Novel analysis of nonlinear seventh-order fractional Kaup–Kupershmidt equation via the Caputo operator

Abstract

In this work, we use two unique methodologies, the homotopy perturbation transform method and Yang transform decomposition method, to solve the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The physical phenomena that arise in chemistry, physics, and engineering are mathematically explained in this equation, in particular, nonlinear optics, quantum mechanics, plasma physics, fluid dynamics, and so on. The provided methods are used to solve the fractional nonlinear seventh-order KK problem along with the Yang transform and fractional Caputo derivative. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approaches and the fractional operator to develop satisfactory approximations to the offered problem. To solve the fractional KK equation, we first use the Yang transform and fractional Caputo derivative. He’s and Adomian polynomials are useful to manage nonlinear terms. It is shown that the suggested approximate solution converges to the exact one. In these approaches, the results are calculated as convergent series. The key advantage of the recommended approaches is that they provide highly precise results with little computational work. The suggested approach results are compared to the precise solution. By comparing the outcomes with the precise solution using graphs and tables we can verify the efficacy of the offered strategies. Also, the outcomes of the suggested methods at various fractional orders are examined, demonstrating that the findings get more accurate as the value moves from fractional order to integer order. Moreover, the offered methods are innovative, simple, and quite accurate, demonstrating that they are effective for resolving differential equations.

1 Introduction

In past decades, fractional calculus has exceeded classical calculus in terms of popularity. In the context of exploration, ordinary calculus has achieved its peak. Thus scientific researchers and mathematicians need to understand fractional calculus. Instead of using the traditional “integer” order, it enables us to explain real-world phenomena more accurately. Several mathematicians participated and made important contributions to the subject, including Riesz, Laplace, Fourier, and others. Modern examples of contemporary definitions of fractional-order derivatives and integrals, which have transformed a new era in the history of fractional derivatives, include the Atangana–Baleanu fractional integral [1], the Caputo fractional derivative [2], and the Caputo–Fabrizio fractional derivative [3]. Several processes in physics, chemistry, engineering, and other disciplines can be accurately modeled using fractional calculus. Moreover, fractional calculus is often used to represent a variety of viscoelastic materials frequency-dependent damping behavior [4], the dynamics of nanoparticle-substrate interfaces [5], economics [6], and much more [7, 8]. At first, scientists were not aware of the significance of this calculus, but in recent years, a growing number of academics have begun using it to study a wide range of natural events. In practical applications, the Caputo fractional derivative is frequently employed because it makes it possible to involve the conventional initial and boundary conditions when creating mathematical models. Furthermore, the Caputo fractional derivative of a constant is equal to zero, just as the integer-order derivative [9].

Several associated scientific and engineering disciplines as well as diverse natural physical phenomena, such as electrodynamics [10], finance [11], and nanotechnology [12] are linked to issues concerning fractional calculus applications. Understanding the features of nonlinear problems that arise in everyday life requires understanding numerical and approximate solutions of fractional partial differential equations (FPDEs). A variety of mathematical methods that have been created and studied [13–15] have been used to obtain precise solutions of NLFDEs. For example, the q-homotopy analysis transform approach for Navier–Stokes equations having fractional-order [16], natural transform decomposition method for fractional modified Boussinesq and approximate long wave equations [17] and fractional-order Kaup–Kupershmidt equation [18], Yang transform decomposition method for time-fractional Fisher’s equation [19] and for time-fractional Noyes–Field model [20], Elzaki homotopy perturbation technique for solving regularized long-wave equations of fractional order [21], variational iteration transform method for fractional third-order Burgers and KdV nonlinear systems [22], Adomian decomposition Laplace transform method for investigating fractional diabetes model involving glucose-insulin alliance scheme [23], general residual power series method for solving fractional differential equations [24] and time-fractional Korteweg–de Vries equation [25, 26], modified Khater method for solving nonlinear fractional Ostrovsky equation [27], existence and stability analysis of fractional stochastic chaotic systems [28, 29], modified $(\frac{G'}{G})$-expansion scheme for traveling wave solutions of fractional Boussinesq equation [30], homotopy perturbation transform method for solving nonlinear fractional differential equations [31, 32], generalized Kudryashov method for nonlinear FPDEs of Burgers type [33], Laplace residual power series approach for solving Black–Scholes option pricing equations having fractional order [34], first integral method for solving fractional Cahn–Allen equation and fractional DSW system [35], and many more [36–40].

A valuable tool for modeling nonlinear processes in engineering and scientific concepts is the differential equation of fractional order. A range of scientific topics has been modeled using partial differential equations, particularly nonlinear ones. Because of FPDEs, scientists were able to discover and model an extensive assortment of important and useful physical challenges along with their effort in the physical sciences [41, 42]. The significance of obtaining approximate solutions has always been remarked. To model future processes, dynamic processes of physical systems collect information and signify the present and previous states of a causal nature. The fractional derivatives can be used as a substitutional technique to clarify how dynamic systems interact with causal behavior in response to their local environment [43, 44]. This inspires us to investigate the physical interpretation of fractional derivative-based system behavior for a class of nonlinear evolution differential equations. In the current study, we develop a series-form solution for a seventh-order nonlinear FPDE in the Caputo sense using two innovative methods based on the Yang transform.

Let us study the time-fractional mathematical models in more detail. The seventh-order modified fractional Korteweg–de Vries equation has the general form

$$ \begin{aligned} &D_{\beta}^{\wp}{\mathcal{M}(\varpi ,{\beta})}+\kappa _{1} \mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})+ \kappa _{2}\mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &+\kappa _{3} \mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ &+{\kappa _{4}\mathcal{M}^{2}(\varpi ,{\beta})M}_{\varpi \varpi \varpi}( \varpi ,{\beta})+\kappa _{5}\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})\\ &+\kappa _{6} \mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &+\kappa _{7}\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &+\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}( \varpi ,{\beta})=0, \end{aligned} $$

(1)

where the constant parameters $\kappa _{i},i=1,2,\ldots ,7$, are finite and cannot be zero, $D_{\beta}^{\wp}$ is the Caputo time-fractional derivative of order ℘, and $\mathcal{M}=\mathcal{M}(\varpi ,{\beta})$ is a function of the dynamic wave profile of the spatiotemporal patterns that will be derived ultimately. Pomeau et al. [45] investigated the stability of the Korteweg–de Vries equation in terms of singular perturbation and came up with the classical model of Equation (1). The fundamental case of practical goals can be described for particular values of the constant parameters $\kappa _{i}$. The time-fractional seventh-order Kaup–Kupershmidt (KK) equation can be expressed as

$$ \begin{aligned} &D_{\beta}^{\wp}{\mathcal{M}(\varpi ,{\beta})}+2016\mathcal{M}^{3}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})+630\mathcal{M}^{3}_{ \varpi}(\varpi ,{\beta})\\ &+2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ +504&\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})+252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &+147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &+42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})+\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})=0, \ \ \ \ \ 0< \wp \leq 1, \end{aligned} $$

(2)

with $\kappa _{1}=2016$, $\kappa _{2}=630$, $\kappa _{3}=2268$, $\kappa _{4}=504$, $\kappa _{5}=252$, $\kappa _{6}=147$, $\kappa _{7}=42$.

On the other hand, the fractional modified seventh-order Korteweg–de Vries equations have been effectively used to represent the mechanisms of nonlinear waves and complex systems occurring in shallow seas, plasma physics, ocean dynamics, capillary-gravitational waves, and nonlinear optics. For understanding the fundamental physics, phase transmission, and dynamic wave propagation behaviors controlling these equations, analytical, approximation, and soliton traveling waveform solutions are required. In the sections that follow, we build approximations of the solutions to the aforementioned equations with an appropriate initial condition based on the Caputo derivative. In this study, the fractional Kaup–Kupershmidt (KK) equation is approximated using the Yang transform (YT) along with the ADM and HPM methods. The Adomian and homotopy polynomials are utilized to break down the nonlinear terms, and the Yang transform is employed to change differential equations into algebraic equations. In the numerical cases, our methods yielded infinite series as an outcome. The series provide exact solutions to the relevant equations when written in closed form. To obtain precise and approximative results in a few simple steps, researchers may employ this study as a core reference to explore these approaches and apply it in numerous applications. The presentation of two novel techniques for the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem with minimal and consecutive steps makes this study unique. The offered methods give a high level of accuracy with minimal processing cost. The structure of the current discussion is as follows: The Caputo differentiation operator and the Yang transform theory are briefly described in Sect. 2. The steps for putting the suggested solutions to the general fractional differential equation into practice are outlined in Sects. 3 and 4. In Sect. 5, we provide the existence result of the proposed techniques. Applications of the techniques to this time-fractional equation with the specified initial condition are explained in Sect. 6. In Sect. 7, the conclusions are outlined.

2 Basic concepts

In this part, we give essential explanations of Caputo derivative in addition with the Yang transform (YT) properties.

2.1 Definition

The Caputo derivative of fractional order is defined as [46, 47]

$$ \ \begin{aligned} &D_{{\beta}}^{\wp}\mathcal{M}(\varpi ,{\beta})= \frac{1}{\Gamma (\ell -\wp )}\int _{0}^{{\beta}}({\beta}-\wp )^{\ell -\wp -1} \mathcal{M}^{(\ell )}(\varpi ,\varphi )d\varphi ,\\ &\quad \ell -1< \wp \leq \ell , \quad \ell \in N. \end{aligned} $$

(3)

2.2 Definition

The fractional Riemann–Liouville integral operator is expressed as [46, 47]

$$ \ I_{{\beta}}^{\wp}\mathcal{M}(\varpi ,{\beta})=\frac{1}{\Gamma (\wp )}\int _{0}^{{ \beta}}({\beta}-\varphi )^{\wp -1}\mathcal{M}(\varpi ,\varphi )d\varphi , \varphi >0 \ \ (\ell -1< \wp \leq \ell ),\quad \ell \in N. $$

(4)

2.3 Definition

The YT of a function $\mathcal{M}({\beta})\in C_{m}$, $m\geq -1$, is [48, 49]

$$ Y\{\mathcal{M}({\beta})\}=F(u) =\int _{0}^{\infty}e^{\frac{-{\beta}}{u}} \mathcal{M}({\beta})d{\beta},\ \ {\beta}>0, $$

(5)

where u is the transform variable.

Some important functions for the YT are

$$ \begin{aligned} Y[1]=&u, \\ Y[{\beta}]=&u^{2}, \\ Y[{\beta}^{q}]=&\Gamma (q+1)u^{q+1}, \end{aligned} $$

(6)

and the inverse YT is

$$ Y^{-1}\{F(u)\}=\mathcal{M}({\beta}). $$

(7)

2.4 Definition

The YT of the nth derivative function is [48]

$$ Y\{\mathcal{M}^{n}({\beta})\}= \frac{F(u)}{u^{n}}-\sum _{\ell =0}^{n-1} \frac{\mathcal{M}^{\ell}(0)}{u^{n-\ell -1}}, \ \ n=1,2,3,\dots \,. $$

(8)

2.5 Definition

The YT of the fractional derivative function is [48]

$$ Y\{\mathcal{M}^{\wp}({\beta})\}= \frac{F(u)}{u^{\wp}}-\sum _{\ell =0}^{n-1} \frac{\mathcal{M}^{\ell}(0)}{u^{\wp -(\ell +1)}}, \ \ n-1< \wp \leq n. $$

(9)

3 The general idea of HPTM

Here we give the general solution of HPTM to analyze FPDE:

$$ D_{{\beta}}^{\wp}\mathcal{M}(\varpi ,{\beta})=\mathcal{P}_{1}[\varpi ] \mathcal{M}(\varpi ,{\beta})+\mathcal{Q}_{1}[\varpi ]\mathcal{M}(\varpi ,{ \beta}), \ \ \varpi >0, \ \ 0< \wp \leq 1, $$

(10)

with initial guess

$$ \mathcal{M}(\varpi ,0)=\xi (\varpi ). $$

Here $D_{{\beta}}^{\wp}=\frac{\partial ^{\wp}}{\partial {\beta}^{\wp}}$ represent the Caputo derivative, and $\mathcal{P}_{1}[\varpi ]$ and $\mathcal{Q}_{1}[\varpi ]$ are linear and nonlinear operators.

By employing YT we obtain

$$\begin{aligned}& \ Y[D_{{\beta}}^{\wp}\mathcal{M}(\varpi ,{\beta})]=Y[\mathcal{P}_{1}[\varpi ] \mathcal{M}(\varpi ,{\beta})+\mathcal{Q}_{1}[\varpi ]\mathcal{M}(\varpi ,{ \beta})], \end{aligned}$$

(11)

$$\begin{aligned}& \frac{1}{u^{\wp}}\{F(u)- u \mathcal{M}(0)\}=Y[\mathcal{P}_{1}[\varpi ] \mathcal{M}(\varpi ,{\beta})+\mathcal{Q}_{1}[\varpi ]\mathcal{M}(\varpi ,{ \beta})]. \end{aligned}$$

(12)

By simplification we have

$$ F(u)=u \mathcal{M}(0)+u^{\wp}Y[\mathcal{P}_{1}[\varpi ]\mathcal{M}( \varpi ,{\beta})+\mathcal{Q}_{1}[\varpi ]\mathcal{M}(\varpi ,{\beta})]. $$

(13)

By implementing the inverse YT we have

$$ \mathcal{M}(\varpi ,{\beta})=\mathcal{M}(0)+Y^{-1}[u^{\wp}Y[\mathcal{P}_{1}[ \varpi ]\mathcal{M}(\varpi ,{\beta})+\mathcal{Q}_{1}[\varpi ]\mathcal{M}( \varpi ,{\beta})]]. $$

(14)

Now by using the idea of HPM we have

$$ \mathcal{M}(\varpi ,{\beta})=\sum _{k=0}^{\infty}\epsilon ^{k} \mathcal{M}_{k}(\varpi ,{\beta}) $$

(15)

with parameter $\epsilon \in [0,1]$.

The nonlinear term is deliberated as

$$ \mathcal{Q}_{1}[\varpi ]\mathcal{M}(\varpi ,{\beta})=\sum _{k=0}^{\infty} \epsilon ^{k}H_{n}(\mathcal{M}). $$

(16)

In addition, $H_{k}(\mathcal{M})$ shows He’s polynomials and is

$$ \ H_{k}(\mathcal{M}_{0},\mathcal{M}_{1},\ldots,\mathcal{M}_{n})= \frac{1}{\Gamma (n+1)}D_{\epsilon}^{k}\left [\mathcal{Q}_{1}\left ( \sum _{k=0}^{\infty}\epsilon ^{i}\mathcal{M}_{i}\right )\right ]_{ \epsilon =0}, $$

(17)

where $D_{\epsilon}^{k}=\frac{\partial ^{k}}{\partial \epsilon ^{k}}$.

By inserting (15) and (16) into (14) we get

$$ \begin{aligned} \sum _{k=0}^{\infty}\epsilon ^{k}\mathcal{M}_{k}(\varpi ,{\beta})&= \mathcal{M}(0)+\epsilon\\ &\quad \times \left (Y^{-1}\left [u^{\wp}Y\{ \mathcal{P}_{1}\sum _{k=0}^{\infty}\epsilon ^{k}\mathcal{M}_{k}( \varpi ,{\beta})+ \sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\} \right ]\right ). \end{aligned} $$

(18)

Equating the coefficients at $\epsilon ^{k}$, we have

$$ \ \begin{aligned} &\epsilon ^{0}:\mathcal{M}_{0}(\varpi ,{\beta})=\mathcal{M}(0), \\ &\epsilon ^{1}:\mathcal{M}_{1}(\varpi ,{\beta})=Y^{-1}\left [u^{\wp}Y( \mathcal{P}_{1}[\varpi ]\mathcal{M}_{0}(\varpi ,{\beta})+H_{0}( \mathcal{M}))\right ], \\ &\epsilon ^{2}:\mathcal{M}_{2}(\varpi ,{\beta})=Y^{-1}\left [u^{\wp}Y( \mathcal{P}_{1}[\varpi ]\mathcal{M}_{1}(\varpi ,{\beta})+H_{1}( \mathcal{M}))\right ], \\ &. \\ &. \\ &. \\ &\epsilon ^{k}:\mathcal{M}_{k}(\varpi ,{\beta})=Y^{-1}\left [u^{\wp}Y( \mathcal{P}_{1}[\varpi ]\mathcal{M}_{k-1}(\varpi ,{\beta})+H_{k-1}( \mathcal{M}))\right ], \\ k>0,k\in N. \end{aligned} $$

(19)

By continuing the same process we get the solution of $\mathcal{M}_{k}(\varpi ,{\beta})$ as

$$ \ \mathcal{M}(\varpi ,{\beta})=\lim _{M\rightarrow \infty}\sum _{k=1}^{M} \mathcal{M}_{k}(\varpi ,{\beta}). $$

(20)

4 The general idea of YTDM

Here we give the general solution of YTDM to analyze FPDE

$$ D_{\beta}^{\wp}{\mathcal{M}(\varpi ,{\beta})}=\mathcal{P}_{1}(\varpi ,{\beta})+ \mathcal{Q}_{1}(\varpi ,{\beta}),\ \ \varpi >0, \ \ 0< \wp \leq 1, $$

(21)

with initial guess

$$ \mathcal{M}(\varpi ,0)=\xi (\varpi ). $$

Here $D_{\beta}^{\wp}=\frac{\partial ^{\wp}}{\partial {\beta}^{\wp}}$ represents the Caputo derivative, and $\mathcal{P}_{1}$ and $\mathcal{Q}_{1}$ are linear and nonlinear operators.

By employing YT we obtain

$$ \begin{aligned} &Y[D_{\beta}^{\wp}{\mathcal{M}(\varpi ,{\beta})}]=Y[ \mathcal{P}_{1}(\varpi ,{\beta})+\mathcal{Q}_{1}(\varpi ,{\beta})], \\ &\frac{1}{u^{\wp}}\{F(u)- u \mathcal{M}(0)\}=Y[\mathcal{P}_{1}( \varpi ,{\beta})+\mathcal{Q}_{1}(\varpi ,{\beta})]. \end{aligned} $$

(22)

By simplification we have

$$ F(u)=u \mathcal{M}(0)+u^{\wp}Y[\mathcal{P}_{1}(\varpi ,{\beta})+ \mathcal{Q}_{1}(\varpi ,{\beta})], $$

(23)

By implementing the inverse YT we have

$$ \mathcal{M}(\varpi ,{\beta})=\mathcal{M}(0)+Y^{-1}[u^{\wp}Y[\mathcal{P}_{1}( \varpi ,{\beta})+\mathcal{Q}_{1}(\varpi ,{\beta})]. $$

(24)

Now the series solution is illustrated as

$$ \mathcal{M}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{ \beta}). $$

(25)

The nonlinear term is deliberated as

$$ \mathcal{Q}_{1}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{A}_{m} $$

(26)

with

$$ \mathcal{A}_{m}=\frac{1}{m!}\left [ \frac{\partial ^{m}}{\partial \ell ^{m}}\left \{\mathcal{Q}_{1}\left ( \sum _{k=0}^{\infty}\ell ^{k}\varpi _{k},\sum _{k=0}^{\infty}\ell ^{k}{ \beta}_{k}\right )\right \}\right ]_{\ell =0}. $$

(27)

By inserting (25) and (26) into (24) we have

$$ \sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{\beta})=\mathcal{M}(0)+Y^{-1} \left [u^{\wp }\left [Y\left \{\mathcal{P}_{1}(\sum _{m=0}^{\infty} \varpi _{m},\sum _{m=0}^{\infty}{\beta}_{m})+\sum _{m=0}^{\infty} \mathcal{A}_{m}\right \}\right ]\right ]. $$

(28)

Equating both sides we have

$$\begin{aligned}& \mathcal{M}_{0}(\varpi ,{\beta})=\mathcal{M}(0), \\& \mathcal{M}_{1}(\varpi ,{\beta})=Y^{-1}\left [u^{\wp }Y\{\mathcal{P}_{1}( \varpi _{0},{\beta}_{0})+\mathcal{A}_{0}\}\right ], \end{aligned}$$

(29)

Finally, we can write the solution in general for $m\geq 1$ as

$$ \mathcal{M}_{m+1}(\varpi ,{\beta})=Y^{-1}\left [u^{\wp }Y\{\mathcal{P}_{1}( \varpi _{m},{\beta}_{m})+\mathcal{A}_{m}\}\right ]. $$

5 Existence results

In this section, we illustrate the existence results of the proposed approach.

Theorem 5.1

Let H be a Hilbert space with accurate solution $\chi (\varpi ,{\beta})$ of (10), and let $\chi (\varpi ,{\beta})$, $\chi _{n}(\varpi ,{\beta})\in H$, and $\wp \in (0,1)$. The resultant solution $\sum _{q=0}^{\infty}\chi _{q}(\varpi ,{\beta})$ goes to $\chi (\varpi ,{\beta})$ if $\chi _{q}(\varpi ,{\beta})\leq \chi _{q-1}(\varpi ,{\beta})$ for all $q>A$, i.e., for all $\varpi >0$, there exists $A>0$ such that $||\chi _{q+n}(\varpi ,{\beta})||\leq \beta$ for all $q,n\in N$.

Proof. Take a sequence of $\sum _{q=0}^{\infty}\chi _{q}(\varpi ,{\beta})$:

$$ \begin{aligned} \mathcal{D}_{0}(\varpi ,{\beta})=&\chi _{0}(\varpi ,{\beta}), \\ \mathcal{D}_{1}(\varpi ,{\beta})=&\chi _{0}(\varpi ,{\beta})+\chi _{1}( \varpi ,{\beta}), \\ \mathcal{D}_{2}(\varpi ,{\beta})=&\chi _{0}(\varpi ,{\beta})+\chi _{1}( \varpi ,{\beta})+\chi _{2}(\varpi ,{\beta}), \\ \mathcal{D}_{3}(\varpi ,{\beta})=&\chi _{0}(\varpi ,{\beta})+\chi _{1}( \varpi ,{\beta})+\chi _{2}(\varpi ,{\beta})+\chi _{3}(\varpi ,{\beta}), \\ &\vdots \\ \mathcal{D}_{q}(\varpi ,{\beta})=&\chi _{0}(\varpi ,{\beta})+\chi _{1}( \varpi ,{\beta})+\chi _{2}(\varpi ,{\beta})+\cdots +\chi _{q}(\varpi ,{\beta}). \end{aligned} $$

(30)

We will show that $\mathcal{D}_{q}(\varpi ,{\beta})$ forms a Cauchy sequence to achieve the chosen result. Also, assume that

$$ \begin{aligned} ||\mathcal{D}_{{q}+1}(\varpi ,{\beta})-\mathcal{D}_{q}( \varpi ,{\beta})||&=||\chi _{{q}+1}(\varpi ,{\beta})||\\ &\leq \wp ||\chi _{q}( \varpi ,{\beta})||\leq \wp ^{2}||\chi _{q-1}(\varpi ,{\beta})||\\ &\leq \wp ^{3}|| \chi _{q-2}(\varpi ,{\beta})||\cdots \\ &\leq \wp _{q+1}||\chi _{0}(\varpi ,{\beta})||. \end{aligned} $$

(31)

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

$$\begin{aligned} \begin{aligned} ||\mathcal{D}_{q}(\varpi ,{\beta})-\mathcal{D}_{n}(\varpi ,{ \beta})||=&||\chi _{{q}+n}(\varpi ,{\beta})||=||\mathcal{D}_{q}(\varpi ,{\beta})- \mathcal{D}_{q-1}(\varpi ,{\beta})\\ &+(\mathcal{D}_{q-1}(\varpi ,{\beta})- \mathcal{D}_{q-2}(\varpi ,{\beta})) \\ &+(\mathcal{D}_{q-2}(\varpi ,{\beta})-\mathcal{D}_{q-3}(\varpi ,{\beta}))+ \cdots +(\mathcal{D}_{n+1}(\varpi ,{\beta})\\ &-\mathcal{D}_{n}(\varpi ,{\beta}))|| \\ \leq &||\mathcal{D}_{q}(\varpi ,{\beta})-\mathcal{D}_{q-1}(\varpi ,{\beta})||+||( \mathcal{D}_{q-1}(\varpi ,{\beta})-\mathcal{D}_{q-2}(\varpi ,{\beta}))|| \\ &+||(\mathcal{D}_{q-2}(\varpi ,{\beta})-\mathcal{D}_{q-3}(\varpi ,{\beta}))||+ \cdots +||(\mathcal{D}_{n+1}(\varpi ,{\beta})\\ &-\mathcal{D}_{n}(\varpi ,{ \beta}))|| \\ \leq &\wp ^{q}||\chi _{0}(\varpi ,{\beta})||+\wp ^{q-1}||\chi _{0}( \varpi ,{\beta})||+\cdots +\wp ^{q+1}||\chi _{0}(\varpi ,{\beta})|| \\ =&||\chi _{0}(\varpi ,{\beta})||(\wp ^{q}+\wp ^{q-1}+\wp ^{q+1}) \\ =&||\chi _{0}(\varpi ,{\beta})||\frac{1-\wp ^{q-n}}{1-\wp ^{q+1}}\wp ^{n+1}. \end{aligned} \end{aligned}$$

(32)

As $0<\wp <1$, and $\chi _{0}(\varpi ,{\beta})$ are bounded, taking $\beta =1-\wp /(1-\wp _{q-n})\wp ^{n+1}||\chi _{0}(\varpi ,{\beta})||$, we get

$$ ||\chi _{q+n}(\varpi ,{\beta})||\leq \beta ,\forall q,n\in N. $$

(33)

Hence $\{\chi _{q}(\varpi ,{\beta})\}_{q=0}^{\infty}$ is a Cauchy sequence in H. Therefore the sequence $\{\chi _{q}(\varpi ,{\beta})\}_{q=0}^{\infty}$ is a convergent sequence with limit $\lim _{q\rightarrow \infty}\chi _{q}(\varpi ,{\beta})=\chi (\varpi ,{\beta}) \in \mathcal{H}$, which completex the proof.

Theorem 5.2

Assume that $\sum _{h=0}^{k}\chi _{h}({\varpi},{\beta})$ is finite and $\chi (\varpi ,{\beta})$ is the series solution. Considering $\wp >0$ with $||\chi _{h+1}(\varpi ,{\beta})||\leq ||\chi _{h}(\varpi ,{\beta})||$, the maximum absolute error is determined as

$$ ||\chi (\varpi ,{\beta})-\sum _{h=0}^{k}\chi _{h}(\varpi ,{\beta})||< \frac{\wp ^{k+1}}{1-\wp}||\chi _{0}(\varpi ,{\beta})||. $$

(34)

Proof. Let $\sum _{h=0}^{k}\chi _{h}(\varpi ,{\beta})$ be finite, which shows that $\sum _{h=0}^{k}\chi _{h}(\varpi ,{\beta})<\infty$. Then

$$ \begin{aligned} ||\chi (\varpi ,{\beta})-\sum _{h=0}^{k}\chi _{h}(\varpi ,{ \beta})||=&||\sum _{h=k+1}^{\infty}\chi _{h}(\varpi ,{\beta})|| \\ \leq &\sum _{h=k+1}^{\infty}||\chi _{h}(\varpi ,{\beta})|| \\ \leq &\sum _{h=k+1}^{\infty}\wp ^{h}||\chi _{0}(\varpi ,{\beta})|| \\ \leq &\wp ^{k+1}(1+\wp +\wp ^{2}+\cdots )||\chi _{0}(\varpi ,{\beta})|| \\ \leq &\frac{\wp ^{k+1}}{1-\wp}||\chi _{0}(\varpi ,{\beta})||, \end{aligned} $$

(35)

which completes the proof.

Theorem 5.3

The result of (21) is unique when $0<(\varphi _{1}+\varphi _{2})( \frac{{{\beta}}^{{\wp}}}{\Gamma ({\wp}+1)})<1$.

Proof. Let $H=(C[J],||\cdot ||)$ be a Banach space of all continuous functionz on J with norm $||\phi ({\beta})||={\max}_{{\beta}\in J}|\phi ({\beta})|$. Let $I:H\rightarrow H$ be a nonlinear mapping, where

$$ \mathcal{M}^{C}_{l+1}=\mathcal{M}^{C}_{0}+Y^{-1}[u^{\wp}Y[\mathcal{P}_{1}( \mathcal{M}_{l}(\varpi ,{\beta}))+\mathcal{Q}_{1}(\mathcal{M}_{l}( \varpi ,{\beta}))]], \ \ l\geq 0. $$

Suppose that $|\mathcal{P}_{1}(\mathcal{M})-\mathcal{P}_{1}(\mathcal{M}^{*})|< \varphi _{1}|\mathcal{M}-\mathcal{M}^{*}|$ and $|\mathcal{Q}_{1}(\mathcal{M})-\mathcal{Q}_{1}(\mathcal{M}^{*})|< \varphi _{2}|\mathcal{M}-\mathcal{M}^{*}|$, where $\mathcal{M}:=\mathcal{M}(\varpi ,{\beta})$ and $\mathcal{M}^{*}:=\mathcal{M}^{*}(\varpi ,{\beta})$ are two separate function values, and $\varphi _{1}$, $\varphi _{2}$ are Lipschitz constants. Then

$$\begin{aligned} ||I\mathcal{M}-I\mathcal{M}^{*}||&\leq \max _{t\in J}|Y^{-1} \Big[u^{\wp}Y[\mathcal{P}_{1}(\mathcal{M})-\mathcal{P}_{1}( \mathcal{M}^{*})] \\ &+u^{\wp}Y[\mathcal{Q}_{1}(\mathcal{M})-\mathcal{Q}_{1}(\mathcal{M}^{*})]| \Big] \\ &\leq \max _{{\beta}\in J}\Big[\varphi _{1}Y^{-1}[u^{\wp}Y[|\mathcal{M}- \mathcal{M}^{*}|]] \\ &+\varphi _{2}Y^{-1}[u^{\wp}Y[|\mathcal{M}-\mathcal{M}^{*}|]]\Big] \\ &\leq \max _{t\in J}(\varphi _{1}+\varphi _{2})\left [Y^{-1}[u^{\wp}Y| \mathcal{M}-\mathcal{M}^{*}|]\right ] \\ &\leq (\varphi _{1}+\varphi _{2})\left [Y^{-1}[u^{\wp}Y||\mathcal{M}- \mathcal{M}^{*}||]\right ] \\ &=(\varphi _{1}+\varphi _{2})( \frac{{{\beta}}^{{\wp}}}{\Gamma ({\wp}+1)})||\mathcal{M}-\mathcal{M}^{*}||. \end{aligned}$$

(36)

Thus I is a contraction if $0<(\varphi _{1}+\varphi _{2})( \frac{{{\beta}}^{{\wp}}}{\Gamma ({\wp}+1)})<1$. The result of (21) is unique by the Banach fixed point theorem.

Theorem 5.4

The result of (21) is convergent.

Proof. Let $\mathcal{M}_{m}=\sum _{r=0}^{m}\mathcal{M}_{r}(\varpi ,{\beta})$. To show that $\mathcal{M}_{m}$ is a Cauchy sequence in H, let

$$ \begin{aligned} ||\mathcal{M}_{m}-\mathcal{M}_{n}||&=\max _{{\beta}\in J}| \sum _{r=n+1}^{m}\mathcal{M}_{r}| \\ &\leq \max _{{\beta}\in J} \left |Y^{-1}\left [u^{\wp}Y\left [\sum _{r=n+1}^{m}( \mathcal{P}_{1}(\mathcal{M}_{r-1})+\mathcal{Q}_{1}(\mathcal{M}_{r-1})) \right ]\right ]\right | \\ &=\max _{{\beta}\in J} \left |Y^{-1}\left [u^{\wp}Y\left [\sum _{r=n+1}^{m-1}( \mathcal{P}_{1}(\mathcal{M}_{r})+\mathcal{Q}_{1}(\mathcal{M}_{r})) \right ]\right ]\right | \\ &\leq \max _{{\beta}\in J}|Y^{-1}[u^{\wp}Y[(\mathcal{P}_{1}(\mathcal{M}_{m-1})- \mathcal{P}_{1}(\mathcal{M}_{n-1})\\ &+\mathcal{Q}_{1}(\mathcal{M}_{m-1})- \mathcal{Q}_{1}(\mathcal{M}_{n-1}))]]| \\ &\leq \varphi _{1}\max _{{\beta}\in J}|Y^{-1}[u^{\wp}Y[(\mathcal{P}_{1}( \mathcal{M}_{m-1})-\mathcal{P}_{1}(\mathcal{M}_{n-1}))]]| \\ &+\varphi _{2}\max _{{\beta}\in J}|Y^{-1}[u^{\wp}Y[(\mathcal{Q}_{1}( \mathcal{M}_{m-1})-\mathcal{Q}_{1}(\mathcal{M}_{n-1}))]]| \\ &=(\varphi _{1}+\varphi _{2})( \frac{{{\beta}}^{{\wp}}}{\Gamma ({\wp}+1)})||\mathcal{M}_{m-1}- \mathcal{M}_{n-1}||, \ \ n=1,2,3,\dots \,. \end{aligned} $$

(37)

Taking $m=n+1$, we have

$$ \begin{aligned} ||\mathcal{M}_{n+1}-\mathcal{M}_{n}||&\leq \varphi || \mathcal{M}_{n}-\mathcal{M}_{n-1}||\\ &\leq \varphi ^{2}||\mathcal{M}_{n-1} \mathcal{M}_{n-2}||\leq \cdots \leq \varphi ^{n}||\mathcal{M}_{1}- \mathcal{M}_{0}||, \end{aligned} $$

(38)

where $\varphi =(\varphi _{1}+\varphi _{2})( \frac{{{\beta}}^{{\wp}}}{\Gamma ({\wp}+1)})$. Similarly, we have

$$ \begin{aligned} ||\mathcal{M}_{m}-\mathcal{M}_{n}||&\leq ||\mathcal{M}_{n+1}- \mathcal{M}_{n}||+||\mathcal{M}_{n+2}\mathcal{M}_{n+1}||+\cdots +|| \mathcal{M}_{m}-\mathcal{M}_{m-1}||, \\ &(\varphi ^{n}+\varphi ^{n+1}+\cdots +\varphi ^{m-1})||\mathcal{M}_{1}- \mathcal{M}_{0}|| \\ &\leq \varphi ^{n}\left (\frac{1-\varphi ^{m-n}}{1-\varphi}\right )|| \mathcal{M}_{1}||, \end{aligned} $$

(39)

As $0<\varphi <1$, we get $1-\varphi ^{m-n}<1$. Hence

$$ \begin{aligned} ||\mathcal{M}_{m}-\mathcal{M}_{n}||\leq \frac{\varphi ^{n}}{1-\varphi}\max _{{\beta}\in J}||\mathcal{M}_{1}||. \end{aligned} $$

(40)

Since $||\mathcal{M}_{1}||<\infty$, $||\mathcal{M}_{m}-\mathcal{M}_{n}|| \rightarrow 0$ as $n \rightarrow \infty$, that is, $\mathcal{M}_{m}$ is a Cauchy sequence in H, and thus the series $\mathcal{M}_{m}$ is convergent.

6 Application

6.1 Example

Consider the seventh-order TK-K equation

$$ \begin{aligned} &D_{\beta}^{\wp}{\mathcal{M}(\varpi ,{\beta})}=-2016 \mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})-630 \mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}(\varpi ,{\beta}) \mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{ \beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})-252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta}), \ \ \ \ \ 0< \wp \leq 1, \end{aligned} $$

(41)

$$ \mathcal{M}(\varpi ,0)=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}( \rho \varpi )). $$

By implementing YT we have

$$ \ \begin{aligned} & Y\left [ \frac{\partial ^{\wp}\mathcal{M}}{\partial {\beta}^{\wp}}\right ]=Y \Bigg[-2016\mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{ \beta})-630\mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})-252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg]. \end{aligned} $$

(42)

It follows that

$$\begin{aligned} & \frac{1}{u^{\wp}}\{F(u)- u \mathcal{M}(0)\}=Y\Bigg[-2016 \mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})-630 \mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}(\varpi ,{\beta}) \mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{ \beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta}) -252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta}) \end{aligned}$$

(43)

$$\begin{aligned} &-147\mathcal{M}_{ \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}( \varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg], \\ & F(u)= u \mathcal{M}(0)+u^{\wp} \Bigg[-2016\mathcal{M}^{3}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})-630\mathcal{M}^{3}_{ \varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})-252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg]. \end{aligned}$$

(44)

By implementing inverse YT, we have

$$\begin{aligned} \ \begin{aligned} &\mathcal{M}(\varpi ,{\beta})=\mathcal{M}(0)+Y^{-1}\Bigg[u^{ \wp}\Bigg\{ Y\Bigg(-2016\mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{ \varpi}(\varpi ,{\beta})-630\mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &-2268 \mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \\ &\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})-504\mathcal{M}^{2}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})-252\mathcal{M}_{ \varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg)\Bigg\} \Bigg], \\ &\mathcal{M}(\varpi ,{\beta})=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}( \rho \varpi ))+Y^{-1}\Bigg[u^{\wp}\Bigg\{ Y\Bigg(-2016\mathcal{M}^{3}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})\\ &-630\mathcal{M}^{3}_{ \varpi}(\varpi ,{\beta}) \\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})-504\mathcal{M}^{2}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})\\ &-252\mathcal{M}_{ \varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta}) \\ &-147\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta})-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg) \Bigg\} \Bigg]. \end{aligned} \end{aligned}$$

(45)

Now by using the idea of HPM with $H_{k}(\mathcal{M})$ represent He’s polynomial

$$\begin{aligned} \begin{aligned} &\sum _{k=0}^{\infty}\epsilon ^{k}\mathcal{M}_{k}( \varpi ,{\beta})=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}(\rho \varpi ))+\epsilon \Bigg(Y^{-1}\Bigg[u^{\wp}Y\Bigg[-2016\Bigg(\sum _{k=0}^{ \infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg)\\ &-630\Bigg(\sum _{k=0}^{ \infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg) \\ &-2268\Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg)-504 \Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg)-252 \Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg)\\ &-147 \Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg) \\ &-42\Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}H_{k}(\mathcal{M})\Bigg)- \Bigg(\sum _{k=0}^{\infty}\epsilon ^{k}\mathcal{M}_{k}(\varpi ,{\beta}) \Bigg)_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}\Bigg]\Bigg] \Bigg). \end{aligned} \end{aligned}$$

(46)

Equating the coefficients at $\epsilon ^{k}$, we have

$$\begin{aligned} &\epsilon ^{0}:\mathcal{M}_{0}(\varpi ,{\beta})= \frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}(\rho \varpi )), \\ &\epsilon ^{1}:\mathcal{M}_{1}(\varpi ,{\beta})=- \frac{4\rho ^{9}{\beta}^{\wp}\tanh (\rho \varpi )\operatorname{sech}^{2}(\rho \varpi )}{3\Gamma (\wp +1)}, \\ &\epsilon ^{2}:\mathcal{M}_{2}(\varpi ,{\beta})= \frac{16\rho ^{16}{\beta}^{2\wp}(\cosh (2\rho \varpi )-2)\operatorname{sech}^{4}(\rho \varpi )}{9\Gamma (2\wp +1)}, \\ &\vdots \end{aligned}$$

By continuing the same process we get the solution in the series form as

$$ \ \begin{aligned} &\mathcal{M}({\varpi},{{\beta}})={\mathcal{M}}_{0}({\varpi},{{ \beta}})+{\mathcal{M}}_{1}({\varpi},{{\beta}})+{\mathcal{M}}_{2}({\varpi},{{ \beta}})+\cdots , \\ &\mathcal{M}({\varpi},{{\beta}})=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}( \rho \varpi ))- \frac{4\rho ^{9}{\beta}^{\wp}\tanh (\rho \varpi )\operatorname{sech}^{2}(\rho \varpi )}{3\Gamma (\wp +1)}\\ &+ \frac{16\rho ^{16}{\beta}^{2\wp}(\cosh (2\rho \varpi )-2)\operatorname{sech}^{4}(\rho \varpi )}{9\Gamma (2\wp +1)}+ \cdots . \end{aligned} $$

Application of YTDM

By implementing YT we have

$$ \begin{aligned} &Y\left \{ \frac{\partial ^{\wp}\mathcal{M}}{\partial {\beta}^{\wp}}\right \} =Y \Bigg[-2016\mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{ \beta})-630\mathcal{M}^{3}_{\varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})-252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg]. \end{aligned} $$

(47)

It follows that

$$\begin{aligned} &\frac{1}{u^{\wp}}\{F(u)- u \mathcal{M}(0)\} =Y\Bigg[-2016 \mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})-630 \mathcal{M}^{3}_{\varpi}(\varpi ,{\beta}) \\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})-504\mathcal{M}^{2}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})\\ &-252\mathcal{M}_{ \varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta}) \end{aligned}$$

(48)

$$\begin{aligned} &-147\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) -42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg], \\ &F(u)= u \mathcal{M}(0)+u^{\wp}Y\Bigg[-2016\mathcal{M}^{3}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})-630\mathcal{M}^{3}_{ \varpi}(\varpi ,{\beta})\\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta}) \\ &-504\mathcal{M}^{2}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta})-252\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-147\mathcal{M}_{\varpi}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta}) \\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg]. \end{aligned}$$

(49)

By implementing the inverse YT we have

$$\begin{aligned} \ \begin{aligned} &\mathcal{M}(\varpi ,{\beta})=\mathcal{M}(0)+Y^{-1}\Bigg[u^{ \wp}\Bigg\{ Y\Bigg[-2016\mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{ \varpi}(\varpi ,{\beta})-630\mathcal{M}^{3}_{\varpi}(\varpi ,{\beta}) \\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})-504\mathcal{M}^{2}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})\\ &-252\mathcal{M}_{ \varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta}) \\ &-147\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta})-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg] \Bigg\} \Bigg], \\ &\mathcal{M}(\varpi ,{\beta})=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}( \rho \varpi ))+Y^{-1}\Bigg[u^{\wp}\Bigg\{ Y\Bigg[-2016\mathcal{M}^{3}( \varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta})\\ &-630\mathcal{M}^{3}_{ \varpi}(\varpi ,{\beta}) \\ &-2268\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})-504\mathcal{M}^{2}(\varpi ,{ \beta})\mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})\\ &-252\mathcal{M}_{ \varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi}( \varpi ,{\beta}) \\ &-147\mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta})\\ &-42\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})-\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg] \Bigg\} \Bigg]. \end{aligned} \end{aligned}$$

(50)

Now the series form is

$$ \mathcal{M}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{ \beta}). $$

(51)

Let us present the nonlinear terms by Adomian polynomials as $\mathcal{M}^{3}(\varpi ,{\beta})\mathcal{M}_{\varpi}(\varpi ,{\beta}) = \sum _{m=0}^{\infty}\mathcal{A}_{m}$, $\mathcal{M}^{3}_{\varpi}(\varpi ,{ \beta})=\sum _{m=0}^{\infty}\mathcal{B}_{m}$, $\mathcal{M}(\varpi ,{\beta}) \mathcal{M}_{\varpi}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi}(\varpi ,{ \beta})=\sum _{m=0}^{\infty}\mathcal{C}_{m}$, $\mathcal{M}^{2}(\varpi , {\beta}) \mathcal{M}_{\varpi \varpi \varpi}(\varpi ,{\beta})=\sum _{m=0}^{\infty} \mathcal{D}_{m}$, $\mathcal{M}_{\varpi \varpi}(\varpi ,{\beta})\mathcal{M}_{ \varpi \varpi \varpi}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{E}_{m}$, $\mathcal{M}_{\varpi}(\varpi ,{\beta}) \mathcal{M}_{\varpi \varpi \varpi \varpi}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{F}_{m}$, $\mathcal{M}(\varpi ,{\beta})\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{G}_{m}$. So we get

$$\begin{aligned} &\sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{\beta})= \mathcal{M}(\varpi ,0) +Y^{-1}\Bigg[u^{\wp }Y\Bigg[-2016\sum _{m=0}^{ \infty}\mathcal{A}_{m}-630\sum _{m=0}^{\infty}\mathcal{B}_{m}\\ &-2268 \sum _{m=0}^{\infty}\mathcal{C}_{m}-504\sum _{m=0}^{\infty} \mathcal{D}_{m} \\ &-252\sum _{m=0}^{\infty}\mathcal{E}_{m}-147\sum _{m=0}^{\infty} \mathcal{F}_{m}-42\sum _{m=0}^{\infty}\mathcal{G}_{m}-\mathcal{M}_{ \varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg] \Bigg], \\ &\sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{\beta})= \frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}(\rho \varpi )) +Y^{-1} \Bigg[u^{\wp }Y\Bigg[-2016\sum _{m=0}^{\infty}\mathcal{A}_{m}\\ &-630 \sum _{m=0}^{\infty}\mathcal{B}_{m}\\ &-2268\sum _{m=0}^{\infty} \mathcal{C}_{m} \\ &-504\sum _{m=0}^{\infty}\mathcal{D}_{m}-252\sum _{m=0}^{\infty} \mathcal{E}_{m}-147\sum _{m=0}^{\infty}\mathcal{F}_{m}-42\sum _{m=0}^{ \infty}\mathcal{G}_{m}-\mathcal{M}_{\varpi \varpi \varpi \varpi \varpi \varpi \varpi}(\varpi ,{\beta})\Bigg]\Bigg]. \end{aligned}$$

(52)

Equating both sides, we have

$$ \mathcal{M}_{0}(\varpi ,{\beta})=\frac{\rho ^{2}}{3}(1-\frac{3}{2}\tanh ^{2}( \rho \varpi )). $$

On $m=0$,

$$ \mathcal{M}_{1}(\varpi ,{\beta})=- \frac{4\rho ^{9}{\beta}^{\wp}\tanh (\rho \varpi )\operatorname{sech}^{2}(\rho \varpi )}{3\Gamma (\wp +1)}; $$

on $m=1$,

$$ \mathcal{M}_{2}(\varpi ,{\beta})= \frac{16\rho ^{16}{\beta}^{2\wp}(\cosh (2\rho \varpi )-2)\operatorname{sech}^{4}(\rho \varpi )}{9\Gamma (2\wp +1)}. $$

By continuing the process, we get the solution in series form as

$$\begin{aligned}& \mathcal{M}(\varpi ,{\beta})=\sum _{m=0}^{\infty}\mathcal{M}_{m}(\varpi ,{ \beta}) =\mathcal{M}_{0}(\varpi ,{\beta})+\mathcal{M}_{1}(\varpi ,{\beta})+ \mathcal{M}_{2}(\varpi ,{\beta})+\cdots ,\\& \begin{aligned} &\mathcal{M}(\varpi ,{\beta})=\frac{\rho ^{2}}{3}(1- \frac{3}{2}\tanh ^{2}(\rho \varpi ))- \frac{4\rho ^{9}{\beta}^{\wp}\tanh (\rho \varpi )\operatorname{sech}^{2}(\rho \varpi )}{3\Gamma (\wp +1)}\\ &+ \frac{16\rho ^{16}{\beta}^{2\wp}(\cosh (2\rho \varpi )-2)\operatorname{sech}^{4}(\rho \varpi )}{9\Gamma (2\wp +1)}+ \cdots . \end{aligned} \end{aligned}$$

Taking $\wp =1$, we obtain

$$ \ \mathcal{M}({\varpi},{{\beta}})=\frac{\rho ^{2}}{3}\Bigg(1-\frac{3}{2} \tanh ^{2}\Bigg(\rho \Bigg[\varpi +\frac{4\rho ^{6}}{3}{\beta}\Bigg] \Bigg)\Bigg). $$

(53)

Physical interpretation of the results

Numerical simulations of the studied problem are included in the paper to demonstrate the reliability and efficiency of the methods under consideration. Additionally, we provide an extensive overview of the graphical solutions found. The obtained results are highly satisfactory and closely match with the precise solutions to the problem under consideration. Figure 1 displays a comparison of the 3D surface plots between the exact solution and the obtained approximate solution. Figure 2 displays a comparison of the 3D surface plots of the approximate solution generated at $\wp =0.8,0.6$. Figure 3 presents the 3D and 2D plots of the derived solution with respect to time β for various fractional orders ℘. We may observe how various fractional orders affect the variation in the solution. From every figure we can see that the employed fractional operator in the model under consideration demonstrates some interesting outcomes and validates the model that notably maintains historical and temporal behavior. The comparison of the precise and secure results with the suggested approaches for various fractional orders ℘ is shown in Table 1. The comparison of achieved results with the suggested procedures for varying fractional orders ℘ is shown in Table 2 in terms of absolute error values. In terms of absolute error values, Table 3 compares the secured findings obtained from the homotopy analysis approach with the suggested methods. The comparative analysis of absolute errors indicates that our techniques converge faster than other methods. Furthermore, we demonstrated that we can approach the exact solution as the number of iterations increases, and the numerical simulations highlight both the applicability and precision of the considered solution technique. This confirmed that the strategies under consideration are appropriate and trustworthy for solving fractional differential equations. As the number of terms increases, the solution converges quickly to the exact solution.

Figure 1

Nature of our approach and exact solution

Figure 2

Nature of our approach solution when $\wp =0.8,0.6$

Figure 3

Nature of our approach solution at numerous values of ℘

Table 1 Behavior of the exact and analytical solutions at various values of ℘

Table 2 Numerical simulation in the sense of absolute error at different values of ℘

Table 3 Analytical solution comparison with homotopy analysis method (HAM) at different values of β and ϖ

7 Conclusions

In this paper, we have tried to investigate the fractional Caputo derivative-based homotopy perturbation transform method and the Yang transform decomposition approach, which are used to approximate the fractional nonlinear seventh-order Kaup–Kupershmidt (KK) problem. The terms are obtained as series solutions using two-step approaches. The Yang transformation is used to first simplify the target problem before applying the decomposition method and perturbation approach to find solutions. We have considered one example of the equation along with the initial condition. The nonlinear terms in the issue investigated are expressed using Adomian polynomials and He’s polynomials. It is easier and more direct to solve fractional problems using the suggested hybrid strategies. The offered approaches give a series of more exact and quickly understandable results. Graphs and tables are created to represent the calculated study results. The tables and graphs show that as the value of σ approaches the classical value 1 of the problem, the approximate solution of the problem converges to its precise solution. The impressive results demonstrate how easy and efficient these methods are and how they may be used to solve other nonlinear issues. Several authors, particularly those working in nonlinear sciences like nonlinear optics and plasma physics, can benefit from the results in evaluating and interpreting their experimental and observational data. The readers can employ hybrid approaches merging with our proposed schemes as a future study direction to attain better outcomes. In conclusion, we prove that the approaches we have developed is highly reliable and applicable to huge study classifications concerning fractional-order nonlinear scientific procedures, which help us better understand the nonlinear compound phenomena in related fields of science and innovation.