An Efficient Analytical Method for Analyzing the Nonlinear Fractional Klein–Fock–Gordon Equations

Abstract

The purpose of this article is to solve a nonlinear fractional Klein–Fock–Gordon equation that involves a recently created non-singular kernel fractional derivative by Caputo–Fabrizio. Motivated by some physical applications related to the fractional Klein–Fock–Gordon equation, we focus our study on this equation and some phenomena rated to it. The findings are crucial and essential for explaining a variety of physical processes. In order to find satisfactory approximations to the offered problems, this work takes into account a modern methodology and fractional operator in this context. We first take the Yang transform of the Caputo–Fabrizio fractional derivative and then implement it to solve fractional Klein–Fock–Gordon equations. We will consider three cases of the nonlinear fractional Klein–Fock–Gordon equation to ensure the applicability and effectiveness of the suggested technique. In order to determine an approximate solution to the fractional Klein–Fock–Gordon equation in the fast convergent series form, we can use the fractional homotopy perturbation transform approach. The numerical simulation is provided to demonstrate the effectiveness and dependability of the suggested method. Furthermore, several fractional orders will be used to describe the behavior of the given solutions. The results achieved demonstrate the high efficiency, ease of use, and applicability of this strategy for resolving other nonlinear issues.

1. Introduction

Fractional calculus (FC) was first introduced more than 324 years ago, but it has only recently drawn the interest of numerous academics working in many different fields of science and engineering. This interest is due to its advantageous qualities, including analyticality, heredity, nonlocality, and memory effect. FC has developed over the last several decades into the most potent instrument for analyzing and describing nonlinear complex processes. Due to the complexity associated with the occurrence of heterogeneities, the idea of derivatives with an arbitrary order has been established [1,2,3]. Non-integer order-differential operators can capture complicated media with diffusion mechanisms. Numerous scholars began working on the fundamentals and applications of fractional calculus as a result of the quick development of computer and mathematical techniques and software [4,5,6]. Numerous pioneering approaches that prescribed the basis for fractional calculus have been made available for diverse definitions of the subject. The FC has been used in practical projects and has been applied to biomathematics [7], chaos theory [8], financial models [9], optics [10], and other fields [11,12,13,14,15,16,17].

Due to the soliton-like solutions found in many mathematical physics models, the idea of solitons is crucial, especially in plasma physics [18,19,20,21,22,23,24,25]. One type of common physical system that represents weakly nonlinear dispersive partial differential equations (PDEs) is called a soliton. The term “generalization” refers to the fractional PDEs that are composed of a non-integer order as opposed to the classical integer-order PDEs. Due to its numerous theoretical and practical applications in applied sciences, engineering problems, and in the modeling of many nonlinear problems in different plasma physics, fractional nonlinear differential equations (FNDEs) have gained a lot of interest in past years [26,27,28]. In a scientific study, the approximate solution of fractional nonlinear differential equations (FNDEs) is a topic that has both theoretical and practical applications. It draws a large number of scholars to create unique methods to find precise and approximative solutions of FNDEs. Numerous scientists and mathematicians have created effective direct strategies for the approximative and closed-form solutions of FNDEs over the last few of years. Several effective methods include the Yang transform decomposition method [29,30], the Sine–Gordon expansion method [31], the fractional variational iteration method [32], the q-homotopy analysis Sumudu transform method [33], the natural transform decomposition method via the Caputo–Fabrizio fractional-order derivative [34,35], the reduced differential transform method [36], the homotopy perturbation Sumudu transform method [37], the Elzaki transform decomposition method [38], the Laplace transform decomposition method via the Atangana–Baleanu fractional derivative [39], the variational iteration method via the Atangana–Baleanu fractional derivative [40], the F-expansion method [41], and many other techniques [42,43,44,45,46,47] have been applied for analyzing the FNDEs.

On the other hand, the Klein–Fock–Gordon (KFG) equation, which was developed by physicists Klein, Fock, and Gordon, explains relativistic electrons. This equation, also referred to as the relativistic wave equation, is a quantized representation of the relativistic energy-momentum relation and is connected to the Schrodinger equation. The KFG equation and the Dirac equation both have theoretical applications. The following fractional-order KFG equation is taken into consideration in our investigation:

$${}^{CF}{D}_{\kappa }^{\varsigma }z(\phi ,\kappa )=z_{\phi \phi }(\phi ,\kappa )+az(\phi ,\kappa )+b{z}^n(\phi ,\kappa ),1<\varsigma \le 2,$$

(1)

along with the initial conditions (ICs): $z(\phi ,\kappa )=f\left(x\right)$ and $z_{\kappa }(\phi ,\kappa )=g\left(x\right),$ where ${}^{CF}{D}_{\kappa }^{\varsigma }$ represents the Caputo–Fabrizio fractional derivative, n is a positive integer, and $(a,b)$ are real constants. The Klein–Fock–Gordon equation appeared in a variety of physical processes, including condensed-matter physics, nonlinear optics, quantum field theory, and the interaction of solitons in a collisionless plasma. Different numerical techniques have been devoted to the study of the KFG equation, such as the homotopy analysis method (HAM) [48], the variational iteration method [49], the modified differential transform method [50], the differential transform method (DTM) [51], the q-homotopy analysis transform method [52], the homotopy analysis transform method [53], and many others [54,55].

The homotopy perturbation method (HPM) [56], which is the combination of the homotopy method and the conventional perturbation methodology, was proposed by He in 1999. This technique has been successfully used for solving both linear and nonlinear problems [57,58]. As HPM does not require a small parameter in the equation, it has fewer limitations than conventional perturbation techniques. The major goal of this study is to solve nonlinear fractional order PDEs by using Yang’s newly introduced integral transform, commonly known as the “Yang Transform”, which was discovered with HPM via the Caputo–Fabrizio fractional derivative. The analysis of fractional differential equations has taken on a new dimension as a result of the Caputo–Fabrizio fractional derivative. The nonsingular kernel of the new derivative is one of its most attractive features. The Caputo–Fabrizio derivative has the same additional motivating characteristics of heterogeneity and configuration [59,60], with different scales to the Caputo and Riemann–Liouville fractional derivatives despite being created through the convolution of an ordinary derivative and an exponential function. Using the suggested approach, we can resolve three well-known nonlinear PDEs. In the context of a swiftly converging series, we acquire a power-series solution, and just a few iterations are designed to accomplish outstanding results. A solution that can be easily determined using these strategies can be reached after only a few iterations, without the need for a method such as discretizing the problem or linearizing the nonlinear problem.

The following is information about the paper’s classification: Section 2 illustrates the fundamental concept pertaining to Caputo–Fabrizio definitions. The Yang–Laplace duality property and the Caputo–Fabrizio fractional derivative are presented in Section 3. We demonstrate the general application of the suggested approach in Section 4, together with convergence and error analysis. In Section 5, the numerical solution of the fractional model is briefly explained, and then it is concluded in Section 6.

2. Preliminaries

We presented some definitions related to our present work. Simply put, we present the exponential decay kernel as $P(\kappa ,\varrho )=exp[-\varsigma (\kappa -\varrho /1-\varsigma \left)\right]$.

Definition 1.

If $\mathbb{J}\left(\kappa \right)\in {\mathbf{H}}^1[0,T],T>0$, the derivative in terms of Caputo–Fabrizio (CF) is stated as:

$${}^{CF}{D}_t^{\varsigma }\left[\mathbb{J}\left(\kappa \right)\right]=\frac{N\left(\varsigma \right)}{1-\varsigma }{\int }_0^{\kappa }{\mathbb{J}}^{{}^{\prime }}\left(\varrho \right)P(\kappa ,\varrho )d\varrho .$$

(2)

where $N\left(\varsigma \right)$ represents the normalization function having $N\left(0\right)=N\left(1\right)=1$. Additionally, if $\mathbb{J}\left(\kappa \right)\notin {\mathbf{H}}^1[0,T]$, then the proposed derivative is stated as:

$${}^{CF}{D}_t^{\varsigma }\left[\mathbb{J}\left(\kappa \right)\right]=\frac{N\left(\varsigma \right)}{1-\varsigma }{\int }_0^{\kappa }[\mathbb{J}\left(\kappa \right)-\mathbb{J}\left(\varrho \right)]P(\kappa ,\varrho )d\varrho .$$

(3)

Definition 2.

The CF integral that has fractional order is stated as:

$${}^{CF}{I}_t^{\varsigma }\left[\mathbb{J}\left(\kappa \right)\right]=\frac{1-\varsigma }{N\left(\varsigma \right)}\mathbb{J}\left(\kappa \right)+\frac{\varsigma }{N\left(\varsigma \right)}{\int }_0^{\kappa }\mathbb{J}\left(\varrho \right)d\varrho ,\kappa \ge 0,\varsigma \in (0,1].$$

(4)

Definition 3.

For $N\left(\varsigma \right)=1$, the Laplace transform of the CF derivative leads to the result:

$$L\left[{}^{CF}{D}_t^{\varsigma }\left[\mathbb{J}\left(\kappa \right)\right]\right]=\frac{\mu L\left[\mathbb{J}\right(\kappa )-\mathbb{J}(0\left)\right]}{\mu +\varsigma (1-\mu )}.$$

(5)

Definition 4.

The Yang transform (YT) of $\mathbb{J}\left(\kappa \right)$ is given as:

$$L\left[\mathbb{J}\left(\kappa \right)\right]=\zeta \left(\mu \right)={\int }_0^{\infty }\mathbb{J}\left(\kappa \right)e^{-\frac{t}{\mu }}d\kappa .\kappa >0,$$

(6)

here, μ is the transform variable.

Remark 1.

Some important functions YT is stated as.

$$\begin{array}{cc}\hfill Y\left[1\right]=& \mu ,\hfill \\ \hfill Y\left[\kappa \right]=& {\mu }^2,\hfill \\ \hfill Y\left[t^q\right]=& \mathsf{\Gamma }(q+1){\mu }^{q+1}.\hfill \end{array}$$

(7)

3. Main Work

First, using the Yang–Laplace duality property, we construct the formula for the YT of the CF fractional derivative. To verify the accuracy and effectiveness of the unique method, we provide a few examples with complete solutions at the end of this section.

Lemma 1 (Laplace-Yang duality).

If $F\left(\mu \right)$ is the Laplace transform of $\mathbb{J}\left(\kappa \right),$ then $\zeta \left(v\right)=F\left(\frac{1}{\mu }\right)$.

Proof. 

By inserting $\frac{\kappa }{u}=\phi$ as in Equation (6), we obtain another form of YT.

$$L\left[\mathbb{J}\left(\kappa \right)\right]=\zeta \left(\mu \right)=\mu {\int }_0^{\infty }\mathbb{J}\left(\mu \phi \right)e^{\phi }d\phi .\phi >0.$$

(8)

Since $L\left[\mathbb{J}\left(\kappa \right)\right]=F\left(\mu \right)$,

$$F\left(\mu \right)=L\left[\mathbb{J}\left(\kappa \right)\right]={\int }_0^{\infty }\mathbb{J}\left(\kappa \right)e^{-\mu \kappa }d\kappa .$$

(9)

Put $\kappa =\phi /\mu$ in (9), we obtain

$$F\left(\mu \right)=\frac{1}{\mu }{\int }_0^{\infty }\mathbb{J}\left(\frac{\phi }{\mu }\right)e^{\phi }d\phi .$$

(10)

Now, from Equation (8), we have

$$F\left(\mu \right)=\zeta \left(\frac{1}{\mu }\right).$$

(11)

Thus, from Equations (6) and (9), we obtain

$$F\left(\frac{1}{\mu }\right)=\zeta \left(\mu \right).$$

(12)

So (11) and (12) present the duality relation between the Laplace and YT. □

Lemma 2.

The YT of the CF derivative of $P\left(\kappa \right)$ is given if $P\left(\kappa \right)$ is a continuous function.

$$Y\left[\mathbb{J}\left(\kappa \right)\right]=\frac{Y\left[\mathbb{J}\right(\kappa )-\mu \mathbb{J}(0\left)\right]}{1+\varsigma (\mu -1)}.$$

(13)

Proof. 

The CF fractional’s Laplace transform is represented by

$$L\left[\mathbb{J}\left(\kappa \right)\right]=\frac{L\left[\mu \mathbb{J}\right(\kappa )-\mathbb{J}(0\left)\right]}{\mu +\varsigma (1-\mu )}.$$

(14)

Additionally, there is a connection between the Laplace and Yang properties, i.e., $\zeta \left(\mu \right)=F\frac{1}{\mu }$. The Equation (14) variable $\mu$ is changed to $\frac{1}{\mu }$ to achieve the desired result, and we obtain

$$\begin{array}{cc}\hfill Y\left[\mathbb{J}\left(\kappa \right)\right]=& \frac{\frac{1}{\mu }Y[\mathbb{J}\left(\kappa \right)-\mathbb{J}\left(0\right)]}{\frac{1}{\mu }+\varsigma (1-\frac{1}{\mu })},\hfill \\ \hfill Y\left[\mathbb{J}\left(\kappa \right)\right]=& \frac{Y\left[\mathbb{J}\right(\kappa )-\mu \mathbb{J}(0\left)\right]}{1+\varsigma (\mu -1)}.\hfill \end{array}$$

(15)

which completes the proof. □

4. Algorithm of the Suggested Technique

In this part, we go over the method for differential equations of fractional order with an exponential-decay kernel. We show a few cases with comprehensive solutions and a comparison to the precise solutions.

4.1. Implementation to Caputo–Fabrizio Fractional Differential Equations

First, using YTHPM, we create a technique to solve general nonlinear Caputo–Fabrizio (CF) fractional partial differential equations. Take a look at a general nonlinear CF PDE with a nonlinear term of $N\left(G\right(\phi ,\kappa \left)\right)$ and a linear term of $L\left(G\right(\phi ,\kappa \left)\right)$, as shown below

$$\left\{\begin{array}{c}{}^{CF}{D}_t^{\varsigma }G(\phi ,\kappa )+L\left(G(\phi ,\kappa )\right)+N\left(G(\phi ,\kappa )\right)=g(\phi ,\kappa ),\hfill \\ G(\phi ,0)=h\left(\phi \right),\hfill \end{array}\right.$$

(16)

where the term $g(\phi ,\kappa )$ denotes the source term. On employing YT to Equation (17), we have

$$\frac{Y\left[G\right(\phi ,\kappa )-\mu G(\phi ,0\left)\right]}{1+\varsigma (\mu -1)}=-Y[L\left(G(\phi ,\kappa )\right)+N\left(G(\phi ,\kappa )\right)]+Y\left[g(\phi ,\kappa )\right],$$

$$Y\left[G\right(\phi ,\kappa \left)\right]=\mu h\left(\phi \right)-(1+\varsigma (\mu -1\left)\right)\left[Y\right[L\left(G\right(\phi ,\kappa \left)\right)+N\left(G\right(\phi ,\kappa \left)\right)\left]\right]+Y\left[g\right(\phi ,\kappa \left)\right].$$

(17)

By employing Yang inverse transform, we obtain

$$G(\phi ,\kappa )=\mathcal{G}(\phi ,\kappa )-Y^{-1}[(1+\varsigma (\mu -1))\left[Y[L\left(G(\phi ,\kappa )\right)+N\left(G(\phi ,\kappa )\right)]\right]+Y\left[g(\phi ,\kappa )\right]],$$

(18)

here, $G(\phi ,\kappa )$ shows the source term and initial condition. Now, by utilizing HPM:

$$G(\phi ,\kappa )=\sum _{q=0}^{\infty }{\rho }^q{G}_q(\phi ,\kappa ).$$

(19)

The decomposition of the nonlinear term $N\left(G\right(\phi ,\kappa \left)\right)$ reads

$$N\left(G(\phi ,\kappa )\right)=\sum _{q=0}^{\infty }{\rho }^q{H}_q\left(G\right),$$

(20)

where $H_q\left(G\right)$ shows the He’s polynomial and is determined as:

$$H_q(G_1,G_2,G_3,\cdots ,G_q)=\frac{1}{\mathsf{\Gamma }(q+1)}\frac{{\partial }^q}{\partial {\rho }^q}{\left[N\left(\sum _{i=0}^{\infty }{\rho }^i{G}_i\right)\right]}_{\rho =0},q=1,2,3,\cdots .$$

(21)

By substituting Equations (19) and (20) in Equation (18), we have

$$\sum _{q=0}^{\infty }{\rho }^q{G}_q(\phi ,\kappa )=\mathcal{G}(\phi ,\kappa )-\rho \left(Y^{-1}\left[(1+\varsigma (\mu -1))Y\left[L\sum _{q=0}^{\infty }{\rho }^q{G}_q(\phi ,\kappa )+N\sum _{q=0}^{\infty }{\rho }^q{H}_q\left(G\right)\right]\right]\right).$$

(22)

On comparing the coefficients $\rho$ in (22), we obtain:

$$\begin{array}{cc}\hfill {\rho }^0:G_0(\phi ,\kappa )=& \mathcal{G}(\phi ,\kappa ),\hfill \\ \hfill {\rho }^1:G_1(\phi ,\kappa )=& Y^{-1}\left[(1+\varsigma (\mu -1))Y\left[L\left(G_0(\phi ,\kappa )\right)+H_0\left(G\right)\right]\right],\hfill \\ \hfill {\rho }^2:G_2(\phi ,\kappa )=& Y^{-1}\left[(1+\varsigma (\mu -1))Y\left[L\left(G_1(\phi ,\kappa )\right)+H_1\left(G\right)\right]\right],\hfill \\ \hfill {\rho }^3:G_3(\phi ,\kappa )=& Y^{-1}\left[(1+\varsigma (\mu -1))Y\left[L\left(G_2(\phi ,\kappa )\right)+H_2\left(G\right)\right]\right],\hfill \\ \hfill & ⋮\hfill \\ \hfill {\rho }^q:G_q(\phi ,\kappa )=& Y^{-1}\left[(1+\varsigma (\mu -1))Y\left[L\left(G_q(\phi ,\kappa )\right)+H_q\left(G\right)\right]\right].\hfill \end{array}$$

(23)

Hence, we obtain the solution as given below:

$$G(\phi ,\kappa )=G_0(\phi ,\kappa )+G_1(\phi ,\kappa )+\cdots .$$

(24)

4.2. Convergence and Error Analysis

The convergence and error analysis of the original problem (16) are dealt with in the following theorems, which are based on the mechanism of the approach [61].

Theorem 1.

Suppose the exact solution of (16) is $G(\phi ,\kappa )$ and let $G(\phi ,\kappa )$, $G_n(\phi ,\kappa )\in H$, and $\alpha \in (0,1)$, where H represents the Hilbert space. The solution obtained ${\sum }_{q=0}^{\infty }{G}_q(\phi ,\kappa )$ will converge $G(\phi ,\kappa )$ if $G_q(\phi ,\kappa )\le G_{q-1}(\phi ,\kappa )\forall q>A$, i.e., for any $\omega >0\exists A>0$, such that $\left|\right|G_{q+n}(\phi ,\kappa )\left|\right|\le \beta ,\forall m,n\in N.$

Proof. 

We take a sequence of ${\sum }_{q=0}^{\infty }{G}_q(\phi ,\kappa ).$

$$\begin{array}{cc}\hfill C_0(\phi ,\kappa )=& G_0(\phi ,\kappa ),\hfill \\ \hfill C_1(\phi ,\kappa )=& G_0(\phi ,\kappa )+G_1(\phi ,\kappa ),\hfill \\ \hfill C_2(\phi ,\kappa )=& G_0(\phi ,\kappa )+G_1(\phi ,\kappa )+G_2(\phi ,\kappa ),\hfill \\ \hfill C_3(\phi ,\kappa )=& G_0(\phi ,\kappa )+G_1(\phi ,\kappa )+G_2(\phi ,\kappa )+G_3(\phi ,\kappa ),\hfill \\ \hfill & ⋮\hfill \\ \hfill C_q(\phi ,\kappa )=& G_0(\phi ,\kappa )+G_1(\phi ,\kappa )+G_2(\phi ,\kappa )+\cdots +G_q(\phi ,\kappa ).\hfill \end{array}$$

(25)

We must demonstrate that $C_q(\phi ,\kappa )$ forms a “Cauchy sequence” in order to achieve the desired outcome. Additionally, let us take

$$\begin{array}{cc}\hfill \left|\right|C_{q+1}(\phi ,\kappa )-C_q(\phi ,\kappa )\left|\right|& =\left|\right|G_{q+1}(\phi ,\kappa )\left|\right|\le \alpha \left|\right|G_q(\phi ,\kappa )\left|\right|\le {\alpha }^2\left|\right|G_{q-1}(\phi ,\kappa )\left|\right|\le {\alpha }^3\left|\right|G_{q-2}(\phi ,\kappa )\left|\right|\cdots \hfill \\ \hfill & \le {\alpha }_{q+1}\left|\right|G_0(\phi ,\kappa )\left|\right|.\hfill \end{array}$$

(26)

For $q,n\in N$, we have

$$\begin{array}{cc}\hfill \left|\right|C_q(\phi ,\kappa )-C_n(\phi ,\kappa )\left|\right|=& \left|\right|G_{q+n}(\phi ,\kappa )\left|\right|=\left|\right|C_q(\phi ,\kappa )-C_{q-1}(\phi ,\kappa )+(C_{q-1}(\phi ,\kappa )-C_{q-2}(\phi ,\kappa ))\hfill \\ \hfill & +(C_{q-2}(\phi ,\kappa )-C_{q-3}(\phi ,\kappa ))+\cdots +(C_{n+1}(\phi ,\kappa )-C_n(\phi ,\kappa ))\left|\right|\hfill \\ \hfill \le & \left|\right|C_q(\phi ,\kappa )-C_{q-1}(\phi ,\kappa )\left|\right|+\left|\right|(C_{q-1}(\phi ,\kappa )-C_{q-2}(\phi ,\kappa ))\left|\right|\hfill \\ \hfill & +\left|\right|(C_{q-2}(\phi ,\kappa )-C_{q-3}(\phi ,\kappa ))\left|\right|+\cdots +\left|\right|(C_{n+1}(\phi ,\kappa )-C_n(\phi ,\kappa ))\left|\right|\hfill \\ \hfill \le & {\alpha }^q\left|\right|G_0(\phi ,\kappa )\left|\right|+{\alpha }^{q-1}\left|\right|G_0(\phi ,\kappa )\left|\right|+\cdots +{\alpha }^{q+1}\left|\right|G_0(\phi ,\kappa )\left|\right|\hfill \\ \hfill =& \left|\right|G_0(\phi ,\kappa )\left|\right|({\alpha }^q+{\alpha }^{q-1}+{\alpha }^{q+1})\hfill \\ \hfill =& \left|\right|G_0(\phi ,\kappa )\left|\right|\frac{1-{\alpha }^{q-n}}{1-{\alpha }^{q+1}}{\alpha }^{n+1}.\hfill \end{array}$$

(27)

As $0<\alpha <1$, and $G_0(\phi ,\kappa )$ are bound, so take $\beta =1-\alpha /(1-{\alpha }_{q-n}){\alpha }^{n+1}\left|\right|G_0(\phi ,\kappa )\left|\right|$, and we obtain

$$\left|\right|G_{q+n}(\phi ,\kappa )\left|\right|\le \beta ,\forall q,n\in N.$$

(28)

Hence, ${\left\{G_q(\phi ,\kappa )\right\}}_{q=0}^{\infty }$ makes a “Cauchy sequence” in H. This proves that the sequence ${\left\{G_q(\phi ,\kappa )\right\}}_{q=0}^{\infty }$ is a convergent sequence with the limit ${lim}_{q\to \infty }{G}_q(\phi ,\kappa )=G(\phi ,\kappa )$ for $\exists G(\phi ,\kappa )\in \mathcal{H}$, which completes the proof. □

Theorem 2.

Let us assume that ${\sum }_{h=0}^k{G}_h(\phi ,\kappa )$ is finite and $G(\phi ,\kappa )$ reflects the series solution that was found. Assuming $\alpha >0$ such that $\left|\right|G_{h+1}(\phi ,\kappa )\left|\right|\le \left|\right|G_h(\phi ,\kappa )\left|\right|$, the maximum absolute error is given by the following relation:

$$\left|\right|G(\phi ,\kappa )-\sum _{h=0}^k{G}_h(\phi ,\kappa )\left|\right|<\frac{{\alpha }^{k+1}}{1-\alpha }\left|\right|G_0(\phi ,\kappa )\left|\right|.$$

(29)

Proof. 

Suppose ${\sum }_{h=0}^k{G}_h(\phi ,\kappa )$ is finite, which implies that ${\sum }_{h=0}^k{G}_h(\phi ,\kappa )<\infty$.

Let us consider

$$\begin{array}{cc}\hfill \left|\right|G(\phi ,\kappa )-\sum _{h=0}^k{G}_h(\phi ,\kappa )\left|\right|=& \left|\right|\sum _{h=k+1}^{\infty }{G}_h(\phi ,\kappa )\left|\right|\hfill \\ \hfill \le & \sum _{h=k+1}^{\infty }\left|\right|G_h(\phi ,\kappa )\left|\right|\hfill \\ \hfill \le & \sum _{h=k+1}^{\infty }{\alpha }^h\left|\right|G_0(\phi ,\kappa )\left|\right|\hfill \\ \hfill \le & {\alpha }^{k+1}(1+\alpha +{\alpha }^2+\cdots )\left|\right|G_0(\phi ,\kappa )\left|\right|\hfill \\ \hfill \le & \frac{{\alpha }^{k+1}}{1-\alpha }\left|\right|G_0(\phi ,\kappa )\left|\right|.\hfill \end{array}$$

(30)

which completes the proof of the theorem. □

5. Applications

The YT-HPM is implemented to nonlinear FKFG equations in this study.

Example 1.

Let us assume the nonlinear FKFG equation in the case when $a=0,b=-1$, and $n=1$ as

$${}^{CF}{D}_{\kappa }^{\varsigma }z(\phi ,\kappa )=\frac{{\partial }^{2}z}{\partial {\phi }^2}-z^2(\phi ,\kappa ),\varsigma \in (0,1],$$

(31)

having initial conditions

$$z(\phi ,0)=1+sin\left(\phi \right)\&z_{\kappa }(\phi ,0)=0.$$

On taking YT of Equation (46), we have

$$Y\left[z(\phi ,\kappa )\right]=vz(\phi ,0)-(1+\varsigma v-\varsigma )Y\left[\frac{{\partial }^{2}z}{\partial {\phi }^2}-z^2(\phi ,\kappa )\right].$$

(32)

By employing inverse YT, we obtain

$$z(\phi ,\kappa )=(1+sin\left(\phi \right))-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\left[\frac{{\partial }^{2}z}{\partial {\phi }^2}-z^2(\phi ,\kappa )\right]\right].$$

(33)

Utilizing the HPT approach, we obtain

$$\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )=(1+sin\left(\phi \right))-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\left[\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )-\sum _{q=0}^{\infty }{\rho }^q{H}_q\left(z\right)\right]\right].$$

(34)

where the nonlinear terms are determined by He’s polynomial $H_q\left(z\right)$ and are calculated as

$$\begin{array}{cc}\hfill H_0\left(z\right)=& {\left(z_0\right)}^2,\hfill \\ \hfill H_1\left(z\right)=& 2z_0{z}_1,\hfill \\ \hfill & ⋮\hfill \end{array}$$

(35)

Thus, by comparing the same powers of $\rho$ in (50), we obtain

$$\begin{array}{cc}\hfill {\rho }^0:z_0(\phi ,\kappa )=& 1+sin\left(\phi \right),\hfill \\ \hfill {\rho }^1:z_1(\phi ,\kappa )=& -Y^{-1}\left[(1+\varsigma v-\varsigma )\left[H_0\left(z\right)-\frac{{\partial }^2}{\partial {\phi }^2}{z}_0\right]\right],\hfill \\ \hfill & =-\kappa +\kappa \varsigma -\frac{{\kappa }^2\varsigma }{2}-3\kappa sin\left(\phi \right)+3\kappa \varsigma sin\left(\phi \right)-\frac{3}{2}{\kappa }^2\varsigma sin\left(\phi \right)-\kappa {sin}^2\left(\phi \right)+\kappa \varsigma {sin}^2\left(\phi \right)\hfill \\ \hfill & -\frac{1}{2}{\kappa }^2\varsigma {sin}^2\left(\phi \right).\hfill \end{array}$$

(36)

Now,

$$\begin{array}{cc}\hfill {\rho }^2:z_2(\phi ,\kappa )=& -Y^{-1}\left[(1+\varsigma v-\varsigma )\left[H_1\left(z\right)-\frac{{\partial }^2}{\partial {\phi }^2}{z}_1\right]\right],\hfill \\ \hfill {\rho }^2:z_2(\phi ,\kappa )=& -(\frac{1}{24}){\kappa }^4\varsigma (-2+2{cos}^2\left(\phi \right)-11sin\left(\phi \right)-10{sin}^2\left(\phi \right)-2{sin}^3\left(\phi \right))+\frac{1}{3}{\kappa }^3+\frac{1}{2}{\kappa }^2(2-\hfill \\ \hfill & 4\varsigma +2{\varsigma }^2-{cos}^2\left(\phi \right)+4\varsigma {cos}^2\left(\phi \right)-2{\varsigma }^2{cos}^2\left(\phi \right)+11sin\left(\phi \right)-22\varsigma sin\left(\phi \right)+11{\varsigma }^{2}sin\left(\phi \right)\hfill \\ \hfill & +10{sin}^2\left(\phi \right)-20\varsigma {sin}^2\left(\phi \right)+10{\varsigma }^2{sin}^2\left(\phi \right)+2{sin}^3\left(\phi \right)-4\varsigma {sin}^3\left(\phi \right)+2{\varsigma }^2{sin}^3\left(\phi \right)).\hfill \end{array}$$

(37)

The solution in series form is stated as below:

$$\begin{array}{cc}\hfill & z(\phi ,\kappa )=1+sin\left(\phi \right)-\kappa +\kappa \varsigma -\frac{{\kappa }^2\varsigma }{2}-3\kappa sin\left(\phi \right)+3\kappa \varsigma sin\left(\phi \right)-\frac{3}{2}{\kappa }^2\varsigma sin\left(\phi \right)-\kappa {sin}^2\left(\phi \right)+\kappa \varsigma {sin}^2\left(\phi \right)\hfill \\ \hfill & -\frac{1}{2}{\kappa }^2\varsigma {sin}^2\left(\phi \right)-(\frac{1}{24}){\kappa }^4\varsigma (-2+2{cos}^2\left(\phi \right)-11sin\left(\phi \right)-10{sin}^2\left(\phi \right)-2{sin}^3\left(\phi \right))+\frac{1}{3}{\kappa }^3+\frac{1}{2}{\kappa }^2(2-\hfill \\ \hfill & 4\varsigma +2{\varsigma }^2-{cos}^2\left(\phi \right)+4\varsigma {cos}^2\left(\phi \right)-2{\varsigma }^2{cos}^2\left(\phi \right)+11sin\left(\phi \right)-22\varsigma sin\left(\phi \right)+11{\varsigma }^{2}sin\left(\phi \right)+10{sin}^2\left(\phi \right)-\hfill \\ \hfill & 20\varsigma {sin}^2\left(\phi \right)+10{\varsigma }^2{sin}^2\left(\phi \right)+2{sin}^3\left(\phi \right)-4\varsigma {sin}^3\left(\phi \right)+2{\varsigma }^2{sin}^3\left(\phi \right))+\cdots .\hfill \end{array}$$

(38)

Example 2.

Let us assume the nonlinear FKFG equation in the case when $a=-\frac{3}{4}$, $b=\frac{3}{2}$ and $n=3$ as

$${}^{CF}{D}_{\kappa }^{\varsigma }z(\phi ,\kappa )=\frac{{\partial }^{2}z}{\partial {\phi }^2}-\frac{3}{4}z(\phi ,\kappa )+\frac{3}{2}{z}^3(\phi ,\kappa ),\varsigma \in (0,1],$$

(39)

having the ICs

$$z(\phi ,0)=sech\left(\phi \right)\&z_{\kappa }(\phi ,0)=\frac{1}{2}sech\left(\phi \right)tanh\left(\phi \right).$$

On taking the YT of Equation (46), we have

$$Y\left[z(\phi ,\kappa )\right]=vz(\phi ,0)-(1+\varsigma v-\varsigma )Y\left[\frac{{\partial }^{2}z}{\partial {\phi }^2}-\frac{3}{4}z(\phi ,\kappa )+\frac{3}{2}{z}^3(\phi ,\kappa )\right].$$

(40)

By employing inverse YT, we have

$$z(\phi ,\kappa )=(sech\left(\phi \right)+\frac{1}{2}\kappa sech\left(\phi \right)tanh\left(\phi \right))-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\left[\frac{{\partial }^{2}z}{\partial {\phi }^2}-\frac{3}{4}z(\phi ,\kappa )+\frac{3}{2}{z}^3(\phi ,\kappa )\right]\right].$$

(41)

Utilizing the HPT, we obtain

$$\begin{array}{cc}\hfill & \sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )=(sech\left(\phi \right)+\frac{1}{2}\kappa sech\left(\phi \right)tanh\left(\phi \right))-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\right[\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )-\hfill \\ \hfill & \frac{3}{4}\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )+\sum _{q=0}^{\infty }{\rho }^q{H}_q\left(z\right)\left]\right].\hfill \end{array}$$

(42)

where the nonlinear terms are determined by He’s polynomial $H_q\left(z\right)$ and are calculated as

$$\begin{array}{cc}\hfill H_0\left(z\right)=& \frac{3}{2}{\left(z_0\right)}^3,\hfill \\ \hfill H_1\left(z\right)=& \frac{9}{2}{z}_0^2{z}_1,\hfill \\ \hfill & ⋮\hfill \end{array}$$

(43)

Thus, by comparing the same powers of $\rho$ in (50), we obtain

$$\begin{array}{cc}\hfill {\rho }^0:z_0(\phi ,\kappa )=& sech\left(\phi \right)+\frac{1}{2}\kappa sech\left(\phi \right)tanh\left(\phi \right),\hfill \\ \hfill {\rho }^1:z_1(\phi ,\kappa )=& -Y^{-1}\left[(1+\varsigma v-\varsigma )\left[H_0\left(z\right)-\frac{{\partial }^2}{\partial {\phi }^2}{z}_0\right]\right],\hfill \\ \hfill & =\frac{3}{390}{\kappa }^5\varsigma {sech}^3\left(\phi \right){tanh}^3\left(\phi \right)+\frac{1}{4}\kappa (-3sech\left(\phi \right)+3\varsigma sech\left(\phi \right)+2{sech}^3\left(\phi \right)-2\varsigma {sech}^3\left(\phi \right)\hfill \\ \hfill & =+4sech\left(\phi \right){tanh}^2\left(\phi \right)-4\varsigma sech\left(\phi \right){tanh}^2\left(\phi \right))+\frac{1}{16}{\kappa }^2(-6\varsigma sech\left(\phi \right)+4{sech}^3\left(\phi \right)-3\hfill \\ \hfill & sech\left(\phi \right)tanh\left(\phi \right)+3\varsigma sech\left(\phi \right)tanh\left(\phi \right)-2{sech}^3\left(\phi \right)tanh\left(\phi \right)+2\varsigma {sech}^3\left(\phi \right)tanh\left(\phi \right)+8\hfill \\ \hfill & \varsigma sech\left(\phi \right){tanh}^2\left(\phi \right)+4sech\left(\phi \right){tanh}^3\left(\phi \right)-4\varsigma sech\left(\phi \right){tanh}^3\left(\phi \right))+\frac{1}{48}{\kappa }^3(-3\varsigma sech\left(\phi \right)\hfill \\ \hfill & tanh\left(\phi \right)-2\varsigma {sech}^3\left(\phi \right)tanh\left(\phi \right)+18{sech}^3\left(\phi \right){tanh}^2\left(\phi \right)-18\varsigma {sech}^3\left(\phi \right){tanh}^2\left(\phi \right)+4\varsigma \hfill \\ \hfill & sech\left(\phi \right){tanh}^3\left(\phi \right))-\frac{3}{64}{\kappa }^4(-2\varsigma {sech}^3\left(\phi \right){tanh}^2\left(\phi \right)-{sech}^3\left(\phi \right){tanh}^2\left(\phi \right)+\varsigma {sech}^3\left(\phi \right)\hfill \\ \hfill & {tanh}^2\left(\phi \right)).\hfill \end{array}$$

(44)

The solution in series form is stated as given below:

$$\begin{array}{cc}\hfill & z(\phi ,\kappa )=sech\left(\phi \right)+\frac{1}{2}\kappa sech\left(\phi \right)tanh\left(\phi \right)+\frac{3}{390}{\kappa }^5\varsigma {sech}^3\left(\phi \right){tanh}^3\left(\phi \right)+\frac{1}{4}\kappa (-3sech\left(\phi \right)+3\varsigma sech\left(\phi \right)\hfill \\ \hfill & +2{sech}^3\left(\phi \right)-2\varsigma {sech}^3\left(\phi \right)+4sech\left(\phi \right){tanh}^2\left(\phi \right)-4\varsigma sech\left(\phi \right){tanh}^2\left(\phi \right))+\frac{1}{16}{\kappa }^2(-6\varsigma sech\left(\phi \right)+4\hfill \\ \hfill & {sech}^3\left(\phi \right)-3sech\left(\phi \right)tanh\left(\phi \right)+3\varsigma sech\left(\phi \right)tanh\left(\phi \right)-2{sech}^3\left(\phi \right)tanh\left(\phi \right)+2\varsigma {sech}^3\left(\phi \right)tanh\left(\phi \right)+8\hfill \\ \hfill & \varsigma sech\left(\phi \right){tanh}^2\left(\phi \right)+4sech\left(\phi \right){tanh}^3\left(\phi \right)-4\varsigma sech\left(\phi \right){tanh}^3\left(\phi \right))+\frac{1}{48}{\kappa }^3(-3\varsigma sech\left(\phi \right)tanh\left(\phi \right)-2\hfill \\ \hfill & \varsigma {sech}^3\left(\phi \right)tanh\left(\phi \right)+18{sech}^3\left(\phi \right){tanh}^2\left(\phi \right)-18\varsigma {sech}^3\left(\phi \right){tanh}^2\left(\phi \right)+4\varsigma sech\left(\phi \right){tanh}^3\left(\phi \right))\hfill \\ \hfill & -\frac{3}{64}{\kappa }^4(-2\varsigma {sech}^3\left(\phi \right){tanh}^2\left(\phi \right)-{sech}^3\left(\phi \right){tanh}^2\left(\phi \right)+\varsigma {sech}^3\left(\phi \right){tanh}^2\left(\phi \right))+\cdots .\hfill \end{array}$$

(45)

Example 3.

Let us assume the cubic nonlinear FKFG equation as

$${}^{CF}{D}_{\kappa }^{\varsigma }z(\phi ,\kappa )={\xi }^2\frac{{\partial }^{2}z}{\partial {\phi }^2}-c^2\frac{3}{4}z(\phi ,\kappa )+\delta ϵ\frac{3}{2}{z}^3(\phi ,\kappa ),\varsigma \in (0,1],$$

(46)

having the ICs

$$z(\phi ,0)=ϵcos\left(k\phi \right)\&z_{\kappa }(\phi ,0)=0.$$

On taking the YT of Equation (46), we have

$$Y\left[z(\phi ,\kappa )\right]=vz(\phi ,0)-(1+\varsigma v-\varsigma )Y\left[{\xi }^2\frac{{\partial }^{2}z}{\partial {\phi }^2}-c^2\frac{3}{4}z(\phi ,\kappa )+\delta ϵ\frac{3}{2}{z}^3(\phi ,\kappa )\right].$$

(47)

By employing inverse YT, we have

$$z(\phi ,\kappa )=ϵcos\left(k\phi \right)-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\left[{\xi }^2\frac{{\partial }^{2}z}{\partial {\phi }^2}-c^2\frac{3}{4}z(\phi ,\kappa )+\delta ϵ\frac{3}{2}{z}^3(\phi ,\kappa )\right]\right].$$

(48)

Utilizing the HPT approach, we obtain

$$\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )=ϵcos\left(k\phi \right)-Y^{-1}\left[(1+\varsigma v-\varsigma )Y\left[{\xi }^2\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )-c^2\sum _{q=0}^{\infty }{\rho }^q{z}_q(\phi ,\kappa )+\delta ϵ\sum _{q=0}^{\infty }{\rho }^q{H}_q\left(z\right)\right]\right].$$

(49)

where the nonlinear terms are determined by He’s polynomial $H_q\left(z\right)$ and are calculated as

$$\begin{array}{cc}\hfill H_0\left(z\right)=& {\left(z_0\right)}^3,\hfill \\ \hfill H_1\left(z\right)=& 3z_0^2{z}_1,\hfill \\ \hfill & ⋮\hfill \end{array}$$

(50)

Thus, by comparing the same powers of $\rho$ in (50), we obtain

$$\begin{array}{cc}\hfill {\rho }^0:z_0(\phi ,\kappa )=& ϵcos\left(k\phi \right),\hfill \\ \hfill {\rho }^1:z_1(\phi ,\kappa )=& -Y^{-1}\left[(1+\varsigma v-\varsigma )\left[H_0\left(z\right)-\frac{{\partial }^2}{\partial {\phi }^2}{z}_0\right]\right],\hfill \\ \hfill & =-c^2\kappa ϵcos\left(k\phi \right)+c^2\kappa \varsigma ϵcos\left(k\phi \right)-\frac{1}{2}{c}^2{\kappa }^2\varsigma ϵcos\left(k\phi \right)-k^2\kappa {\xi }^{2}cos\left(k\phi \right)+k^2\kappa \varsigma \hfill \\ \hfill & {\xi }^2ϵcos\left(k\phi \right)-\frac{1}{2}{k}^2{\kappa }^2\varsigma {\xi }^2ϵcos\left(k\phi \right)+\delta {ϵ}^4{cos}^3\left(k\phi \right)-\kappa \varsigma \delta {ϵ}^4{cos}^3\left(k\phi \right)+\frac{1}{2}{\kappa }^2\varsigma \delta {ϵ}^4{cos}^3\left(k\phi \right).\hfill \end{array}$$

(51)

The solution in series form is stated as below:

$$\begin{array}{cc}\hfill & z(\phi ,\kappa )=ϵcos\left(k\phi \right)-c^2\kappa ϵcos\left(k\phi \right)+c^2\kappa \varsigma ϵcos\left(k\phi \right)-\frac{1}{2}{c}^2{\kappa }^2\varsigma ϵcos\left(k\phi \right)-k^2\kappa {\xi }^{2}cos\left(k\phi \right)+k^2\kappa \varsigma \hfill \\ \hfill & {\xi }^2ϵcos\left(k\phi \right)-\frac{1}{2}{k}^2{\kappa }^2\varsigma {\xi }^2ϵcos\left(k\phi \right)+\delta {ϵ}^4{cos}^3\left(k\phi \right)-\kappa \varsigma \delta {ϵ}^4{cos}^3\left(k\phi \right)+\frac{1}{2}{\kappa }^2\varsigma \delta {ϵ}^4{cos}^3\left(k\phi \right)+\cdots .\hfill \end{array}$$

(52)

Results and Discussion

The numerical results for the FKFG equation are shown with FHPTM via the CF derivative in this section. With the help of Maple, the above problems are displayed in tabular and graphical form.

Table 1. Solution by means of proposed method with distinct $\phi$ and $\kappa$ for various values of fractional orders $\varsigma$ of Example 1.

$\mathit{\phi }$	$\mathit{\kappa }$	$\mathit{\varsigma }=1.50$	$\mathit{\varsigma }=1.75$	$\mathit{\varsigma }=2$
1	0.25	2.17213	2.40359	2.63506
2		2.26513	2.51422	2.76330
3		1.25387	1.33280	1.41173
4		0.18869	0.15053	0.11238
5		−0.03370	−0.08605	−0.13840
1	0.50	2.10600	2.50279	2.89959
2		2.19396	2.62097	3.04797
3		1.23132	1.36663	1.50193
4		0.19959	0.13418	0.06878
5		−0.01875	−0.10849	−0.19823
1	0.75	1.64307	2.13906	2.63506
2		1.69579	2.22955	2.76330
3		1.07346	1.24260	1.41173
4		0.27590	0.19414	0.11238
5		0.08594	−0.02623	−0.13840
1	1	0.78334	1.31241	1.84147
2		0.77061	1.33995	1.90929
3		0.78030	0.96071	1.14112
4		0.41761	0.33040	0.24319
5		0.28038	0.16073	0.04107

compares the approximations for the solutions to Example 1 for various values of $\phi$ and $\kappa$. The numerical results for the FKFG equation via the FHPTM via the CF derivative are similarly shown in

Table 2. Solution by means of proposed method with distinct $\phi$ and $\kappa$ for various values of fractional orders $\varsigma$ of Example 2.

$\mathit{\phi }$	$\mathit{\kappa }$	$\mathit{\varsigma }=1.50$	$\mathit{\varsigma }=1.75$	$\mathit{\varsigma }=2$
1	0.25	0.70409	0.70644	0.70786
2		0.29278	0.29488	0.29615
3		0.10966	0.11050	0.11101
4		0.04044	0.04075	0.04094
5		0.01488	0.01499	0.01506
1	0.50	0.74101	0.75029	0.75714
2		0.31593	0.32018	0.32331
3		0.11904	0.12056	0.12168
4		0.04393	0.04449	0.04490
5		0.01617	0.01637	0.01652
1	0.75	0.74194	0.76299	0.78064
2		0.33602	0.34199	0.34700
3		0.12849	0.13031	0.13184
4		0.04753	0.04817	0.04872
5		0.01749	0.01773	0.01793
1	1	0.68815	0.72400	0.75745
2		0.35248	0.35967	0.36637
3		0.13849	0.14027	0.14193
4		0.05142	0.05203	0.05259
5		0.01894	0.01916	0.01936

and

Table 3. Solution by means of proposed method with distinct $\phi$ and $\kappa$ for various values of fractional orders $\varsigma$ of Example 3.

$\mathit{\phi }$	$\mathit{\kappa }$	$\mathit{\varsigma }=1.50$	$\mathit{\varsigma }=1.75$	$\mathit{\varsigma }=2$
1	0.25	0.64712	0.63908	0.63105
2		−0.46521	−0.46173	−0.45824
3		−1.64344	−1.59146	−1.53948
4		−0.84229	−0.82774	−0.81318
5		0.29944	0.29850	0.29756
1	0.50	0.60998	0.59620	0.58243
2		−0.44819	−0.44222	−0.43624
3		−1.41572	−1.32661	−1.23751
4		−0.77664	−0.75168	−0.72672
5		0.29400	0.29239	0.29078
1	0.75	0.58662	0.56940	0.55218
2		−0.43715	−0.42968	−0.42221
3		−1.27711	−1.16573	−1.05434
4		−0.73593	−0.70474	−0.67354
5		0.29017	0.28816	0.28615
1	1	0.57703	0.55866	0.54030
2		−0.43208	−0.42411	−0.41614
3		−1.22761	−1.10880	−0.98999
4		−0.72019	−0.68691	−0.65364
5		0.28795	0.28580	0.28366

, with distinct $\phi$ and $\kappa$ for various values of fractional-order $\varsigma$. The approximate solutions $z(\phi ,\kappa )$ for various fractional-order $\varsigma$ values are shown in Figure 1, Figure 2 and Figure 3, respectively. This illustrates how the graphical behavior varies on the order of the fractional derivative. For Examples 1–3, all of the figures depict the approximate surface of solution $z(\phi ,\kappa )$ for various values of fractional order. It is clear from the figures that the solution is dependent on the order of the fractional derivative.

6. Conclusions

In this article, we have made an effort to understand and analyze the Caputo–Fabrizio fractional derivative-based fractional homotopy perturbation transform method, which is used to provide a rough solution to the nonlinear Klein–Fock–Gordon problem. We have taken into account three instances of the FKFG equation together with various initial conditions. The suggested approach provides more precise and speedily decipherable solutions in the form of a series. Tables and graphs have been used to present the calculated results. The vast results demonstrate the effectiveness and simplicity of this strategy, which may also be used to address other nonlinear issues. The obtained results can help numerous authors to study and interpret their experimental and observation data, especially the authors that specialize in nonlinear sciences such as plasma physics and nonlinear optics.