# Numerical Analysis of the Klein-Gordon Equations by Using the New Iteration Transform Method

Authors: Ahmad Haji Zadeh; Kavikumar Jacob; Nehad Ali Shah; Jae Dong Chung
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper presents an analysis based on a mixture of the Laplace transform and the new iteration method to obtain new approximate results of the fractional-order Klein-Gordon equations in the Caputo-Fabrizio sense. So, a general system to investigate the approximate results of the fractional-order Klein-Gordon equations is obtained. This technique’s effectiveness is demonstrated by comparing the actual results of the fractional-order equations suggested with the results achieved.


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## Body

## 1. Introduction

Fractional partial differential equations (FPDEs) are critical tools for analyzing and simulating numerous narrative models in physics and mathematical models, such as electrical circuits, fluid dynamics, damping, induction, mathematical biology, ad relaxation, (Klimek, 2005; Baleanu et al., 2009; Kilbas et al., 2010; Jumarie, 2009; Mainardi, 2010; Ortigueira, 2010). Fractional derivatives provide more precise representations of real-world problems than integer-order derivatives; they are regarded as an effective technique for describing such physical problems. The subject of fractional calculus is an important and valuable branch of mathematics that plays a critical and severe role in explaining complex dynamic behavior in a wide range of application areas, helps to understand the essence of the matter as well as simplify the control design without any lack of inherited behavior, and describes even more complex structures [1, 2].The Klein-Gordon equations (KGEs) play an important role in physics, nonlinear optics, quantum field theory and solid state physics, plasma physics, kinematics, mathematical biology, and the recurrence of the initial state. The modeling of many phenomena, including the behavior of elementary particles and dislocation of crystals propagation, is the important applications of KGEs. To study solitons [3], examining nonlinear wave equations [4] and condensed matter physics equations gained the attention of scholars. In the previous few years, mathematicians have made many considerable efforts to find the solutions to these equations. There are many methods introduced to find the solution of these equations such as the radial basis functions [5], B-spline collocation method [5], auxiliary approach [6], and exponential-type potential, and there are some more methods mentioned in [7–11] for the solution of these equations. To solve the KGEs of a nonlinear type got tremendous attention of scholar, and a verity of methods were developed as mentioned in [12–14]. Some other methods are the stationary solution [15], the Homotopy perturbation technique [16], the tanh technique [17], the variation iteration technique [18], the traveling wave solutions, and so on.In the recent paper, we are applying new iterative transform method to KGEs of both linear and nonlinear orders of the following form:(1)∂ϱμ∂τϱ−∂2μ∂ζ2+Ψ=gζ,1<ρ≤2,with the boundary conditions
(2)μζ,0=0μτζ,0=kζ.Daftardar-Gejji and Jafari developed a new iterative approach for solving nonlinear equations in 2006 [19, 20]. Jafari et al. first applied the Laplace transformation in the iterative technique. They proposed a new straightforward technique called the iterative Laplace transform method (ILTM) [21] to look for the numerical solution of the FPDE system. The iterative Laplace transform method was used to solve linear and nonlinear partial differential equations such as the time-fractional Fokker-Planck equation [22], Zakharov-Kuznetsov equation [23], and Fornberg-Whitham equation [24]. The Elzaki transform was used to modify the iterative technique, known as the new iterative transform method.The new iterative transform method is implemented to investigate the fractional-order of the Klein-Gordon equations. The solution of the fractional-order problems and integral-order models is calculated applying the current techniques. The proposed approach is also helpful for dealing with other fractional-orders of linear and nonlinear PDEs.


## 2. Fractional Calculus

This section provides some fundamental concepts of fractional calculus.Definition 1.
The Liouville-Caputo operator (C) is given as [25]
(3)DIϱuζ,I=1Γn−β∫0II−θn−1unζ,θdθ,n−1<ρ<n,where unζ,θ is the derivative of integer nth order of uζ,I, n=1,2,⋯∈N and n−1<ϱ≤n. If 0<ϱ≤1; then, we defined the Laplace transformation for the Caputo fractional derivative as follows:
(4)LDIϱuζ,Is=sϱLuζ,Is−sϱ−1uζ,0.Definition 2.
The Caputo-Fabrizio operator (CF) is define as given [25]:
(5)DIϱuζ,I=2−ϱMϱ2n−ϱ∫0Iexp−ϱI−θn−ϱunζ,θdθ,n<ϱ≤n+1.
Mϱ is a normalization form, and M0=M1=1. The exponential law is used as the nonsingular kernel in this fractional operator.
If0<ρ≤1, then we define the Caputo-Fabrizio of the Laplace transformation for the fractional derivative is given as
(6)LDIϱuζ,Is=sLuζ,Is−uζ,0s+ϱ1−s.


## 3. The Iterative Transform Method Basic Procedure

Consider a particular type of a FPDE.(7)Dτϱυζ,τ+Mυζ,τ+Nυζ,τ=hζ,τ,n∈N,n−1<ϱ≤n,where the functions of linear and nonlinear are M and N, respectively.With the initial condition(8)υkζ,0=gkζ,k=0,1,2⋯n−1,implementing the Laplace transformation of Equation (7), we have
(9)LDτϱυζ,τ+LMυζ,τ+Nυζ,τ=Lhζ,τ.Applying the Laplace differentiation is given to(10)Lυζ,τ=1sυζ,0+s+ϱ1−ss2Lhζ,τ−s+ϱ1−ss2LMυζ,τ+Nυζ,τ,using the inverse Laplace transformation of Equation (10) into
(11)υζ,τ=L−11sυζ,0+s+ϱ1−ss2Lhζ,τ−L−1s+ϱ1−ss2LMυζ,τ+Nυζ,τ.As through the iterative technique, we have(12)υζ,τ=∑m=0∞υmζ,τ.Further, the operatorM is linear; therefore
(13)M∑m=0∞υmζ,τ=∑m=0∞Mυmζ,τ,and the operator N is nonlinear; we have the following
(14)N∑m=0∞υmζ,τ=υ0ζ,τ+M∑k=0mυkζ,τ−N∑k=0mυkζ,τ.Putting Equations (12)–(14) in Equation (11), we obtain
(15)∑m=0∞υmζ,τ=L−11sυζ,0+s+ϱ1−ss2Lhζ,τ−L−1s+ϱ1−ss2EM∑k=0mυkζ,τ−N∑k=0mυkζ,τ.The new iterative transform method is defined as(16)υ0ζ,τ=L−11sυζ,0+s+ϱ1−ss2Lgζ,τ,υ1ζ,τ=−L−1s+ϱ1−ss2LMυ0ζ,τ+Nυ0ζ,τ,υm+1ζ,τ=−L−1s+ϱ1−ss2L−M∑k=0mυkζ,τ−N∑k=0mυkζ,τ,m≥1.Finally, Equations (7) and (8) provide the m-terms solution in a series form given as
(17)υζ,τ≅υ0ζ,τ+υ1ζ,τ+υ2ζ,τ+⋯+υmζ,τ,m=1,2,⋯.


## 4. Applications of the Proposed Method

### 4.1. Example

Consider the fractional-order Klein-Gordon equation [18]
(18)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μζ,τ=00<ζτ<00<ϱ≤1,with the initial conditions
(19)μζ,0=0,μτζ,0=ζ.Applying the Laplace transform to Equation (18), we have
(20)sϱLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μζ,τ,(21)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ.Applying the inverse Laplace transform of Equation (21), we have
(22)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ.Now, by using the suggested analytical method, we get(23)μ0ζ,τ=ζτ,μ1ζ,τ=L−1s+ρ1−ss2L∂2μ0ζ,τ∂ζ2−μ0ζ,τ=−ζτ26τρ+3−3ρ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ1ζ,τ=ζτ312010τϱ−10τϱ2−40ϱ+20+20ϱ2+τ2ϱ2,μ3ζ,τ=L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ,μ3ζ,τ=−ζτ45040630ϱ2−630ϱ+210−210ϱ3+τ3ϱ3+21τ2ϱ2−21τ2ϱ3−252τϱ2+126τϱ+126τϱ3,μ4ζ,τ=L−1s+ϱ1−ss2L∂2μ3ζ,τ∂ζ2−μ3ζ,τ,μ4ζ,τ=ζτ5362880τϱ+6−6ϱ∗τ3ϱ3−30τ2ϱ3+30τ2ϱ2+252τϱ3−504τϱ2+252τϱ,−504ϱ3+1512ϱ2−1512ϱ+504,⋮μn+1ζ,τ=L−1s+ϱ1−sns2n+2L∂2μnζ,τ∂ζ2−μnζ,τ.The series form result is(24)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=ζτ−ζτ26τϱ+3−3ϱ+ζτ312010τϱ−10τϱ2−40ϱ+20+20ϱ2+τ2ϱ2−ζτ45040630ϱ2−630ϱ+210−210ϱ3+τ3ϱ3+21τ2ϱ2−21τ2ϱ3−252τϱ2+126τϱ+126τϱ3+ζτ5362880τϱ+6−6ϱτ3ϱ3−30τ2ϱ3+30τ2ϱ2+252τϱ3−504τϱ2+252τϱ−504ϱ3+1512ϱ2−1512ϱ+504−⋯.The problem has the exact solution atϱ=1:
(25)μζ,τ=ζsinτ.In Figure1, the exact and the approximate solutions of example 1 at ϱ=1 are shown, and the second graph shows the 3D graph of different fractional-order ρ, respectively. From the given graphs, it can be shown that both the approximate and exact solutions are in close relation with each other. Also, in Figure 2, the 2D figure of the approximate solutions of problem 1 is analysis at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 1
(a) Exact and an approximate graph of problem 1 and (b) different fractional-order graphs of problem 1.
(a)(b)Figure 2
The different fractional-orders with respect toζ and τ of problem 1.


### 4.2. Example

Consider the fractional-order Klein-Gordon equation [18]:
(26)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μζ,τ=2sinζ0<ζτ<00<ϱ≤1,with the initial conditions
(27)μζ,0=sinζ,μτζ,0=1.We apply the Laplace transformation to Equation (26), and we get
(28)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μζ,τ+2sinζ,(29)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ+2sinζ.Now, using the inverse Laplace transformation of Equation (29), we have
(30)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ+2sinζ.Now, by using the suggested analytical method, we get(31)μ0ζ,τ=sinζ+τ,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ0ζ,τ+2sinζ=−τ26τϱ+3−3ϱ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ1ζ,τ+2sinζ,μ2ζ,τ=τ212020τ−60τϱ+20τϱ2+τ3ϱ2−60+60ϱ+10τ2ϱ−10τ2ϱ2,μ3ζ,τ=L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+2sinζ,μ3ζ,τ=−τ250404200τϱ−1680τ−1680τϱ2−336τ3ϱ2+126ϱτ3+126τ3ϱ3+ϱ3τ5+2520−2520ϱ+1470τ2ϱ2−1470τ2ϱ+210τ2−210τ2ϱ3+21ϱ2τ4−21τ4ϱ3,μ4ζ,τ=L−1s+ϱ1−ss2L∂2μ3ζ,τ∂ζ2−μ3ζ,τ+2sinζ,μ4ζ,τ=τ2362880181440τϱ2+181440τ−423360τϱ+81648τ3ϱ2−39312τ3ϱ3−39312ϱτ3+3024τ3+3024τ3ϱ4+36ϱ3τ6−36ϱ4τ6−1080ϱ3τ5+432ϱ2τ5+432τ5ϱ4+τ7ϱ4−181440+11440ϱ+226800τ2ϱ+45360τ2ϱ3−226800τ2ϱ2−45360τ2−10584ϱ2τ4+10584τ4ϱ3−2016τ4ϱ4+2016ϱτ4,⋮μnζ,τ=L−1s+ϱ1−sns2n+2L∂2μnζ,τ∂ζ2−μnζ,τ+2sinζ.The series form result is(32)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ==sinζ+τ−τ26τϱ+3−3ϱ+τ212020τ−60τϱ+20τϱ2+τ3ϱ2−60+60ϱ+10τ2ϱ−10τ2ϱ2+−τ250404200τϱ−1680τ−1680τϱ2−336τ3ϱ2+126ϱτ3+126τ3ϱ3+ϱ3τ5+2520−2520ϱ+1470τ2ϱ2−1470τ2ϱ+210τ2−210τ2ϱ3+21ϱ2τ4−21τ4ϱ3+τ2362880181440τϱ2+181440τ−423360τϱ+81648τ3ϱ2−39312τ3ϱ3−39312ϱτ3+3024τ3+3024τ3ϱ4+36ϱ3τ6−36ϱ4τ6−1080ϱ3τ5+432ϱ2τ5+432τ5ϱ4+τ7ϱ4−181440+11440ρ+226800τ2ϱ+45360τ2ϱ3−226800τ2ϱ2−45360τ2−10584ϱ2τ4+10584τ4ϱ3−2016τ4ϱ4+2016ϱτ4+⋯.The problem has the exact solution atϱ=1:
(33)μζ,τ=sinζ+sinτ.In Figure3, the exact and the approximate solutions of example 2 at ϱ=1 are shown. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. Also, in Figure 4, the 2D graph of the approximate results of problem 2 is investigated at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 3
The exact and an approximate graph of problem 2.Figure 4
The different fractional-orders with respect toζ and τ of problem 2.


### 4.3. Example

Consider the fractional-order nonlinear Klein-Gordon equation [18]:
(34)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μ2ζ,τ=ζ2τ20<ζτ<00<ϱ≤1,with the initial conditions
(35)μζ,0=0,μτζ,0=ζ.Using the Laplace transform to Equation (34), we get
(36)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μ2ζ,τ+ζ2τ2,(37)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μ2ζ,τ+ζ2τ2.Applying the inverse Laplace transform of Equation (37), we have
(38)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+ζ2τ2.Now, by using the suggested an approximate method, we get(39)μ0ζ,τ=ζτ,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ02ζ,τ+ζ2τ2=0,⋮μnζ,τ=L−1s+ϱ1−ss2L∂2μnζ,τ∂ζ2−μn2ζ,τ+ζ2τ2.The series form result is(40)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=ζτ+0+⋯.The problem has the exact solution atϱ=2:
(41)μζ,τ=ζτ.Figure5 compares the exact solution and approximate solution of example 3 for the nonlinear fractional-order Klein-Gordon equation at ϱ=1. The figure shows the close relationship between the exact and an approximate solution.Figure 5
The exact and an approximate graph of problem 3.


### 4.4. Example

Consider the fractional-order nonlinear Klein-Gordon equation [18]:
(42)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μ2ζ,τ=2ζ2−2τ2+ζ4τ40<ζτ<00<ϱ≤1,with the initial conditions
(43)μζ,0=μτζ,0=0.Using the Laplace transform to Equation (42), we get
(44)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4,(45)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4.Applying the inverse Laplace transform of Equation (45), we have
(46)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4.Now, by using the suggested analytical method, we get(47)μ0ζ,τ=0,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ02ζ,τ+2ζ2−2τ2+ζ4τ4,μ1ζ,τ=τ30−5ϱτ3+20τ2ϱζ4τ5ϱ+60ζ2−60ϱζ2+30ζ2τϱ+6τ4ζ4−6τ4ζ4ϱ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ12ζ,τ+2ζ2−2τ2+ζ4τ4,μ2ζ,τ=−τ162116200−32432400τϱ2−2702700τ3ϱ2−32432400τ2ϱ+21621600τ2ϱ2−228228τ9ϱ3ζ4+456456τ9ϱ2ζ4−228228τ9ϱζ4+12012τ9ζ6ϱ3−32760ϱ2τ10ζ4+32760ϱ3τ10ζ4−1365τ11ϱ3ζ4+22932τ11ζ8ϱ3+22932τ11ζ8ϱ+5005τ9ϱ3+100100ϱ2τ8−100100ϱ3τ8−1158300τ7ϱ2+579150τ7ϱ−480480τ8ζ4+58968τ10ζ8+1853280τ6ζ6+3088800τ6ϱ2−3088800τ6ϱ+21621600τ2ζ4+579150ϱ3τ7−8648640τ4ζ2−6486480ζ2τ5+1029600τ6+240240ϱ2τ8ζ6−240240ϱ3τ8ζ6+1312740τ7ζ6ϱ3−2625480τ7ζ6ϱ2−96525τ7ϱ3ζ2+1312740τ7ζ6ϱ−115830τ7ϱ2ζ2−6486480ϱ3τ4ζ4+6486480ϱ2τ4ζ4−43243200τ3ϱ2ζ4+21621600τ3ϱ3ζ4+21621600τ3ϱζ4+99ζ8ϱ3τ13−1853280τ6ϱζ2+308880τ6ϱ2ζ2+1544400τ6ϱ3ζ2+540540τ5ϱ3ζ4−6846840τ5ϱ3ζ2+6126120τ5ϱζ2+7207200τ5ϱ2ζ2+2772ϱ2τ12ζ8−2772ϱ3τ12ζ8+480480τ8ϱ3ζ4−1441440τ8ϱ2ζ4+1441440τ8ϱζ4−58968τ10ζ8ϱ3−176904τ10ζ8ϱ−1853280τ6ζ6ϱ3+5559840τ6ζ6ϱ2−5559840τ6ζ6ϱ+64864800τ2ϱ2ζ4−21621600τ2ϱ3ζ4−64864800τ2ϱζ4+8648640τ4ϱ3ζ2−25945920τ4ϱ2ζ2+176904τ10ζ8ϱ2+25945920ζ2ϱτ4−45864τ11ζ8ϱ2+2702700ϱτ3−32432400ζ2−540540ζ4τ5ϱ−16216200ζ2τϱ+3243240τ4ζ4ρ−1029600ρ3τ6−3243240τ4ζ4−32432400τ+32432400ρζ2+64864800τϱ+10810800τ2,⋮μnζ,τ=L−1s+ϱ1−ss2L∂2μnζ,τ∂ζ2−μn2ζ,τ+2ζ2−2τ2+ζ4τ4.The series form result is(48)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=τ30−5ϱτ3+20τ2ϱζ4τ5ϱ+60ζ2−60ϱζ2+30ζ2τϱ+6τ4ζ4−6τ4ζ4ϱ−τ162116200−32432400τϱ2−2702700τ3ϱ2−32432400τ2ϱ+21621600τ2ϱ2−228228τ9ϱ3ζ4+456456τ9ϱ2ζ4−228228τ9ϱζ4+12012τ9ζ6ϱ3−32760ϱ2τ10ζ4+32760ϱ3τ10ζ4−1365τ11ϱ3ζ4+22932τ11ζ8ϱ3+22932τ11ζ8ϱ+5005τ9ϱ3+100100ϱ2τ8−100100ϱ3τ8−1158300τ7ϱ2+579150τ7ϱ−480480τ8ζ4+58968τ10ζ8+1853280τ6ζ6+3088800τ6ϱ2−3088800τ6ϱ+21621600τ2ζ4+579150ϱ3τ7−8648640τ4ζ2−6486480ζ2τ5+1029600τ6+240240ϱ2τ8ζ6−240240ϱ3τ8ζ6+1312740τ7ζ6ϱ3−2625480τ7ζ6ϱ2−96525τ7ϱ3ζ2+1312740τ7ζ6ϱ−115830τ7ϱ2ζ2−6486480ϱ3τ4ζ4+6486480ϱ2τ4ζ4−43243200τ3ϱ2ζ4+21621600τ3ϱ3ζ4+21621600τ3ϱζ4+99ζ8ϱ3τ13−1853280τ6ϱζ2+308880τ6ϱ2ζ2+1544400τ6ϱ3ζ2+540540τ5ϱ3ζ4−6846840τ5ϱ3ζ2+6126120τ5ϱζ2+7207200τ5ϱ2ζ2+2772ϱ2τ12ζ8−2772ϱ3τ12ζ8+480480τ8ϱ3ζ4−1441440τ8ϱ2ζ4+1441440τ8ϱζ4−58968τ10ζ8ϱ3−176904τ10ζ8ϱ−1853280τ6ζ6ϱ3+5559840τ6ζ6ϱ2−5559840τ6ζ6ϱ+64864800τ2ϱ2ζ4−21621600τ2ϱ3ζ4−64864800τ2ϱζ4+8648640τ4ϱ3ζ2−25945920τ4ϱ2ζ2+176904τ10ζ8ϱ2+25945920ζ2ϱτ4−45864τ11ζ8ϱ2+2702700ϱτ3−32432400ζ2−540540ζ4τ5ϱ−16216200ζ2τϱ+3243240τ4ζ4ϱ−1029600ϱ3τ6−3243240τ4ζ4−32432400τ+32432400ϱζ2+64864800τϱ+10810800τ2+⋯.The problem has the exact solution atϱ=1:
(49)μζ,τ=ζ2τ2.In Figure6, the exact and the approximate solutions of example 4 at ϱ=1 are shown, and the second graph shows the 3D graph of different fractional-order ρ, respectively. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 6
The exact and an approximate graph and different fractional-order graphs of problem 4.


## 4.1. Example

Consider the fractional-order Klein-Gordon equation [18]
(18)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μζ,τ=00<ζτ<00<ϱ≤1,with the initial conditions
(19)μζ,0=0,μτζ,0=ζ.Applying the Laplace transform to Equation (18), we have
(20)sϱLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μζ,τ,(21)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ.Applying the inverse Laplace transform of Equation (21), we have
(22)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ.Now, by using the suggested analytical method, we get(23)μ0ζ,τ=ζτ,μ1ζ,τ=L−1s+ρ1−ss2L∂2μ0ζ,τ∂ζ2−μ0ζ,τ=−ζτ26τρ+3−3ρ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ1ζ,τ=ζτ312010τϱ−10τϱ2−40ϱ+20+20ϱ2+τ2ϱ2,μ3ζ,τ=L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ,μ3ζ,τ=−ζτ45040630ϱ2−630ϱ+210−210ϱ3+τ3ϱ3+21τ2ϱ2−21τ2ϱ3−252τϱ2+126τϱ+126τϱ3,μ4ζ,τ=L−1s+ϱ1−ss2L∂2μ3ζ,τ∂ζ2−μ3ζ,τ,μ4ζ,τ=ζτ5362880τϱ+6−6ϱ∗τ3ϱ3−30τ2ϱ3+30τ2ϱ2+252τϱ3−504τϱ2+252τϱ,−504ϱ3+1512ϱ2−1512ϱ+504,⋮μn+1ζ,τ=L−1s+ϱ1−sns2n+2L∂2μnζ,τ∂ζ2−μnζ,τ.The series form result is(24)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=ζτ−ζτ26τϱ+3−3ϱ+ζτ312010τϱ−10τϱ2−40ϱ+20+20ϱ2+τ2ϱ2−ζτ45040630ϱ2−630ϱ+210−210ϱ3+τ3ϱ3+21τ2ϱ2−21τ2ϱ3−252τϱ2+126τϱ+126τϱ3+ζτ5362880τϱ+6−6ϱτ3ϱ3−30τ2ϱ3+30τ2ϱ2+252τϱ3−504τϱ2+252τϱ−504ϱ3+1512ϱ2−1512ϱ+504−⋯.The problem has the exact solution atϱ=1:
(25)μζ,τ=ζsinτ.In Figure1, the exact and the approximate solutions of example 1 at ϱ=1 are shown, and the second graph shows the 3D graph of different fractional-order ρ, respectively. From the given graphs, it can be shown that both the approximate and exact solutions are in close relation with each other. Also, in Figure 2, the 2D figure of the approximate solutions of problem 1 is analysis at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 1
(a) Exact and an approximate graph of problem 1 and (b) different fractional-order graphs of problem 1.
(a)(b)Figure 2
The different fractional-orders with respect toζ and τ of problem 1.


## 4.2. Example

Consider the fractional-order Klein-Gordon equation [18]:
(26)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μζ,τ=2sinζ0<ζτ<00<ϱ≤1,with the initial conditions
(27)μζ,0=sinζ,μτζ,0=1.We apply the Laplace transformation to Equation (26), and we get
(28)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μζ,τ+2sinζ,(29)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ+2sinζ.Now, using the inverse Laplace transformation of Equation (29), we have
(30)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μζ,τ∂ζ2−μζ,τ+2sinζ.Now, by using the suggested analytical method, we get(31)μ0ζ,τ=sinζ+τ,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ0ζ,τ+2sinζ=−τ26τϱ+3−3ϱ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ1ζ,τ+2sinζ,μ2ζ,τ=τ212020τ−60τϱ+20τϱ2+τ3ϱ2−60+60ϱ+10τ2ϱ−10τ2ϱ2,μ3ζ,τ=L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+2sinζ,μ3ζ,τ=−τ250404200τϱ−1680τ−1680τϱ2−336τ3ϱ2+126ϱτ3+126τ3ϱ3+ϱ3τ5+2520−2520ϱ+1470τ2ϱ2−1470τ2ϱ+210τ2−210τ2ϱ3+21ϱ2τ4−21τ4ϱ3,μ4ζ,τ=L−1s+ϱ1−ss2L∂2μ3ζ,τ∂ζ2−μ3ζ,τ+2sinζ,μ4ζ,τ=τ2362880181440τϱ2+181440τ−423360τϱ+81648τ3ϱ2−39312τ3ϱ3−39312ϱτ3+3024τ3+3024τ3ϱ4+36ϱ3τ6−36ϱ4τ6−1080ϱ3τ5+432ϱ2τ5+432τ5ϱ4+τ7ϱ4−181440+11440ϱ+226800τ2ϱ+45360τ2ϱ3−226800τ2ϱ2−45360τ2−10584ϱ2τ4+10584τ4ϱ3−2016τ4ϱ4+2016ϱτ4,⋮μnζ,τ=L−1s+ϱ1−sns2n+2L∂2μnζ,τ∂ζ2−μnζ,τ+2sinζ.The series form result is(32)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ==sinζ+τ−τ26τϱ+3−3ϱ+τ212020τ−60τϱ+20τϱ2+τ3ϱ2−60+60ϱ+10τ2ϱ−10τ2ϱ2+−τ250404200τϱ−1680τ−1680τϱ2−336τ3ϱ2+126ϱτ3+126τ3ϱ3+ϱ3τ5+2520−2520ϱ+1470τ2ϱ2−1470τ2ϱ+210τ2−210τ2ϱ3+21ϱ2τ4−21τ4ϱ3+τ2362880181440τϱ2+181440τ−423360τϱ+81648τ3ϱ2−39312τ3ϱ3−39312ϱτ3+3024τ3+3024τ3ϱ4+36ϱ3τ6−36ϱ4τ6−1080ϱ3τ5+432ϱ2τ5+432τ5ϱ4+τ7ϱ4−181440+11440ρ+226800τ2ϱ+45360τ2ϱ3−226800τ2ϱ2−45360τ2−10584ϱ2τ4+10584τ4ϱ3−2016τ4ϱ4+2016ϱτ4+⋯.The problem has the exact solution atϱ=1:
(33)μζ,τ=sinζ+sinτ.In Figure3, the exact and the approximate solutions of example 2 at ϱ=1 are shown. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. Also, in Figure 4, the 2D graph of the approximate results of problem 2 is investigated at different fractional-order ρ for ζ and τ. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 3
The exact and an approximate graph of problem 2.Figure 4
The different fractional-orders with respect toζ and τ of problem 2.


## 4.3. Example

Consider the fractional-order nonlinear Klein-Gordon equation [18]:
(34)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μ2ζ,τ=ζ2τ20<ζτ<00<ϱ≤1,with the initial conditions
(35)μζ,0=0,μτζ,0=ζ.Using the Laplace transform to Equation (34), we get
(36)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μ2ζ,τ+ζ2τ2,(37)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μ2ζ,τ+ζ2τ2.Applying the inverse Laplace transform of Equation (37), we have
(38)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+ζ2τ2.Now, by using the suggested an approximate method, we get(39)μ0ζ,τ=ζτ,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ02ζ,τ+ζ2τ2=0,⋮μnζ,τ=L−1s+ϱ1−ss2L∂2μnζ,τ∂ζ2−μn2ζ,τ+ζ2τ2.The series form result is(40)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=ζτ+0+⋯.The problem has the exact solution atϱ=2:
(41)μζ,τ=ζτ.Figure5 compares the exact solution and approximate solution of example 3 for the nonlinear fractional-order Klein-Gordon equation at ϱ=1. The figure shows the close relationship between the exact and an approximate solution.Figure 5
The exact and an approximate graph of problem 3.


## 4.4. Example

Consider the fractional-order nonlinear Klein-Gordon equation [18]:
(42)∂CFϱ+1μζ,τ∂τϱ+1−∂2μζ,τ∂ζ2+μ2ζ,τ=2ζ2−2τ2+ζ4τ40<ζτ<00<ϱ≤1,with the initial conditions
(43)μζ,0=μτζ,0=0.Using the Laplace transform to Equation (42), we get
(44)s2s+ϱ1−sLμζ,τ=μ0ζ,0s−1+μτζ,0s−2+L∂2μζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4,(45)Lμζ,τ=s−1μζ,0+s−2μτζ,0+s+ϱ1−ss2L∂2μζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4.Applying the inverse Laplace transform of Equation (45), we have
(46)μζ,τ=L−1s−1μζ,0+s−2μτζ,0+L−1s+ϱ1−ss2L∂2μ2ζ,τ∂ζ2−μ2ζ,τ+2ζ2−2τ2+ζ4τ4.Now, by using the suggested analytical method, we get(47)μ0ζ,τ=0,μ1ζ,τ=L−1s+ϱ1−ss2L∂2μ0ζ,τ∂ζ2−μ02ζ,τ+2ζ2−2τ2+ζ4τ4,μ1ζ,τ=τ30−5ϱτ3+20τ2ϱζ4τ5ϱ+60ζ2−60ϱζ2+30ζ2τϱ+6τ4ζ4−6τ4ζ4ϱ,μ2ζ,τ=L−1s+ϱ1−ss2L∂2μ1ζ,τ∂ζ2−μ12ζ,τ+2ζ2−2τ2+ζ4τ4,μ2ζ,τ=−τ162116200−32432400τϱ2−2702700τ3ϱ2−32432400τ2ϱ+21621600τ2ϱ2−228228τ9ϱ3ζ4+456456τ9ϱ2ζ4−228228τ9ϱζ4+12012τ9ζ6ϱ3−32760ϱ2τ10ζ4+32760ϱ3τ10ζ4−1365τ11ϱ3ζ4+22932τ11ζ8ϱ3+22932τ11ζ8ϱ+5005τ9ϱ3+100100ϱ2τ8−100100ϱ3τ8−1158300τ7ϱ2+579150τ7ϱ−480480τ8ζ4+58968τ10ζ8+1853280τ6ζ6+3088800τ6ϱ2−3088800τ6ϱ+21621600τ2ζ4+579150ϱ3τ7−8648640τ4ζ2−6486480ζ2τ5+1029600τ6+240240ϱ2τ8ζ6−240240ϱ3τ8ζ6+1312740τ7ζ6ϱ3−2625480τ7ζ6ϱ2−96525τ7ϱ3ζ2+1312740τ7ζ6ϱ−115830τ7ϱ2ζ2−6486480ϱ3τ4ζ4+6486480ϱ2τ4ζ4−43243200τ3ϱ2ζ4+21621600τ3ϱ3ζ4+21621600τ3ϱζ4+99ζ8ϱ3τ13−1853280τ6ϱζ2+308880τ6ϱ2ζ2+1544400τ6ϱ3ζ2+540540τ5ϱ3ζ4−6846840τ5ϱ3ζ2+6126120τ5ϱζ2+7207200τ5ϱ2ζ2+2772ϱ2τ12ζ8−2772ϱ3τ12ζ8+480480τ8ϱ3ζ4−1441440τ8ϱ2ζ4+1441440τ8ϱζ4−58968τ10ζ8ϱ3−176904τ10ζ8ϱ−1853280τ6ζ6ϱ3+5559840τ6ζ6ϱ2−5559840τ6ζ6ϱ+64864800τ2ϱ2ζ4−21621600τ2ϱ3ζ4−64864800τ2ϱζ4+8648640τ4ϱ3ζ2−25945920τ4ϱ2ζ2+176904τ10ζ8ϱ2+25945920ζ2ϱτ4−45864τ11ζ8ϱ2+2702700ϱτ3−32432400ζ2−540540ζ4τ5ϱ−16216200ζ2τϱ+3243240τ4ζ4ρ−1029600ρ3τ6−3243240τ4ζ4−32432400τ+32432400ρζ2+64864800τϱ+10810800τ2,⋮μnζ,τ=L−1s+ϱ1−ss2L∂2μnζ,τ∂ζ2−μn2ζ,τ+2ζ2−2τ2+ζ4τ4.The series form result is(48)μζ,τ=μ0ζ,τ+μ1ζ,τ+μ2ζ,τ+μ3ζ,τ+⋯μnζ,τ,μζ,τ=τ30−5ϱτ3+20τ2ϱζ4τ5ϱ+60ζ2−60ϱζ2+30ζ2τϱ+6τ4ζ4−6τ4ζ4ϱ−τ162116200−32432400τϱ2−2702700τ3ϱ2−32432400τ2ϱ+21621600τ2ϱ2−228228τ9ϱ3ζ4+456456τ9ϱ2ζ4−228228τ9ϱζ4+12012τ9ζ6ϱ3−32760ϱ2τ10ζ4+32760ϱ3τ10ζ4−1365τ11ϱ3ζ4+22932τ11ζ8ϱ3+22932τ11ζ8ϱ+5005τ9ϱ3+100100ϱ2τ8−100100ϱ3τ8−1158300τ7ϱ2+579150τ7ϱ−480480τ8ζ4+58968τ10ζ8+1853280τ6ζ6+3088800τ6ϱ2−3088800τ6ϱ+21621600τ2ζ4+579150ϱ3τ7−8648640τ4ζ2−6486480ζ2τ5+1029600τ6+240240ϱ2τ8ζ6−240240ϱ3τ8ζ6+1312740τ7ζ6ϱ3−2625480τ7ζ6ϱ2−96525τ7ϱ3ζ2+1312740τ7ζ6ϱ−115830τ7ϱ2ζ2−6486480ϱ3τ4ζ4+6486480ϱ2τ4ζ4−43243200τ3ϱ2ζ4+21621600τ3ϱ3ζ4+21621600τ3ϱζ4+99ζ8ϱ3τ13−1853280τ6ϱζ2+308880τ6ϱ2ζ2+1544400τ6ϱ3ζ2+540540τ5ϱ3ζ4−6846840τ5ϱ3ζ2+6126120τ5ϱζ2+7207200τ5ϱ2ζ2+2772ϱ2τ12ζ8−2772ϱ3τ12ζ8+480480τ8ϱ3ζ4−1441440τ8ϱ2ζ4+1441440τ8ϱζ4−58968τ10ζ8ϱ3−176904τ10ζ8ϱ−1853280τ6ζ6ϱ3+5559840τ6ζ6ϱ2−5559840τ6ζ6ϱ+64864800τ2ϱ2ζ4−21621600τ2ϱ3ζ4−64864800τ2ϱζ4+8648640τ4ϱ3ζ2−25945920τ4ϱ2ζ2+176904τ10ζ8ϱ2+25945920ζ2ϱτ4−45864τ11ζ8ϱ2+2702700ϱτ3−32432400ζ2−540540ζ4τ5ϱ−16216200ζ2τϱ+3243240τ4ζ4ϱ−1029600ϱ3τ6−3243240τ4ζ4−32432400τ+32432400ϱζ2+64864800τϱ+10810800τ2+⋯.The problem has the exact solution atϱ=1:
(49)μζ,τ=ζ2τ2.In Figure6, the exact and the approximate solutions of example 4 at ϱ=1 are shown, and the second graph shows the 3D graph of different fractional-order ρ, respectively. From the given figures, it can be seen that both the approximate and exact solutions are in close contact with each other. It is demonstrated that the outcomes of time-fractional problems converge to an integer-order effect as the time-fractional evaluation to integer-order.Figure 6
The exact and an approximate graph and different fractional-order graphs of problem 4.


## 5. Conclusion

In this paper, the iterative transformation method is implemented to achieve approximate analytical results of the fractional-order Klein-Gordon equations, which is widely applied in problems for spatial effects in applied sciences. In physical models, the technique yields series form results that converge very quickly. The obtained results in this article are expected to be important for further analysis of the sophisticated nonlinear models. The calculations of this method are very simple and straightforward. As a result, we conclude that this technique can be used to solve a variety of nonlinear fractional-order partial differential equation systems.



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