The Natural Transform Decomposition Method for Solving Fractional Klein-Gordon Equation

Mohamed Elbadri^1,2

¹Department of Mathematics, Faculty of Sciences and Arts, Jouf University, Tubarjal, Saudi Arabia.
²Department of Mathematics, University of Gezira, Wad Madani, Sudan.

DOI: 10.4236/am.2023.143014                

Abstract

In this paper, a coupling of the natural transform method and the Admoian decomposition method called the natural transform decomposition method (NTDM), is utilized to solve the linear and nonlinear time-fractional Klein-Gordan equation. The (NTDM), is introduced to derive the approximate solutions in series form for this equation. Solutions have been drawn for several values of the time power. To identify the strength of the method, three examples are presented.

Keywords

Natural Transform, Adomian Decomposition Method, Fractional Klein-Gordon Equation

1. Introduction

The fractional calculus has acquired importance in applied mathematics, and it has been used to model several fields in physical science such as viscoelasticity control, diffusion, etc. Fractional differential equations appear in different regions of engineering and physics science, such as continuum traffic flow [1] , the oscillation of earthquake [2] , fluid-dynamic traffic [3] and so on. The fractional order and integer order differential equations have been studied and solved by utilizing several methods [4] - [12] . Many problems in science and engineering have been modeled by the Klein-Gordon equation, such as classical and quantum mechanics, solitons and condensed physics. The Klein-Gordon equation has been extensively solved by many researchers using several analytic methods [13] [14] [15] [16] . The product of this equation has been properly described by a fractional version of them. The integral transform methods are the most important methods for solving many problems, some of them have been coupled with the other analytical methods to obtain analytic solutions [17] [18] [19] [20] and it has proven effective in solving problems. The natural transform method [21] [22] [23] [24] is an example of the integral transform methods. It is a moderate and effective new method for solving differential equations. The natural transform decomposition method (NTDM) is a method combined of the natural transform and a domain decomposition method. It has been used to solve the fraction model [25] - [31] . The main aim of this work is to employ the natural transform decomposition method to solve the linear and nonlinear Klein-Gordon equations of fractional order.

2. Natural Transform

Definition 1 [21] [22] [23] [24] The natural transform of a function $g\left(t\right)$ defined by the integral:

${\mathbb{N}}^{+}\left[g\left(t\right)\right]=\phi (s,u)={\displaystyle {\int }_0^{\infty }}g\left(ut\right){\text{e}}^{-st}\text{d}t,u>0,s>0,$ (1)

where u and s are the natural variables.

Property 1

${\mathbb{N}}^{+}\left[t^{\gamma }\right]=\frac{u^{\gamma }\Gamma \left[\gamma +1\right]}{s^{\gamma +1}},\gamma >-1.$ (2)

Theorem 1 If $n\in N$ where $n-1\le \gamma <n$ and $\phi \left(s\mathrm{,}u\right)$ is natural transform of the function $g\left(t\right)$ , then the natural transform of Caputo fractional derivative of $\frac{{\partial }^{\gamma }g\left(x\mathrm{,}t\right)}{\partial t^{\gamma }}$ is given by

${\mathbb{N}}^{+}\left[\frac{{\partial }^{\gamma }g\left(x,t\right)}{\partial t^{\gamma }}\right]=\frac{s^{\gamma }}{u^{\gamma }}\phi \left(s,u\right)-{\displaystyle \underset{k=0}{\overset{n-1}{\sum }}}\frac{s^{\gamma -\left(k+1\right)}}{u^{\gamma }-k}{\left[\frac{{\partial }^{\gamma }g\left(x,t\right)}{\partial t^{\gamma }}\right]}_{t=0}.$ (3)

Definition 2 The inverse natural transform of $\phi \left(s\mathrm{,}u\right)$ is defined by

${\mathbb{N}}^{-}\left[\phi \left(s,u\right)\right]=g\left(t\right)=\frac{1}{2\pi i}{\displaystyle {\int }_{\epsilon -i\infty }^{\epsilon +i\infty }}\phi \left(s,u\right){\text{e}}^{\frac{st}{u}}\text{d}t,u>0,s>0.$ (4)

3. Analysis of Method

In this part, we apply (NTMD) to the following general fractional Klein Gordon equation of the form:

$\begin{array}{l}\frac{{\partial }^{\gamma }V\left(x,t\right)}{\partial t^{\gamma }}=a\frac{{\partial }^{2}V\left(x,t\right)}{\partial x^2}+bV\left(x,t\right)+\mathcal{F}\left(V\left(x,t\right)\right)+\mu \left(x,t\right),\\ 1<\gamma \le 2\text{and}x,t>0,\end{array}$ (5)

subject to:

$V\left(x,0\right)=f_1\left(x\right),V_t\left(x,0\right)=f_2\left(x\right),$ (6)

where $a\mathrm{,}b$ are constants, $\mu \left(x\mathrm{,}t\right)$ is a source term and $\mathcal{F}\left(V\left(x\mathrm{,}t\right)\right)$ is a nonlinear function. Taking natural transform to Equation (5), we get

$\begin{array}{l}\frac{s^{\gamma }}{u^{\gamma }}\phi \left(x\mathrm{,}s\mathrm{,}u\right)-\frac{s^{\gamma -1}}{u^{\gamma }}V\left(x\mathrm{,0}\right)-\frac{s^{\gamma -2}}{u^{\gamma -1}}{V}_t\left(x\mathrm{,0}\right)\\ ={\mathbb{N}}^{+}\left[a\frac{{\partial }^{2}V\left(x\mathrm{,}t\right)}{\partial x^2}+bV\left(x\mathrm{,}t\right)+\mathcal{F}\left(V\left(x\mathrm{,}t\right)\right)+\mu \left(x\mathrm{,}t\right)\right]\mathrm{.}\end{array}$ (7)

Substitution the initial condition (6) into Equation (7), we get

$\begin{array}{l}\phi \left(x\mathrm{,}s\mathrm{,}u\right)=\frac{f_1\left(x\right)}{s}+\frac{u{f}_2\left(x\right)}{s^2}\\ +\frac{s^{\gamma }}{u^{\gamma }}{\mathbb{N}}^{+}\left[a\frac{{\partial }^{2}V\left(x\mathrm{,}t\right)}{\partial x^2}+bV\left(x\mathrm{,}t\right)+\mathcal{F}\left(V\left(x\mathrm{,}t\right)\right)+\mu \left(x\mathrm{,}t\right)\right]\mathrm{.}\end{array}$ (8)

Operating with inverse natural transform of (8) gives

$\begin{array}{l}V\left(x\mathrm{,}t\right)\\ =\psi \left(x\mathrm{,}t\right)+{\mathbb{N}}^{-}\left[\frac{s^{\gamma }}{u^{\gamma }}{\mathbb{N}}^{+}\left[a\frac{{\partial }^2{x}^{2}V\left(x\mathrm{,}t\right)}{\partial x^2}+bV\left(x\mathrm{,}t\right)+\mathcal{F}\left(V\left(x\mathrm{,}t\right)\right)+h\left(x\mathrm{,}t\right)\right]\right]\mathrm{,}\end{array}$ (9)

where $\psi \left(x\mathrm{,}t\right)$ represents the term deriving from the source term and the directed initial conditions. The natural decomposition method represents solution as infinite series

$V\left(x,t\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{V}_n\left(x,t\right),$ (10)

and the term $\mathcal{F}\left(V\left(x\mathrm{,}t\right)\right)$ is as follows

$\mathcal{F}\left(V\left(x,t\right)\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{A}_n,$ (11)

where $A_n$ can be calculated by

$A_n=\frac{1}{n!}\frac{{\text{d}}^n}{\text{d}{\beta }^n}{\left[N{\displaystyle \underset{i=0}{\overset{n}{\sum }}}{\beta }^i{V}_i\right]}_{\beta =0}.$ (12)

Substitution Equation (10) and Equation (11) into Equation (9), yields

${\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{V}_n\left(x,t\right)=\psi \left(x,t\right)+{\mathbb{N}}^{-}\left[\frac{s^{\gamma }}{u^{\gamma }}{\mathbb{N}}^{+}\left[a{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{{\partial }^2{V}_n\left(x,t\right)}{\partial x^2}+b{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{V}_n\left(x,t\right)+{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{A}_n\right]\right].$ (13)

We obtain the recursive relation

$V_0\left(x,t\right)=\psi \left(x,t\right)$

$V_{n+1}\left(x,t\right)={\mathbb{N}}^{-}\left[\frac{s^{\gamma }}{u^{\gamma }}{\mathbb{N}}^{+}\left[a\frac{{\partial }^2{V}_n\left(x,t\right)}{\partial x^2}+b{V}_n\left(x,t\right)+A_n\right]\right].$ (14)

The approximate solution can be written as a series form

$V\left(x,t\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{V}_n\left(x,t\right)$ (15)

4. Applications

Now, we explain the efficacy of the method by the three examples:

Example 1 Consider the linear inhomogeneous fractional Klein-Gordon equation:

$\frac{{\partial }^{\gamma }V\left(x,t\right)}{\partial t^{\gamma }}=\frac{{\partial }^{2}V\left(x,t\right)}{\partial x^2}+V\left(x,t\right)+6x^{3}t+\left(x^3-6x\right)t^3,t>0,x\in ℝ\text{and}1<\gamma \le 2,$ (16)

subject to:

$V\left(x,0\right)=0,V_t\left(x,0\right)=0$

Solution:

Subsequent the discussion presented above, the system of Equation (13) becomes:

$V_0\left(x,t\right)=\frac{6x^3{t}^{\gamma +1}}{\Gamma \left[\gamma +2\right]}+\frac{6\left(x^3-6x\right)t^{\gamma +3}}{\Gamma \left[\gamma +4\right]},$

$V_{n+1}\left(x,t\right)={\mathbb{N}}^{-}\left[\frac{u^{\gamma }}{s^{\gamma }}{\mathbb{N}}^{+}\left[{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}\frac{{\partial }^2{V}_n\left(x,t\right)}{\partial x^2}-{\displaystyle \underset{n=0}{\overset{\infty }{\sum }}}{V}_n\left(x,t\right)\right]\right],n\ge 0.$ (17)

This gives

$V_1\left(x,t\right)=\frac{36x^3{t}^{2\gamma +1}}{\Gamma \left[2\gamma +2\right]}-\frac{6x^3{t}^{2\gamma +1}}{\Gamma \left[2\gamma +2\right]}+\frac{72x{t}^{2\gamma +3}}{\Gamma \left[2\gamma +4\right]}-\frac{6x^3{t}^{2\gamma +3}}{\Gamma \left[2\gamma +4\right]}$

$V_2\left(x,t\right)=-\frac{72x{t}^{3\gamma +1}}{\Gamma \left[3\gamma +2\right]}+\frac{6x^3{t}^{3\gamma +1}}{\Gamma \left[3\gamma +2\right]}+\frac{6x^3{t}^{3\gamma +3}}{\Gamma \left[3\gamma +4\right]}-\frac{108x{t}^{3\gamma +3}}{\Gamma \left[3\gamma +4\right]}$

$V_3\left(x,t\right)=\frac{108x{t}^{4\gamma +1}}{\Gamma \left[4\gamma +2\right]}-\frac{6x^3{t}^{4\gamma +2}}{\Gamma \left[4\gamma +2\right]}+\frac{144x{t}^{4\gamma +3}}{\Gamma \left[4\gamma +4\right]}-\frac{6x^3{t}^{4\gamma +3}}{\Gamma \left[4\gamma +4\right]}$

$⋮$

Therefore, the approximate solution is given by (Figure 1 and Figure 2):

$\begin{array}{c}V\left(x,t\right)=\frac{6x^3{t}^{\gamma +1}}{\Gamma \left[\gamma +2\right]}+\frac{6(x^3-6x)t^{\gamma +3}}{\Gamma \left[\gamma +4\right]}+\frac{36x^3{t}^{2\gamma +1}}{\Gamma \left[2\gamma +2\right]}-\frac{6x^3{t}^{2\gamma +1}}{\Gamma \left[2\gamma +2\right]}+\frac{72x{t}^{2\gamma +3}}{\Gamma \left[2\gamma +4\right]}\\ -\frac{6x^3{t}^{2\gamma +3}}{\Gamma \left[2\gamma +4\right]}-\frac{72x{t}^{3\gamma +1}}{\Gamma \left[3\gamma +2\right]}+\frac{6x^3{t}^{3\gamma +1}}{\Gamma \left[3\gamma +2\right]}+\frac{6x^3{t}^{3\gamma +3}}{\Gamma \left[3\gamma +4\right]}-\frac{108x{t}^{3\gamma +3}}{\Gamma \left[3\gamma +4\right]}\\ +\frac{108x{t}^{4\gamma +1}}{\Gamma \left[4\gamma +2\right]}-\frac{6x^3{t}^{4\gamma +2}}{\Gamma \left[4\gamma +2\right]}+\frac{144x{t}^{4\gamma +3}}{\Gamma \left[1\gamma +4\right]}-\frac{6x^3{t}^{4\gamma +3}}{\Gamma \left[1\gamma +4\right]}+\cdots \end{array}$

Then the result at $\gamma =2$ is

$V\left(x,t\right)=x^3{t}^3$

Example 2 Consider the non-linear inhomogeneous fractional Klein-Gordon equation:

$\frac{{\partial }^{\gamma }V\left(x,t\right)}{\partial x^{\gamma }}=\frac{{\partial }^{2}V\left(x,t\right)}{\partial x^2}-V^2\left(x,t\right)+2x^2-2t^2+x^4{t}^4,t>0,x\in ℝ\text{and}1<\gamma \le 2,$ (18)

subject to:

$V\left(x,0\right)=0,V_t\left(x,0\right)=0.$ (19)

(a)(b)(c)

Figure 1. The approximate solutions for Example 1 when (a) $\gamma =1.5$ ; (b) $\gamma =1.75$ ; (c) $\gamma =1.90$ .

Figure 2. The approximate and exact solutions for Example 1 when $\gamma =1.5,\gamma =1.75,\gamma =1.90$ .

Solution: Proceeding as in Example 1, Equation (13) becomes:

$V_0\left(x,t\right)=\frac{2x^2{t}^{\gamma }}{\Gamma \left[\gamma +1\right]}-\frac{4t^{\gamma +2}}{\Gamma \left[\gamma +3\right]}+\frac{24x^4{t}^{\gamma +4}}{\Gamma \left[\gamma +5\right]}$

$V_{n+1}={\mathbb{N}}^{-}\left[\frac{u^{\gamma }}{s^{\gamma }}{\mathbb{N}}^{+}\left[{\partial }^2{V}_n\left(x\mathrm{,}t\right)\partial x^2-A_n\right]\right]\mathrm{,}n\ge 0.$ (20)

This gives

$\begin{array}{c}{V}_1\left(x,t\right)=\frac{4t^{2\gamma }}{\Gamma \left[2\gamma +1\right]}+\frac{288x^2{t}^{2\gamma +4}}{\Gamma \left[2\gamma +5\right]}-\frac{4x^4{t}^{3\gamma +4}\Gamma \left[2\gamma +1\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[3\gamma +1\right]}\\ +\frac{16x^2{t}^{3\gamma +2}\Gamma \left[2\gamma +3\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[3\gamma +3\right]}-\frac{16t^{3\gamma +4}\Gamma \left[2\gamma +5\right]}{{\Gamma }^2\left[\gamma +3\right]\Gamma \left[3\gamma +5\right]}\\ -\frac{96x^6{t}^{3\gamma +4}\Gamma \left[2\gamma +5\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +5\right]}+\frac{192x^4{t}^{3\gamma +6}\Gamma \left[2\gamma +7\right]}{\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +7\right]}\\ -\frac{576x^8{t}^{3\gamma +3}\Gamma \left[2\gamma +9\right]}{{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]}.\end{array}$

$\begin{array}{c}{V}_2\left(x\mathrm{,}t\right)=\frac{576t^{3\gamma +4}}{\Gamma \left[3\gamma +5\right]}-\frac{48x^2{t}^{4\gamma }\Gamma \left[2\gamma +1\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[4\gamma +1\right]}-\frac{16x^3{t}^{4\gamma }\Gamma \left[3\gamma +1\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +1\right]}\\ -\frac{2880x^4{t}^{4\gamma +4}\Gamma \left[2\gamma +5\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[4\gamma +5\right]}-\frac{192x^5{t}^{4\gamma +4}\Gamma \left[3\gamma +5\right]}{\Gamma \left[\gamma +5\right]\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +5\right]}\\ -\frac{1152x^5{t}^{4\gamma +4}\Gamma \left[3\gamma +5\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +5\right]}+\frac{2304x^2{t}^{4\gamma +6}\Gamma \left[2\gamma +7\right]}{\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[4\gamma +7\right]}\\ +\frac{2304x^2{t}^{4\gamma +6}\Gamma \left[2\gamma +7\right]}{\Gamma \left[\gamma +3\right]\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +7\right]}+\frac{16x^4{t}^{5\gamma }\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +1\right]}{{\Gamma }^3\left[\gamma +1\right]\Gamma \left[3\gamma +1\right]\Gamma \left[5\gamma +1\right]}\\ -\frac{32x^5{t}^{5\gamma +2}\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +3\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[3\gamma +1\right]\Gamma \left[5\gamma +3\right]}\end{array}$

$\begin{array}{c}-\frac{64x^5{t}^{5\gamma +2}\Gamma \left[2\gamma +3\right]\Gamma \left[4\gamma +3\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[3\gamma +3\right]\Gamma \left[5\gamma +3\right]}\\ +\frac{192x^9{t}^{5\gamma +4}\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +5\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +1\right]\Gamma \left[5\gamma +5\right]}\\ +\frac{128x^3{t}^{5\gamma +4}\Gamma \left[2\gamma +3\right]\Gamma \left[4\gamma +5\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +3\right]\Gamma \left[3\gamma +3\right]\Gamma \left[5\gamma +5\right]}\\ +\frac{64x^3{t}^{5\gamma +4}\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +5\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +3\right]\Gamma \left[3\gamma +5\right]\Gamma \left[5\gamma +5\right]}\\ +\frac{348x^9{t}^{5\gamma +4}\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +5\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +5\right]\Gamma \left[5\gamma +5\right]}\end{array}$

$\begin{array}{c}-\frac{768x^7{t}^{5\gamma +6}\Gamma \left[2\gamma +3\right]\Gamma \left[4\gamma +7\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +3\right]\Gamma \left[5\gamma +7\right]}\\ -\frac{128x^5{t}^{5\gamma +6}\Gamma \left[2\gamma +7\right]\Gamma \left[4\gamma +7\right]}{{\Gamma }^3\left[\gamma +3\right]\Gamma \left[3\gamma +5\right]\Gamma \left[3\gamma +1\right]\Gamma \left[5\gamma +7\right]}\\ -\frac{768x^7{t}^{5\gamma +6}\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +7\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +5\right]\Gamma \left[5\gamma +7\right]}\\ -\frac{768x^7{t}^{5\gamma +6}\Gamma \left[2\gamma +7\right]\Gamma \left[4\gamma +7\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +3\right]\Gamma \left[5\gamma +7\right]}\\ +\frac{4608x^{1}1t^{5\gamma +8}\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +9\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +5\right]\Gamma \left[5\gamma +9\right]}\end{array}$

$\begin{array}{c}+\frac{768x^5{t}^{5\gamma +8}\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +9\right]}{\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +5\right]\Gamma \left[5\gamma +9\right]}\\ +\frac{1536x^5{t}^{5\gamma +8}\Gamma \left[2\gamma +7\right]\Gamma \left[4\gamma +9\right]}{{\Gamma }^2\left[\gamma +3\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +7\right]\Gamma \left[5\gamma +9\right]}\\ +\frac{230x^{11}{t}^{5\gamma +8}\Gamma \left[2\gamma +9\right]\Gamma \left[4\gamma +9\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]\Gamma \left[5\gamma +9\right]}\\ -\frac{9216x^{11}{t}^{5\gamma +10}\Gamma \left[2\gamma +7\right]\Gamma \left[4\gamma +11\right]}{\Gamma \left[\gamma +3\right]{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +7\right]\Gamma \left[5\gamma +11\right]}\end{array}$

$\begin{array}{c}-\frac{4608x^9{t}^{5\gamma +10}\Gamma \left[2\gamma +9\right]\Gamma \left[4\gamma +11\right]}{\Gamma \left[\gamma +3\right]{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]\Gamma \left[5\gamma +11\right]}\\ +\frac{27648x^{13}{t}^{5\gamma +12}\Gamma \left[2\gamma +9\right]\Gamma \left[4\gamma +13\right]}{{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]\Gamma \left[5\gamma +13\right]}\\ +\frac{32t^{4\gamma +12}\Gamma \left[2\gamma +3\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[5\gamma +3\right]}+\frac{32256x^6{t}^{4\gamma +8}\Gamma \left[2\gamma +9\right]}{{\Gamma }^2\left[\gamma +5\right]\Gamma \left[4\gamma +9\right]}\\ -\frac{32t^{4\gamma +12}\Gamma \left[2\gamma +3\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[5\gamma +3\right]}-\frac{13824x^7{t}^{4\gamma +8}\Gamma \left[3\gamma +9\right]}{\Gamma \left[\gamma +5\right]\Gamma \left[2\gamma +5\right]\Gamma \left[4\gamma +9\right]}\end{array}$

$⋮$

Therefore, the series solution is given by (Figure 3 and Figure 4):

(a)(b)(c)

Figure 3. The approximate solutions for Example 2 when (a) $\gamma =1.8$ ; (b) $\gamma =1.9$ ; (c) $\gamma =2$ .

Figure 4. The approximate and exact solutions for Example 1 when $\gamma =1.8,\gamma =1.9,\gamma =2$ .

$\begin{array}{c}V\left(x,t\right)=\frac{2x^2{t}^{\gamma }}{\Gamma \left[\gamma +1\right]}-\frac{4t^{\gamma +2}}{\Gamma \left[\gamma +3\right]}+\frac{24x^4{t}^{\gamma +4}}{\Gamma \left[\gamma +5\right]}+\frac{4t^{2\gamma }}{\Gamma \left[2\gamma +1\right]}+\frac{288x^2{t}^{2\gamma +4}}{\Gamma \left[2\gamma +5\right]}\\ -\frac{4x^4{t}^{3\gamma +4}\Gamma \left[2\gamma +1\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[3\gamma +1\right]}+\frac{16x^2{t}^{3\gamma +2}}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[3\gamma +3\right]}\\ -\frac{16t^{3\gamma +4}\Gamma \left[2\gamma +5\right]}{{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]}-\frac{96x^6{t}^{3\gamma +4}\Gamma \left[2\gamma +5\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +5\right]}\\ +\frac{192x^4{t}^{3\gamma +6}\Gamma \left[2\gamma +7\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[3\gamma +7\right]}-\frac{576x^8{t}^{3\gamma +3}\Gamma \left[2\gamma +9\right]}{{\Gamma }^2\left[\gamma +5\right]\Gamma \left[3\gamma +9\right]}+\frac{576t^{3\gamma +4}}{\Gamma \left[3\gamma +5\right]}\\ -\frac{48x^2{t}^{4\gamma }\Gamma \left[2\gamma +1\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[4\gamma +1\right]}-\frac{16x^3{t}^{4\gamma }\Gamma \left[3\gamma +1\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[2\gamma +1\right]\Gamma \left[4\gamma +1\right]}\\ -\frac{2880x^4{t}^{4\gamma +4}\Gamma \left[2\gamma +5\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +5\right]\Gamma \left[4\gamma +5\right]}+\cdots \end{array}$

For $\gamma =2$ , the exact solution is

$V\left(x,t\right)=x^2{t}^2$

Example 3 Consider the non-linear inhomogeneous fractional Klein-Gordon equation with cubic nonlinearity:

$\begin{array}{l}\frac{{\partial }^{\gamma }V\left(x,t\right)}{\partial x^{\gamma }}+\frac{{\partial }^{2}V\left(x,t\right)}{\partial x^2}+V\left(x,t\right)+V^3(x,t)=2x+x{t}^2+x^3{t}^6,\\ t>0,x\in ℝ\text{and}1<\gamma \le 2\end{array}$ (21)

Subject to

$V\left(x,0\right)=0,V_t\left(x,0\right)=0$

Using the previous aforesaid, we get

$V_0\left(x,t\right)=\frac{2x{t}^{\gamma }}{\Gamma \left[\gamma +1\right]}+\frac{2x^4{t}^{\gamma +2}}{\Gamma \left[\gamma +3\right]}+\frac{720x^4{t}^{\gamma +6}}{\Gamma \left[\gamma +7\right]}$

(a)(b)(c)

Figure 5. The approximate solutions for Example 3 when (a) $\gamma =1.75$ ; (b) $\gamma =1.85$ ; (c) $\gamma =1.95$ .

Figure 6. The approximate and exact solutions for Example 3 when $\gamma =1.75,\gamma =1.85,\gamma =1.95$ .

$\begin{array}{c}{V}_1\left(x,t\right)=-\frac{2x{t}^{2\gamma }}{\Gamma \left[2\gamma +1\right]}-\frac{2x{t}^{2\gamma +2}}{\Gamma \left[2\gamma +3\right]}-\frac{4320x{t}^{2\gamma +6}}{\Gamma \left[2\gamma +7\right]}-\frac{720x^3{t}^{2\gamma +6}}{\Gamma \left[2\gamma +7\right]}\\ -\frac{8x^3{t}^{4\gamma }\Gamma \left[3\gamma +1\right]}{{\Gamma }^3\left[\gamma +1\right]\Gamma \left[4\gamma +1\right]}-\frac{24x^3{t}^{4\gamma +2}\Gamma \left[3\gamma +3\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[4\gamma +3\right]}\\ -\frac{24x^3{t}^{4\gamma +4}\Gamma \left[3\gamma +5\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +3\right]\Gamma \left[4\gamma +5\right]}-\frac{8x^3{t}^{4\gamma +6}\Gamma \left[3\gamma +7\right]}{{\Gamma }^3\left[\gamma +3\right]\Gamma \left[4\gamma +7\right]}\\ -\frac{8640x^5{t}^{4\gamma +6}\Gamma \left[3\gamma +7\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +7\right]}-\frac{17280x^5{t}^{4\gamma +8}\Gamma \left[3\gamma +9\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +9\right]}\\ -\frac{8640x^5{t}^{4\gamma +10}\Gamma \left[3\gamma +11\right]}{{\Gamma }^2\left[\gamma +3\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +11\right]}-\frac{3110400x^7{t}^{4\gamma +12}\Gamma \left[4\gamma +13\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +13\right]}\\ -\frac{3110400x^5{t}^{4\gamma +14}\Gamma \left[3\gamma +15\right]}{\Gamma \left[\gamma +3\right]{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +15\right]}-\frac{373248x^9{t}^{4\gamma +18}\Gamma \left[3\gamma +19\right]}{{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +19\right]}.\end{array}$

Therefore, the series solution is given by (Figure 5 and Figure 6):

$\begin{array}{c}V\left(x,t\right)=\frac{2x{t}^{\gamma }}{\Gamma \left[\gamma +1\right]}+\frac{2x4t^{\gamma +2}}{\Gamma \left[\gamma +3\right]}+\frac{720x^4{t}^{\gamma +6}}{\Gamma \left[\gamma +7\right]}-\frac{2x{t}^{2\gamma }}{\Gamma \left[2\gamma +1\right]}-\frac{2x{t}^{2\gamma +2}}{\Gamma \left[2\gamma +3\right]}\\ -\frac{4320x{t}^{2\gamma +6}}{\Gamma \left[2\gamma +7\right]}-\frac{720x^3{t}^{2\gamma +6}}{\Gamma \left[2\gamma +7\right]}-\frac{8x^3{t}^{4\gamma }\Gamma \left[3\gamma +1\right]}{{\Gamma }^3\left[\gamma +1\right]\Gamma \left[4\gamma +1\right]}\\ -\frac{24x^3{t}^{4\gamma +2}\Gamma \left[3\gamma +3\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[4\gamma +3\right]}-\frac{24x^3{t}^{4\gamma +4}\Gamma \left[3\gamma +5\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +3\right]\Gamma \left[4\gamma +5\right]}\\ -\frac{8x^3{t}^{4\gamma +6}\Gamma \left[3\gamma +7\right]}{{\Gamma }^3\left[\gamma +3\right]\Gamma \left[4\gamma +7\right]}-\frac{8640x^5{t}^{4\gamma +6}\Gamma \left[3\gamma +7\right]}{{\Gamma }^2\left[\gamma +1\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +7\right]}\end{array}$

$\begin{array}{c}-\frac{17280x^5{t}^{4\gamma +8}\Gamma \left[3\gamma +9\right]}{\Gamma \left[\gamma +1\right]\Gamma \left[\gamma +3\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +9\right]}-\frac{8640x^5{t}^{4\gamma +10}\Gamma \left[3\gamma +11\right]}{{\Gamma }^2\left[\gamma +3\right]\Gamma \left[\gamma +7\right]\Gamma \left[4\gamma +11\right]}\\ -\frac{3110400x^7{t}^{4\gamma +12}\Gamma \left[4\gamma +13\right]}{\Gamma \left[\gamma +1\right]{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +13\right]}-\frac{3110400x^5{t}^{4\gamma +14}\Gamma \left[3\gamma +15\right]}{\Gamma \left[\gamma +3\right]{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +15\right]}\\ -\frac{373248x^9{t}^{4\gamma +18}\Gamma \left[3\gamma +19\right]}{{\Gamma }^2\left[\gamma +7\right]\Gamma \left[4\gamma +19\right]}+\cdots \end{array}$

For $\gamma =2$ , the exact solution is

$V\left(x,t\right)=x{t}^2$

5. Conclusion

In this research, we successfully applied (NTDM) to obtain an approximate solution for the linear and nonlinear fractional Klein Gordon equation. We have tested the method on three examples, which revealed that the method is highly effective. Indeed, in Figure 4, we see that when $\gamma =2$ , the approximate solution is almost identical to the exact solution.

Conflicts of Interest

The author declares that they have no competing interests.

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