [9]. utt (x, t) − uxx (x, t) + u(x, t) = 2 cos(x)

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With the initial conditions

(14)

u(x, 0) = cos(x) , ut (x, 0) = 1

The traditional ADM is expressed as follow in Eq. (15) ∞ X n=0

∞  ∞ X X  un (x, t) = f (x) + g(x)t + L−1 [2 cos(x)] + L−1  unxx (x, t) − un (x, t) n=0

4

n=0

(15)

Journal Pre-proof New Submission To solve the problem by MADM, let us rewrite Eq. (15) as follows ∞  ∞  ∞ X X  X  un (x, t) = f (x) + g(x)t + L−1  an tn  + L−1 [2 cos(x)] − pL−1  an tn  n=0 n=0 n=0  ∞ X 
unxx (x, t) − un (x, t)  + L−1  n=0

We have following recursive relation ∞  X  n  u0 = f (x) + g(x)t + L  an t  n=0 ∞  X  −1 −1  n  un+1 = L [2 cos(x)] − pL  an t  n=0 ∞  X
 −1  + L  unxx (x, t) − un (x, t)  ; n ≥ 0 −1

n=0

0

f

h i u0 (x, t) = cos(x) + t + L−1 a0 + a1 t + a2 t2 + a3 t3 + ... Z tZ th i = cos(x) + t + a0 + a1 t + a2 t2 + a3 t3 + ... dtdt 0

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We obtain

1 1 1 1 = cos(x) + t + a0 t2 + a1 t3 + a2 t4 + + a3 t5 + ... 2 6 12 20

0

0

Z tZ t

rep

h i u1 (x, t) = L−1 [2 cos(x)] − pL−1 a0 + a1 t + a2 t2 + a3 t3 + ...    1 1 1 1 + L−1 − cos(x) − cos(x) + t + a0 t2 + a1 t3 + a2 t4 + a3 t5 + ... 2 6 12 20 Z tZ t Z tZ th i [2 cos(x)] dtdt − p = a0 + a1 t + a2 t2 + a3 t3 + ... dtdt 0

0

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Setting u1 (x, t) = 0 , we find

lP

 1 1 1 1 + −2 cos(x) − t − a0 t2 − a1 t3 − a2 t4 − a3 t5 + ... dtdt 2 6 12 20 0 0     1 1 1 1 1 = − pa0 t2 + − − pa1 t3 + − a0 − pa2 t4 2 6 6 24 12   1 1 a1 − pa3 t5 + ... + − 120 20

Jo u

    1 1 1 1 1 − pa0 t2 + − − pa1 t3 + − a0 − pa2 t4 2 6 6 2 12   1 1 + − a1 − pa3 t5 + ... = 0 120 20

It can be shown that a0 = 0, a1 = − p1 , a2 = 0, a3 = 6p12 , ... Substituting the initial condition and the value of an , n = 0, 1, 2, ... in Eq. (11), we get " # 1 1 u(x, t) = cos(x) + t + L−1 0 − t + 0 + 2 t3 + ... p 6p 5

Journal Pre-proof New Submission Let p = 1, we get   1 u(x, t) = cos(x) + t + L−1 −t + t3 + ... 6   1 3 1 5 = cos(x) + t + − t + t + ... 3! 5!   1 3 1 5 = cos(x) + t − t + t + ... 3! 5! = cos(x) + sin(t) It found that the calculated solution has accuracy and precision as same as the conventional method [9]. However, the calculation method has less complexity and fast convergence. Example 3. Consider the following nonlinear Klein-Gordon equation [9]. utt (x, t) − uxx (x, t) + u2 = 1 + 2xt + x2 t2 With the initial conditions

(16)

u(x, 0) = 1 , ut (x, 0) = x

The traditional ADM is expressed as follow in Eq. (16) −1

un (x, t) = f (x) + g(x)t + L

h

2 2

i

−1

1 + 2xt + x t + L

f

∞ X

n=0

roo

n=0

∞  ∞ X X   unxx (x, t) − An   n=0

n=0

We have following recursive relation

rep

To solve the problem by the MADM, let us rewrite Eq. (16) as follows ∞  ∞  ∞ X X  X  −1  n −1 2 2 −1 n    un (x, t) = f (x) + g(x)t + L  an t  + L [1 + x t + 2xt] − pL  an t  n=0 n=0 n=0 ∞  ∞ X X   + L−1  unxx (x, t) − An  n=0

∞  X  n  u0 = f (x) + g(x)t + L  an t  n=0 ∞  X  −1 2 2 −1  n un+1 = L [1 + x t + 2xt] − pL  an t  n=0  −1  + L unxx (x, t) − An ; n ≥ 0

h i u0 (x, t) = 1 + xt + L−1 a0 + a1 t + a2 t2 + a3 t3 + ... Z tZ th i = 1 + xt + a0 + a1 t + a2 t2 + a3 t3 + ... dtdt

Jo u

We obtain

rna

lP

−1

0

0

1 1 1 1 = 1 + xt + a0 t2 + a1 t3 + a2 t4 + a3 t5 + ... 2 6 12 20

6

(17)

Journal Pre-proof New Submission ∞  X    u1 (x, t) = L−1 1 + 2xt + x2 t2 − pL−1  an tn  + L−1 u0xx (x, t) − A0 n=0 ∞  X i h i h   −1  n 2 2 −1  an t  + L−1 0 − u20 = L 1 + 2xt + x t − pL  n=0 h i h i −1 2 2 −1 = L 1 + 2xt + x t − pL a0 + a1 t + a2 t2 + a3 t3 + ...    1 1 1 1 − L−1 1 + xt + a0 t2 + a1 t3 + ... 1 + xt + a0 t2 + a1 t3 + ... 2 6 2 6 Z tZ th Z tZ th i i a0 + a1 t + a2 t2 + a3 t3 + ... dtdt = 1 + 2xt + x2 t2 dtdt − p i

h

0

0

0

0

Z tZ t  1 − 1 + 2xt + x2 t2 + a0 t2 + a0 xt3 + a1 t3 + ... dtdt 3 0 0 1 1 1 1 = − pa0 t2 − pa1 t3 − pa2 t4 − pa3 t5 + ... 2 6 12 20 1 1 1 − a0 t4 − a0 xt5 − a1 t5 + ... 12 20 60

Setting u1 (x, t) = 0 , we find

roo

f

  1 1 1 1 − pa0 t2 − pa1 t3 + − a0 − pa2 t4 2 6 12 12   1 1 1 + − a0 x − a1 − pa3 t5 + ... = 0 20 60 20

rep

It can be shown that a0 = 0, a1 = 0, a2 = 0, a3 = 0, ... Substituting the initial condition and the value of an , n = 0, 1, 2, ... in Eq. (11), we get u(x, t) = 1 + xt + L−1 [0] = 1 + xt

It found that the calculated solution has accuracy and precision as same as the conventional method [9]. However, the calculation method has less complexity and fast convergence.

lP

Example 4. Consider the following nonlinear Klein-Gordon equation [6]. utt (x, t) − uxx (x, t) + u2 (x, t) = x2 t2

rna

With the initial conditions

(18)

u(x, 0) = 0 , ut (x, 0) = x

The following expression is shown in Eq. (18) ∞ X

−1

un (x, t) = f (x) + g(x)t + L

h

2 2

i

x t +L

n=0

Jo u

−1

∞  ∞ X X   unxx (x, t) − An   n=0

(19)

n=0

In Eq. (18) An are Adomian polynomials that represent nonlinear term. So Adomian polynomials are given as follow An = u2 (x, t)

7

Journal Pre-proof New Submission The few component of the Adomian polynomial are given as follow A0 = u20 A1 = 2u0 u1 .. . To solve the problem by MADM, let us rewrite Eq. (19) as follows ∞  ∞  ∞ X X  X  −1  n −1 2 2 −1 n    un (x, t) = f (x) + g(x)t + L  an t  + L [x t ] − pL  an t  n=0 n=0 n=0 ∞  ∞ X X   + L−1  unxx (x, t) − An  n=0

We have following recursive relation

∞  X  u0 = f (x) + g(x)t + L−1  an tn  n=0 ∞  X  n −1 2 2 −1   an t  un+1 = L [x t ] − pL  n=0   + L−1 unxx (x, t) − An ; n ≥ 0

f

We obtain

n=0

0

0

roo

h i u0 (x, t) = xt + L−1 a0 + a1 t + a2 t2 + a3 t3 + ... Z tZ th i = xt + a0 + a1 t + a2 t2 + a3 t3 + ... dtdt

rep

1 1 1 1 = xt + a0 t2 + a1 t3 + a2 t4 + a3 t5 + ... 2 6 12 20 ∞  h i X    −1 2 2 −1  n  u1 (x, t) = L +x t − pL  an t  + L−1 u0xx (x, t) − A0 n=0 Z tZ th i = −p a0 + a1 t + a2 t2 + a3 t3 + ... dtdt 0

0

rna

lP

Z tZ t  1 + −a0 xt3 − a1 xt3 + ... dtdt 3 0 0 1 1 1 = − pa0 t2 − pa1 t3 − pa2 t4 + ... 2 6 12

Setting u1 (x, t) = 0 , we find

Jo u

1 1 1 − pa0 t2 − pa1 t3 − pa2 t4 + ... = 0 2 6 12 It can be shown that a0 = 0, a1 = 0, a2 = 0, a3 = 0, ... Substituting the initial condition and the value of an , n = 0, 1, 2, ... in Eq. (11), we get u(x, t) = xt + L−1 [0] = xt

It found that the calculated solution has accuracy and precision as same as the conventional method [6]. However, the calculation method has less complexity and fast convergence. 8

Journal Pre-proof New Submission 4. Conclusion In this paper, modified Adomian decomposition method has been applied to homogeneous linear, inhomogeneous linear and nonlinear Klein-Gordon equation with initial conditions. The results show that the modified Adomian decomposition method is powerful, convergence solution faster than traditional Adomian decomposition method and easy to calculation because of this method only requires the calculation of the first Adomian polynomial. This is the advantage of this method that reduces the steps of calculation and gives accurate results when compared to other methods. The support evidence which show that modified Adomian decomposition has advantage than traditional Adomian decomposition method one is illustrated via the four examples of application which will be very clear with nonlinear Klein-Gordon equation. From the efficiency of this method, the author has the idea to apply modified Adomian decomposition method to solve the problem of tsunami which is a disaster in one nature that causes damage to humans and the environment. In order to be able to find the wave velocity that will hit the shore leading to the preparation can prevent future disasters. 5. Acknowledgement The authors would like to thank Division of Mathematics, Faculty of Science, Maejo University for research financial support. References

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[1] C. Sowmiya, R. Raja, J. Cao, G. Rajchakit, Impulsive discrete-time BAM neural networks with random parameter uncertainties and timevarying leakage delays: an asymptotic stability analysis, Nonlinear Dynamics, 91 (2018) 2571–2592. [2] C. Maharajan, R. Raja, J. Cao, G. Rajchakit, A. Alsaedi, Impulsive Cohen-Grossberg BAM neural networks with mixed time-delays: An exponential stability analysis issue, Neuro Computing, 275 (2018) 2588–2602. [3] C. Sowmiya, R. Raja, J. Cao, G. Rajchakit, A. Alsaedi, Enhanced robust nite-time passivity for Markovian jumping discrete-time BAM neural networks with leakage delay, Advances in Dierence Equations, 2017. [4] B. Sundara Vadivooa, R. Rajab, R. Aly Seadawyc, G. Rajchakitd, Nonlinear integro-differential equations with small unknown parameters: A controllability analysis problem, Mathematics and Computers in Simulation, 155 (2019) 15–26. [5] S. N. Datta, A. Ghosh, R. Chakraborty, Variational aspects of the Klein–Gordon equation, Indian Journal of Physics, 89 (2015) 181-187. [6] M. A. Ramadan, and M. S. Al-Luhaibi, Application of Sumudu decomposition method for solving linear and nonlinear Klein-Gordon equations, International Journal of Soft Computing and Engineering (IJSCE), 3 (2016) 138-140. [7] M. Hussain, and M. Khanf, A variational iterative method for solving the linear and nonlinear Klein-Gordon equations, Applied Mathematical Sciences, 4 (2010) 1931-1940. [8] H. Hosseinzadeh, H. Jafari, and M. Roohani, Application of laplace decomposition method for solving Klein-Gordon equation, World Applied Sciences Journal, 8 (2010) 809-813. [9] S. Kulkarni, and K. Takale, Application of Adomian decomposition method for solving linear and nonlinear Klein-Gordon equations, International Journal of Engineering, Contemporary Mathematics and Sciences, 1 (2015) 21-27. [10] A. K. Adio, Natural decomposition method for solving the linear and nonlinear Klein Gordon equations, International Journal of Research in Applied, 4 (2016) 59-72. [11] D. Kumar, J. Singh, D. Baleanu, A hybrid computational approach for Klein–Gordon equations on Cantor sets, Nonlinear Dynamics, 87 (2017) 511-517. [12] B. Gr´ebert, E. Paturel, KAM for the Klein Gordon equation on SdSd, Bollettino dell’Unione Matematica Italiana, 9 (2016) 237-288. [13] D. A. Nugraha, A. Suparmi, C. Cari and B. N. Pratiwi, Asymptotic iteration method for solution of the Kratzer potential in D-dimensional Klein-Gordon equation, Journal of Physics: Conference Series, 820 (2017) 1-8. [14] Y. Luo, X. Li and C. Gu, Fourth-order compact and energy conservative scheme for solving nonlinear Klein-Gordon equation, Numerical Methods for Partial Differential Equations, (2017) 1283–1304 [15] J. Biaza, and K. Hosseini, A modified Adomian decomposition method for singular initial value Emden-Fowler type equations, International Journal of Applied Mathematical Research, 5 (2016) 69-72.

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