A Novel Approach to Solving Fractional-Order Kolmogorov and Rosenau–Hyman Models through the q-Homotopy Analysis Transform Method

Abstract

In this study, a novel method called the q-homotopy analysis transform method (q-HATM) is proposed for solving fractional-order Kolmogorov and Rosenau–Hyman models numerically. The proposed method is shown to have fast convergence and is demonstrated using test examples. The validity of the proposed method is confirmed through graphical representation of the obtained results, which also highlights the ability of the method to modify the solution’s convergence zone. The q-HATM is an efficient scheme for solving nonlinear physical models with a series solution in a considerable admissible domain. The results indicate that the proposed approach is simple, effective, and applicable to a wide range of physical models.

1. Introduction

Fractional calculus (FC) is a branch of mathematical analysis that deals with derivatives and integrals of a non-integer order. The fractional derivative of a function represents its rate of change with respect to a non-integer order derivative operator. Fractional calculus has found applications in various fields such as physics, engineering, finance, and biology. The core principle of FC is that natural phenomenon modeling is done via fractional operators rather than integer operators. As a result, fractional calculus focuses on phenomena that standard theory cannot model [1,2,3,4,5]. Fractional partial differential equations (FPDEs) have received a number of notable contributions in the past. In many different domains, such as unification of diffusion, dynamical systems, wave propagation phenomenon, heat transfer, control theory, image processing, mixed convection flows, and mechanical systems, these equations are more useful for analyzing and describing a variety of phenomena [6,7,8,9,10,11,12,13]. In recent times, a variety of nonlinear fractional differential equations that do not possess exact analytical solutions have been approximately solved by means of numerical methods such as the variational iteration method (VIM), residual power series method (RPSM), reproducing kernel method (RKM), Laplace Adomian decomposition method (LADM), Laplace variational iteration method (LVIM) and Adomian decomposition method (ADM) for further details on the approaches and numerical strategies used to solve fractional differential equations [14,15,16,17,18,19,20].

Fractional calculus is a rapidly growing field of mathematics that deals with fractional-order differentiation and integration. The application of fractional calculus in modeling real-world systems has been a topic of research for several years [21,22,23]. The Fractional-Order Kolmogorov and Rosenau–Hyman models are two such models that have gained significant attention in recent years. The Fractional-Order Kolmogorov model is a fractional generalization of the classical Kolmogorov model, which is a stochastic process that describes the evolution of a probability density function. The Fractional-Order Kolmogorov model is used to model complex systems such as biological, ecological, and financial systems [24,25,26].

The Rosenau–Hyman model is another fractional-order model that is used to describe the dynamics of complex systems. This model is particularly useful for systems that exhibit long-range interactions, such as plasma physics, fluid dynamics, and geophysics. Both models have demonstrated their effectiveness in capturing the behavior of complex systems that cannot be fully explained by classical models. In this context, this article will explore the Fractional-Order Kolmogorov and Rosenau–Hyman models, their properties, and applications [27,28,29,30,31,32,33].

Fractional calculus (FC) is a classical extension of calculus that deals with arbitrary order differentiation and integration. It focuses on phenomena that standard theory cannot model using fractional operators rather than integer operators. Fractional partial differential equations (FPDEs) have received significant attention in various domains, such as diffusion, wave propagation, and heat transfer, due to their usefulness in analyzing and describing a variety of phenomena. However, no technique gives an explicit solution for FPDEs due to the complexity of fractional calculus [1,2,3,4,5]. Numerical techniques such as the variational iteration method (VIM), residual power series method (RPSM), Adomian decomposition method (ADM), Laplace Adomian decomposition method (LADM), reproducing kernel method (RKM) and Laplace variational iteration method (LVIM) have been used to solve these equations approximately [6,7,8,9,10,11,12,13].

The homotopy analysis method (HAM) is a powerful technique for solving differential and integral equations of fractional and classical order. Unlike other methods, HAM does not require perturbation or linearization and has been successfully applied to various nonlinear models in science and technology. However, the drawback of HAM is that it requires a significant amount of computer memory and processing time. To address this issue, some researchers have suggested combining HAM with previously used transform methods. By using transform methods, it is possible to simplify the complexity of the problem, which can reduce the computational burden of HAM. Furthermore, the combination of HAM with transform methods can provide more accurate solutions and improve the convergence rate. Overall, the combination of HAM with transform methods is a promising direction for future research in this area. It has the potential to make HAM more efficient and effective for solving various nonlinear models, which can have a significant impact on scientific and technological advancements [14,15,16,17,18,19,20].

In the current work, q-HATM was taken into consideration when trying to solve challenges that were predicted inside the FC framework. Singh et al. [34], using the Laplace transform and the q-homotopy analysis approach, suggest this method. This solution technique deals with numerous operators that may help regulate the convergence province and we modify the accuracy of the produced findings. It is not necessary to meet any of the aforementioned objectives. The suggested method is innovative in that it provides a straightforward method for locating the solution, a sizable convergence zone, and a non-local influence in the discovered solution. The suggested method manipulates and regulates the acquired solution, which, contrary to other conventional methods, swiftly converges to the analytical solution in a constrained acceptable zone.

2. Preliminaries

Here some basic definitions about fractional derivative and Laplace transform [35,36,37,38,39] and integrals [40] are discussed here.

Definition 1.

Let $\mathcal{U}\in {\mathbb{H}}^1(\mu ,ϵ)(ϵ>\mu ),\varkappa \in [0,1]$ be differentiable, then the Atangana-Baleanu derivative of order ϰ in Caputo sense is given by

$${}_{\mu }^{ABC}{D}_{\varsigma }^{\varkappa }\left(\mathcal{U}\left(\varsigma \right)\right)=\frac{\mathfrak{N}\left[\varkappa \right]}{1-\varkappa }\frac{d}{d\varsigma }{\int }_{\mu }^{\varsigma }{\mathcal{U}}^{\prime }\left(\phi \right)E_{\varkappa }\left[\varkappa \frac{{(\varsigma -\phi )}^{\varkappa }}{\varkappa -1}\right]d\phi ,$$

(1)

where, the function $\mathfrak{N}$ is a normalization of the function satisfies $\mathfrak{N}\left(0\right)=\mathfrak{N}\left(1\right)=1$. Noting that

$$E_{\varkappa }\left({\varsigma }^{\varkappa }\right)=\sum _{\vartheta =0}^{\infty }\frac{{\varsigma }^{\varkappa \vartheta }}{\Gamma (\varkappa \vartheta +1)}.$$

Definition 2.

The AB fractional integral is defined by

$${}_{\mu }^{AB}{I}_{\varsigma }^{\varkappa }\left(\mathcal{U}\left(\varsigma \right)\right)=\frac{1-\varkappa }{\mathfrak{N}\left[\varkappa \right]}\mathcal{U}\left(\varsigma \right)+\frac{\varkappa }{\mathfrak{N}\left[\varkappa \right]\Gamma \left(\varkappa \right)}{\int }_{\mu }^{\varsigma }\mathcal{U}\left(\phi \right){(\varsigma -\phi )}^{\varkappa -1}d\phi .$$

(2)

Definition 3.

Fractional derivative Laplace transform (LT) is given by

$$\mathcal{L}{[}_{\mu }^{ABC}{D}_{\varsigma }^{\varkappa }\left(\mathcal{U}\left(\varsigma \right)\right)=\frac{\mathfrak{N}\left[\varkappa \right]}{1-\varkappa }\frac{s^{\varkappa }\mathcal{L}\left[\mathcal{U}\left(\varsigma \right)\right]-s^{\varkappa -1}\mathcal{U}\left(0\right)}{s^{\varkappa }+\frac{\varkappa }{(1-\varkappa )}},0<\varkappa \le 1.$$

(3)

3. Methodology

The general methodology of q-HATM [41,42,43,44] for fractional Kolmogorov IVP

$${}^{ABC}{D}_{\varsigma }^{\varkappa }\mathcal{U}(\varrho ,\varsigma )=\mathrm{R}\left[\mathcal{U}(\varrho ,\varsigma )\right]+\mathrm{N}\left[\mathcal{U}(\varrho ,\varsigma )\right],0<\varkappa \le 1,$$

(4)

with initial condition

$$\mathcal{U}(\varrho ,0)=f\left(\varrho \right),$$

(5)

where ${}^{ABC}{D}_{\varsigma }^{\varkappa }\mathcal{U}(\varrho ,\varsigma )$ symbolise the AB derivative of $\mathcal{U}(\varrho ,\varsigma )$, $\mathrm{R}$ and $\mathrm{N}$ is linear and nonlinear functions.

On using the LT on Equation (4), we have after simplification

$$\mathcal{L}\left[\mathcal{U}(\varrho ,\varsigma )\right]=\frac{f\left(\varrho \right)}{s}+\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left(1-\varkappa +\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[\mathrm{R}\left[\mathcal{U}\right(\varrho ,\varsigma \left)\right]+\mathrm{N}\left[\mathcal{U}\right(\varrho ,\varsigma \left)\right]\right].$$

(6)

The non-linear operator is defined as follows

$$\begin{array}{cc}\hfill N\left[\varphi \right(\varrho ,\varsigma ;q\left)\right]=& \mathcal{L}\left[\mathcal{U}(\varrho ,\varsigma )\right]-\frac{f\left(\varrho \right)}{s}\hfill \\ \hfill & +\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left(1-\varkappa +\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[\mathrm{R}\varphi (\varrho ,\varsigma ;q)+\mathrm{N}\varphi (\varrho ,\varsigma ;q)\right].\hfill \end{array}$$

(7)

Here, $\varphi (\varrho ,\varsigma ;q)$ is the real-valued function with respect to $\varrho$, $\varsigma$ and $q\in [0,\frac{1}{n}]$. Now, we define a homotopy as follows

$$(1-nq)\mathcal{L}[\varphi (\varrho ,\varsigma ;q)-{\mathcal{U}}_0(\varrho ,\varsigma )]=\hslash q\mathfrak{N}\left[\varphi (\varrho ,\varsigma ;q)\right],$$

(8)

where ℏ is an auxiliary parameter, $\mathcal{L}$ is LT, $q\in [0,\frac{1}{n}]$$(n\ge 1)$ is the embedding parameter. For $q=0$ and $q=\frac{1}{B}$, the below hold true

$$\varphi (\varrho ,\varsigma ;0)={\mathcal{U}}_0(\varrho ,\varsigma ),\varphi (\varrho ,\varsigma ;\frac{1}{n})=\mathcal{U}(\varrho ,\varsigma ).$$

(9)

Thus, by intensifying q from 0 to $\frac{1}{n}$, the solution $\varphi (\varrho ,\varsigma ;q)$ varies from initial guess ${\mathcal{U}}_0(\varrho ,\varsigma )$ to $\mathcal{U}(\varrho ,\varsigma )$. We defining $\varphi (\varrho ,\varsigma ;q)$ with respect to q by using the Taylor theorem, we get

$$\varphi (\varrho ,\varsigma ;q)={\mathcal{U}}_0(\varrho ,\varsigma )+\sum _{m=1}^{\infty }{\mathcal{U}}_m(\varrho ,\varsigma )q^m,$$

(10)

where

$${\mathcal{U}}_m=\frac{1}{m!}\frac{{\partial }^m\varphi (\varrho ,\varsigma ;q)}{\partial q}{|}_{q=0}.$$

(11)

The series (8) converges at $q=\frac{1}{n}$ for the proper choice of ${\mathcal{U}}_0(\varrho ,\xi ,\varsigma )$, n and ℏ. Then

$$\mathcal{U}(\varrho ,\varsigma )={\mathcal{U}}_0(\varrho ,\varsigma )+\sum _{m=1}^{\infty }{\mathcal{U}}_m(\varrho ,\varsigma ){\left(\frac{1}{n}\right)}^m.$$

(12)

Taking the derivative of Equation (8) with respect to the embedding parameter q and then putting $q=0$, later dividing by $m!$, we obtain

$$\mathcal{L}[\mathcal{U}(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\Re }_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right),$$

(13)

where the vectors are defined as

$${\overrightarrow{\mathcal{U}}}_m=[{\mathcal{U}}_0(\varrho ,\varsigma ),{\mathcal{U}}_1(\varrho ,\varsigma ),\cdots ,{\mathcal{U}}_m(\varrho ,\varsigma )].$$

(14)

On applying inverse LT on Equation (13), one can get

$${\mathcal{U}}_m(\varrho ,\xi ,\varsigma )=k_m{\mathcal{U}}_{m-1}(\varrho ,\xi ,\varsigma )+\hslash {\mathcal{L}}^{-1}\left[{\Re }_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right)\right],$$

(15)

where

$$\begin{array}{cc}\hfill {\Re }_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right)=& \mathcal{L}\left[{\mathcal{U}}_{m-1}(\varrho ,\varsigma )\right]-\left(1-\frac{k_m}{n}\right)\left(\frac{f\left(\varrho \right)}{s}\right)+\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left(1-\varkappa +\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[\mathrm{N}\mathcal{U}(\varrho ,\varsigma )\right],\hfill \end{array}$$

(16)

and

$$k_m=\left\{\begin{array}{cc}0,\hfill & m\le 1,\hfill \\ n,\hfill & m>1.\hfill \end{array}\right.$$

(17)

Using the Equations (15) and (16), one can get the series of ${\mathcal{U}}_m(\varkappa ,\varsigma )$. Lastly, the series q-HATM solution is defined as

$$\mathcal{U}(\varrho ,\varsigma )=\sum _{m=0}^{\infty }{\mathcal{U}}_m(\varrho ,\varsigma ).$$

(18)

4. Numerical Problems

Problem 1.

Consider the following non-linear time-fractional Kolmogorov IVP:

$$\left\{\begin{array}{cc}\hfill & D_{\varsigma }^{\varkappa }\mathcal{U}(\varrho ,\varsigma )=(\varrho +1)D_{\varrho }\mathcal{U}(\varrho ,\varsigma )+{\varrho }^2{e}^{\varsigma }{D}_{\varrho }^2\mathcal{U}(\varrho ,\varsigma ),0<\varkappa \le 1,(\varrho ,\varsigma )\in [0,1]\times \mathbb{R},\hfill \\ \hfill & \mathcal{U}(\varrho ,0)=\varrho +1.\hfill \end{array}\right.$$

(19)

The exact solution at $\varkappa =1$ is given by

$$\mathcal{U}(\varrho ,\varsigma )=(\varrho +1)e^{\varsigma }.$$

(20)

Applying the Laplace transform on Equation (19) and using initial condition, we get

$$\mathcal{L}\left(\mathcal{U}\right)=\frac{\left(\varrho +1\right)}{s}+\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[(\varrho +1)\frac{\partial \mathcal{U}}{\partial \varrho }+{\varrho }^2{e}^{\varsigma }\frac{{\partial }^2\mathcal{U}}{\partial {\varrho }^2}\right].$$

(21)

The non-linear operator is defined as

$$\begin{array}{cc}\hfill \mathfrak{N}\left[\mathrm{\Phi }\right(\varrho ,\varsigma ;q\left)\right]=& \mathcal{L}\left(\mathrm{\Phi }(\varrho ,\varsigma ;q)\right)-\frac{\left(\varrho +1\right)}{s}\hfill \\ \hfill & -\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[(\varrho +1)\frac{\partial \mathrm{\Phi }(\varrho ,\varsigma ;q)}{\partial \varrho }+{\varrho }^2{e}^{\varsigma }\frac{{\partial }^2\mathrm{\Phi }(\varrho ,\varsigma ;q)}{\partial {\varrho }^2}\right].\hfill \end{array}$$

(22)

The mth order deformation equation define by the assist of suggested method as follows

$$\mathcal{L}[{\mathcal{U}}_m(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\mathcal{R}}_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right),$$

(23)

where ${\mathcal{R}}_m\left(\overrightarrow{\mathcal{U}}\right)$ is

$$\begin{array}{cc}\hfill {\mathcal{R}}_m\left(\overrightarrow{\mathcal{U}}\right)=& \mathcal{L}\left({\mathcal{U}}_{m-1}\right)-\left(1-\frac{k_m}{n}\right)\mathcal{L}\left(\varrho +1\right)\hfill \\ \hfill & -\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[(\varrho +1)\frac{\partial {\mathcal{U}}_{m-1}}{\partial \varrho }+{\varrho }^2{e}^{\varsigma }\frac{{\partial }^2{\mathcal{U}}_{m-1}}{\partial {\varrho }^2}\right].\hfill \end{array}$$

(24)

Applying inverse LT on Equation (31), we get

$${\mathcal{U}}_m(\varrho ,\varsigma )=k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )+\hslash \mathcal{L}\left[{\mathcal{R}}_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right)\right].$$

(25)

Solving the above equations, we get

$$\begin{array}{cc}\hfill {\mathcal{U}}_0(\varrho ,\varsigma )=& \varrho +1,\hfill \\ \hfill {\mathcal{U}}_1(\varrho ,\varsigma )=& \frac{\hslash \left(-1-\varrho +\left(-\frac{(\varrho +1){\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}+\varrho +1\right)\varkappa \right)}{\mathfrak{N}\left[\varkappa \right]},\hfill \\ \hfill {\mathcal{U}}_2(\varrho ,\varsigma )=& \frac{n\hslash \left(-1-\varrho +\left(-\frac{(\varrho +1){\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}+\varrho +1\right)\varkappa \right)}{\mathfrak{N}\left[\varkappa \right]}+\frac{{\hslash }^2}{\mathfrak{N}{\left[\varkappa \right]}^2}(1+{\varkappa }^2\left(1+\frac{{\varsigma }^{2\varkappa }}{\Gamma (2\varkappa +1)}\right)+\mathfrak{N}\left[\varkappa \right](-1+\varkappa )\hfill \\ \hfill & +\left(-2+\frac{(-\mathfrak{N}\left[\varkappa \right]-2\varkappa +2){\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\right)\varkappa )(\varrho +1),\hfill \\ \hfill {\mathcal{U}}_3(\varrho ,\varsigma )=& n(\frac{n\hslash \left(-1-\varrho +\left(-\frac{(\varrho +1){\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}+\varrho +1\right)\varkappa \right)}{\mathfrak{N}\left[\varkappa \right]}+\frac{{\hslash }^2}{\mathfrak{N}{\left[\varkappa \right]}^2}(1+{\varkappa }^2\left(1+\frac{{\varsigma }^{2\varkappa }}{\Gamma (2\varkappa +1)}\right)+\mathfrak{N}\left[\varkappa \right](-1+\varkappa )\hfill \\ \hfill & +\left(-2+\frac{(-\mathfrak{N}\left[\varkappa \right]-2\varkappa +2){\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\right)\varkappa \left)(\varrho +1)\right)+\hslash \left(\frac{1}{\mathfrak{N}{\left[\varkappa \right]}^2}\right(\hslash (n+2\hslash +{\varkappa }^2(2\hslash +n+\frac{(2\hslash +n){\varsigma }^{2\varkappa }}{\Gamma (2\varkappa +1)}\hfill \\ \hfill & +\varrho (2\hslash +n))+\left(-4\hslash -2n+\frac{(4\hslash +2n-\mathfrak{N}\left[\varkappa \right](2\hslash +n)-2\varkappa (2\hslash +n)){\varsigma }^{\varkappa }(\varrho +1)}{\Gamma (\varkappa +1)}-2\varrho (2\hslash +n)\right)\varkappa \hfill \\ \hfill & +\varrho (2\hslash +n)\left)\right)+\frac{1}{\mathfrak{N}{\left[\varkappa \right]}^3}(\hslash (\varrho +1)((-1+\varkappa )\left(\mathfrak{N}{\left[\varkappa \right]}^2(\hslash +n)+\frac{3\hslash {\varkappa }^2{\varsigma }^{2\varkappa }}{\Gamma (2\varkappa +1)}\right)+(\left(1-\frac{{\varsigma }^{3\varkappa }}{\Gamma (3\varkappa +1)}\right){\varkappa }^3\hfill \\ \hfill & -3{\varkappa }^3+3\left(-\frac{{(-1+\varkappa )}^2{\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}+1\right)\varkappa -1)\hslash )\left)\right),\hfill \\ \hfill & ⋮\hfill \end{array}$$

(26)

In Figure 1, the exact and analytical solutions graph at $\varkappa =1$ for Problem 1. Figure 2, three dimensional various fractional order graph of ϰ.

Table 1. Absolute error of Exact and fourth terms approximate solutions for Problem 1 with various $\varsigma$ and $\varrho$ for $\varkappa =1$.

$\mathit{\varsigma }$	$\mathit{\varrho }$	${\mathit{Approximate}}_{\mathit{\varkappa }=1}$	Exact	AE
0.1	0	1.105166667	1.105170918	4.251 × ${10}^{-6}$
	0.3	1.326200000	1.326205102	5.102 × ${10}^{-6}$
	0.5	1.547233333	1.547239285	5.952 × ${10}^{-6}$
	0.7	1.547239285	1.768273469	6.802 × ${10}^{-6}$
	0.9	1.989300000	1.989307652	7.652 × ${10}^{-6}$
	1	2.210333333	2.210341836	8.503 × ${10}^{-6}$
0.3	0	1.221333333	1.221402758	6.9425 × ${10}^{-5}$
	0.3	1.465600000	1.465683310	8.3310 × ${10}^{-5}$
	0.5	1.709866667	1.709963861	9.7194 × ${10}^{-5}$
	0.7	1.954133333	1.954244413	1.11080 × ${10}^{-4}$
	0.9	2.198400000	2.198524964	1.24964 × ${10}^{-4}$
	1	2.442666667	2.442805516	1.38849 × ${10}^{-4}$

, absolute error of Exact and fourth terms approximate solutions for Problem 1 with various ς and ϱ for $\varkappa =1$.

Problem 2.

Consider the following non-linear time-fractional Rosenau–Hyman IVP:

$$\left\{\begin{array}{cc}\hfill & D_{\varsigma }^{\varkappa }\mathcal{U}(\varrho ,\varsigma )=\mathcal{U}(\varrho ,\varsigma )D_{\varrho }^3\mathcal{U}(\varrho ,\varsigma )+\mathcal{U}(\varrho ,\varsigma )D_{\varrho }\mathcal{U}(\varrho ,\varsigma )+3D_{\varrho }\mathcal{U}(\varrho ,\varsigma )D_{\varrho }^2\mathcal{U}(\varrho ,\varsigma ),0<\varkappa \le 1,(\varrho ,\varsigma )\in [0,1]\times \mathbb{R},\hfill \\ \hfill & \mathcal{U}(\varrho ,0)=-\frac{8\mathcal{C}}{3}{\cos }^2\left(\frac{\varrho }{4}\right).\hfill \end{array}\right.$$

(27)

The exact solution at $\varkappa =1$ is given by

$$\mathcal{U}(\varrho ,\varsigma )=-\frac{8}{3}{\cos }^2\left(\frac{\varrho -\mathcal{C}\varsigma }{4}\right).$$

(28)

Applying the Laplace transform on Equation (27) and using initial condition, we get

$$\mathcal{L}\left(\mathcal{U}\right)=\frac{\left(-\frac{8\mathcal{C}}{3}{\cos }^2\left(\frac{\varrho }{4}\right)\right)}{s}+\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}\left[\mathcal{U}{D}_{\varrho }^3\mathcal{U}+\mathcal{U}{D}_{\varrho }\mathcal{U}+3D_{\varrho }\mathcal{U}{D}_{\varrho }^2\mathcal{U}\right].$$

(29)

The non-linear operator is defined as

$$\begin{array}{cc}\hfill \mathfrak{N}\left[\mathrm{\Phi }\right(\varrho ,\varsigma ;q\left)\right]=& \mathcal{L}\left(\mathrm{\Phi }(\varrho ,\varsigma ;q)\right)-\frac{\left(-\frac{8\mathcal{C}}{3}{\cos }^2\left(\frac{\varrho }{4}\right)\right)}{s}-\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}[\mathrm{\Phi }(\varrho ,\varsigma ;q)D_{\varrho }^3\mathrm{\Phi }(\varrho ,\varsigma ;q)\hfill \\ \hfill & +\mathrm{\Phi }(\varrho ,\varsigma ;q)D_{\varrho }\mathrm{\Phi }(\varrho ,\varsigma ;q)+3D_{\varrho }\mathrm{\Phi }(\varrho ,\varsigma ;q)D_{\varrho }^2\mathrm{\Phi }(\varrho ,\varsigma ;q)].\hfill \end{array}$$

(30)

The mth order deformation equation define by the assist of suggested method as follows

$$\mathcal{L}[{\mathcal{U}}_m(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\mathcal{R}}_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right),$$

(31)

where ${\mathcal{R}}_m\left(\overrightarrow{\mathcal{U}}\right)$ is

$$\begin{array}{cc}\hfill {\mathcal{R}}_m\left(\overrightarrow{\mathcal{U}}\right)=& \mathcal{L}\left({\mathcal{U}}_{m-1}\right)-\left(1-\frac{k_m}{n}\right)\mathcal{L}\left(-\frac{8\mathcal{C}}{3}{\cos }^2\left(\frac{\varrho }{4}\right)\right)-\frac{1}{\mathfrak{N}\left[\varkappa \right]}\left((1-\varkappa )+\frac{\varkappa }{s^{\varkappa }}\right)\mathcal{L}[\sum _{i=0}^{m-1}{\mathcal{U}}_i{D}_{\varrho }^3{\mathcal{U}}_{m-i-1}\hfill \\ \hfill & +\sum _{i=0}^{m-1}{\mathcal{U}}_i{D}_{\varrho }{\mathcal{U}}_{m-i-1}+3\sum _{i=0}^{m-1}{D}_{\varrho }{\mathcal{U}}_i{D}_{\varrho }^2{\mathcal{U}}_{m-i-1}]\hfill \end{array}$$

(32)

Applying inverse LT on Equation (31), we get

$${\mathcal{U}}_m(\varrho ,\varsigma )=k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )+\hslash \mathcal{L}\left[{\mathcal{R}}_m\left({\overrightarrow{\mathcal{U}}}_{m-1}\right)\right].$$

(33)

Solving the above equations, we get

$$\begin{array}{cc}\hfill {\mathcal{U}}_0(\varrho ,\varsigma )=& -\frac{8\mathcal{C}}{3}{\cos }^2\left(\frac{\varrho }{4}\right),\hfill \\ \hfill {\mathcal{U}}_1(\varrho ,\varsigma )=& \frac{4\hslash {\mathcal{C}}^2\cos \left(\frac{\varrho }{4}\right)\sin \left(\frac{\varrho }{4}\right)\left(1+\varkappa \left(-1+\frac{{\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\right)\right)}{3\mathfrak{N}\left[\varkappa \right]},\hfill \\ \hfill {\mathcal{U}}_2(\varrho ,\varsigma )=& \frac{4n\hslash {\mathcal{C}}^2\cos \left(\frac{\varrho }{4}\right)\sin \left(\frac{\varrho }{4}\right)\left(1+\varkappa \left(-1+\frac{{\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\right)\right)}{3\mathfrak{N}\left[\varkappa \right]}\hfill \\ \hfill & +\frac{1}{3}\left({\hslash }^2\right({\mathcal{C}}^3(\frac{{\varsigma }^{2\varkappa }{\varkappa }^2\left(\cos {\left(\frac{\varrho }{4}\right)}^2-\sin {\left(\frac{\varrho }{4}\right)}^2\right)}{\Gamma (2\varkappa +1)}+\frac{2(-1+\varkappa )\left(-2\cos {\left(\frac{\varrho }{4}\right)}^2+1\right)\varkappa {\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\hfill \\ \hfill & +\left(2\cos {\left(\frac{\varrho }{4}\right)}^2-1\right){(-1+\varkappa )}^2\left)\right)+\frac{4{\mathcal{C}}^2\cos \left(\frac{\varrho }{4}\right)\sin \left(\frac{\varrho }{4}\right)\left(1+\varkappa \left(-1+\frac{{\varsigma }^{\varkappa }}{\Gamma (\varkappa +1)}\right)\right)}{\mathfrak{N}\left[\varkappa \right]}),\hfill \\ \hfill & ⋮.\hfill \end{array}$$

(34)

Figure 3, the analytical solutions of different fractional order ϰ of Problem 2. Figure 4, the analytical solutions of different fractional order ϰ of Problem 2.

Table 2. The absolute error of exact and q-HATM of Problem 2 of different fractional order of $\varsigma$ and $\varrho$ for $\varkappa =1$.

$\mathit{\varsigma }$	$\mathit{\varrho }$	${\mathit{Approximate}}_{\mathit{\varkappa }=1}$	Exact	AE
0.1	0	10.56035556	10.56035508	4.8 × ${10}^{-8}$
	0.3	10.64002126	10.64002221	9.5 × ${10}^{-7}$
	0.5	10.66666431	10.66666667	2.36 × ${10}^{-7}$
	0.7	10.64001846	10.64002221	3.75 × ${10}^{-7}$
	0.9	10.56034998	10.56035508	5.10 × ${10}^{-7}$
	1	10.42845487	10.40846247	6.400 × ${10}^{-7}$
0.2	0	10.24568889	10.24565863	3.026 × ${10}^{-6}$
	0.3	10.42833314	10.48246127	1.616 × ${10}^{-6}$
	0.5	10.45689126	10.56035508	7.043 × ${10}^{-6}$
	0.7	10.63991712	10.64002221	1.0408 × ${10}^{-5}$
	0.9	10.66651797	10.66666667	1.3770 × ${10}^{-5}$
	1	10.63983139	10.46003331	1.8972 × ${10}^{-5}$

, absolute error of exact and q-HATM of Problem 2 of different fractional order of ς and ϱ for $\varkappa =1$.

5. Conclusions

The present research has effectively solved the fractional-order Kolmogorov and Rosenau–Hyman models using q-HATM without any difficulty. The q-HATM has numerous advantages and efficiency due to its ability to provide a broad convergence area, uncomplicated solution process, and high precision in obtaining accurate results. The q-HATM approach is more robust and systematic than other analytical methods, and it can be used to investigate nonlinear mathematical models that describe real-world issues. The auxiliary parameter h used in the proposed method explains the non-local convergence and provides an illuminated solution in series form without any ambiguity.