Application of the q-Homotopy Analysis Transform Method to Fractional-Order Kolmogorov and Rosenau–Hyman Models within the Atangana–Baleanu Operator

Abstract

The q-homotopy analysis transform method (q-HATM) is a powerful tool for solving differential equations. In this study, we apply the q-HATM to compute the numerical solution of the fractional-order Kolmogorov and Rosenau–Hyman models. Fractional-order models are widely used in physics, engineering, and other fields. However, their numerical solutions are difficult to obtain due to the non-linearity and non-locality of the equations. The q-HATM overcomes these challenges by transforming the equations into a series of linear equations that can be solved numerically. The results show that the q-HATM is an effective and accurate method for solving fractional-order models, and it can be used to study a wide range of phenomena in various fields.

1. Introduction

Mathematical models known as fractional-order partial differential equations (FPDEs) are used to represent physical phenomena that exhibit complex dynamics and non-local effects. FPDEs extend the classical theory of partial differential equations by allowing the use of non-integer orders of differentiation, which more accurately capture the non-local and non-linear behavior of many physical systems. These equations have been increasingly used in recent years in fields such as physics, engineering, finance, and biology, and have proven to be powerful tools for modeling complex processes and analyzing their behavior. This article will explore the concept of fractional-order partial differential equations and their applications, as well as the challenges associated with their analysis and numerical solution [1,2,3,4,5].

Symmetry is a fundamental concept in physics and mathematics that describes the invariance of a system under certain transformations. In the context of fractional-order models such as the fractional-order Kolmogorov and Rosenau–Hyman models, symmetry plays an important role in understanding the behavior of the system. In summary, symmetry is a base in mathematics and physics that plays an important role in understanding the behavior of fractional-order models, such as the fractional-order Kolmogorov and Rosenau–Hyman models. The fractional-order Kolmogorov model exhibits scaling symmetry, while the Rosenau–Hyman model exhibits reflection symmetry. These symmetries are a consequence of the fractional derivative and the symmetry of the differential operator, respectively, and play an important role in the analysis and interpretation of the models.

The fractional-order Kolmogorov and Rosenau–Hyman equations are two important mathematical models that have gained significant attention in recent years due to their ability to describe a wide range of physical phenomena. These equations are based on the idea of fractional calculus, which extends the concepts of differentiation and integration to non-integer orders [6,7].

The fractional-order Kolmogorov equation is a generalization of the classical Kolmogorov diffusion equation, which is commonly used to model diffusive processes in physics, chemistry, and biology. The fractional-order version takes into account the memory effects and long-range correlations that are present in many complex systems, making it a more accurate and flexible model [8,9,10,11].

The Rosenau–Hyman equation is another important model that has been used to study a wide range of phenomena, including fluid dynamics, turbulence, and nonlinear waves. This equation incorporates fractional derivatives to account for the non-local effects that arise in these systems, making it particularly useful in situations where standard models fail to capture the full complexity of the system. Both the fractional-order Kolmogorov equation and the Rosenau–Hyman equation have attracted significant interest in recent years, not only for their mathematical elegance and theoretical properties, but also for their practical applications in various fields [12,13,14,15].

The q-homotopy analysis transform method (q-HATM) is a powerful and efficient mathematical tool used for solving nonlinear partial differential equations (PDEs). It is an extension of the traditional homotopy analysis transform method (HATM), which was introduced by Liao in 1992. The q-HATM uses a parameter q, which can be any real number, to improve the convergence of the series solution obtained through the HATM. The q-HATM has been successfully applied to solve a wide range of problems in various fields of science and engineering, including fluid mechanics, heat transfer, and nonlinear optics. This method is easy to implement and can produce accurate results even for highly nonlinear and singular problems. In this paper, we will discuss the q-HATM and its application in solving nonlinear PDEs [16,17,18,19,20].

2. Basic Definitions

Definition 1. 

The Aboodh transform on function is given as

$$B=\left\{\mathit{V}\left(\varrho \right):\exists M,n_1,n_2>0,\left|\mathit{V}\left(\varrho \right)\right|<M{e}^{-\epsilon \varrho }\right\}$$

and is expressed as [21,22]

$$A\left\{\mathit{V}\left(\varrho \right)\right\}=\frac{1}{\epsilon }{\int }_0^{\infty }\mathit{V}\left(\varrho \right)e^{-\epsilon \varrho }d\varrho ,\varrho >0\mathit{and}{n}_1\le \epsilon \le n_2$$

Theorem 1. 

Consider a set B containing the Laplace transformation G and Aboodh transformation F of the function $\mathit{V}\left(\varrho \right)$, denoted by $G\left(\epsilon \right)$ and $F\left(\epsilon \right)$, respectively [23,24]. The equation linking these two transformations, which states that $G\left(\epsilon \right)$ is equal to $F\left(\epsilon \right)$ divided by ε.

A more general transformation, called the ZZ transform, was introduced by Zain Ul Abadin Zafar [25], which extends the Aboodh and Laplace integral transforms. The ZZ transform is defined as follows:

Definition 2. 

To define the Atangana-Baleanu Caputo (ABC) operator of a functions $\mathit{V}(\phi ,\varrho )$ from the space $H^1(a,b)$, the following expression is used, where υ is a constant belonging to the interval $(0,1)$ [26]

$${ABC}_a{D}_{\varrho }^{\upsilon }\mathit{V}(\phi ,\varrho )=\frac{B\left(\upsilon \right)}{1-\upsilon }{\int }_a^{\varrho }{\mathit{V}}^{\prime }(\phi ,\varrho )E_{\upsilon }\left(\frac{-\upsilon {(\varrho -\eta )}^{\upsilon }}{1-\upsilon }\right)d\eta .$$

Definition 3. 

The Riemann-Liouville derivative of AB, which is represented by $\mathit{V}(\phi ,\varrho )$, belongs to the space $H^1(a,b)$. This derivative can be expressed for any values of υ with in the intervals $(0,1)$ [26]

$${}_a^{ABR}{D}_{\varrho }^{\upsilon }\mathit{V}(\phi ,\eta )=\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{d}{d\varrho }{\int }_a^{\varrho }\mathit{V}(\phi ,\eta )E_{\upsilon }\left(\frac{-\upsilon {(\varrho -\eta )}^{\upsilon }}{1-\upsilon }\right)d\eta$$

The function $B\left(\upsilon \right)$ possesses the characteristic of yielding a result of 1 for input values of 0 and 1. Moreover, $B\left(\upsilon \right)$ consistently returns a value greater than a for any input value of υ greater than 0.

Theorem 2. 

The definitions of Laplace transform for the ABC derivative and the AB Riemann-Liouville operator can be stated as follows [26]

$$L\left\{{}_a^{ABC}{D}_{\varrho }^{\upsilon }\mathit{V}(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{{\epsilon }^{\upsilon }L\left\{\mathit{V}(\phi ,\varrho )\right\}-{\epsilon }^{\upsilon -1}\mathit{V}(\phi ,0)}{{\epsilon }^{\upsilon }+\frac{\upsilon }{1-\upsilon }}$$

(1)

and

$$L\left\{\begin{array}{c}ABR\hfill \end{array}\right.D_{\varrho }^{\upsilon }\mathit{V}(\phi ,\varrho )\left(\epsilon \right)=\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{{\epsilon }^{\upsilon }L\left\{\mathit{V}(\phi ,\varrho )\right\}}{{\epsilon }^{\upsilon }+\frac{\upsilon }{1-\upsilon }}$$

(2)

Theorem 3. 

A new form of the AB Riemann-Liouville derivative, known as the Aboodh-transform AB Riemann-Liorouville operat [24]

$$G\left(\epsilon \right)=A\left\{\begin{array}{c}ABR\hfill \\ a\hfill \end{array}{D}_{\varrho }^{\upsilon }\mathit{V}(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{1}{\epsilon }\left[\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{{\epsilon }^{\upsilon }L\left\{\mathit{V}(\phi ,\varrho )\right\}}{{\epsilon }^{\upsilon }+\frac{\upsilon }{1-\upsilon }}\right]$$

(3)

Proof. 

We can derive the required solution by utilizing Theorem 1. The following theorem outlines the relationships among the ZZ and Aboodh transforms. □

Theorem 4. 

The definition for the ABC derivative AT can be stated as [24]

$$\begin{array}{cc}\hfill G\left(\epsilon \right)=& A\left\{\begin{array}{c}ABC\hfill \\ a\hfill \end{array}{D}_{\varrho }^{\upsilon }\mathit{V}(\phi ,\varrho )\right\}\left(\epsilon \right)=\frac{1}{\epsilon }\left[\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{{\epsilon }^{\upsilon }L\left\{\mathit{V}(\phi ,\varrho )\right\}-{\epsilon }^{\upsilon -1}\mathit{V}(\phi ,0)}{{\epsilon }^{\upsilon }+\frac{\upsilon }{1-\upsilon }}\right]\hfill \end{array}$$

(4)

Proof. 

We can obtain the solution by applying Theorem 1 and using Equation (1). □

Theorem 5. 

We achieve the Aboodh and ZZ transforms for $\mathit{V}\left(\varrho \right)\in B$, which are denoted by $G\left(s\right)$ and $Z(k,s)$, is given as [24]

$$Z(k,s)=\frac{s^2}{k^2}G\left(\frac{s}{k}\right)$$

Proof. 

According to the ZZ-transformation definitions,

$$Z(k,s)=s{\int }_0^{\infty }\mathit{V}\left(k\varrho \right)e^{-s\varrho }d\varrho$$

(5)

We can obtain Equation (5) by substituting $k\varrho =\varrho$.

$$Z(k,s)=\frac{s}{k}{\int }_0^{\infty }\mathit{V}\left(\varrho \right)e^{-\frac{\pi z}{k}}d\varrho$$

(6)

Equation (6) can be restated as follows:

$$Z(k,s)=\frac{s}{k}F\left(\frac{s}{k}\right)$$

(7)

We can utilize Theorem 1 to rephrase Equation (7) as the Laplace transform of $\mathit{V}\left(\varrho \right)$, denoted as F.

$$Z(k,s)=\frac{s}{k}\frac{F\left(\frac{s}{k}\right)}{\left(\frac{s}{k}\right)}\times \left(\frac{s}{k}\right)={\left(\frac{s}{k}\right)}^{2}G\left(\frac{s}{k}\right)$$

(8)

The function $\mathit{V}\left(\varrho \right)$ undergoes a transformation represented by $G(.)$, known as the Aboodh transform. □

Theorem 6. 

The transform of $\mathit{V}\left(\varrho \right)={\varrho }^{\upsilon -1}$ under the $\mathit{ZZ}$ operator can be expressed as:

$$Z(k,s)=\Gamma \left(\upsilon \right){\left(\frac{k}{s}\right)}^{\upsilon -1}$$

(9)

Theorem 7. 

Assuming that υ and ω are complex numbers with a real part greater than zero, we can define the $\mathit{ZZ}$ transformation of $E_{\upsilon }\left(\omega {\varrho }^{\upsilon }\right)$ as shown in Equation (10) [24]:

$$Z(k,s)={\left(1-\omega {\left(\frac{k}{s}\right)}^{\upsilon }\right)}^{-1}$$

(10)

Here, $Z(k,s)$ is the $\mathit{ZZ}$ transformed expression of $E_{\upsilon }\left(\omega {\varrho }^{\upsilon }\right)$.

Theorem 8. 

The ABC operator can be transformed using the ZZ transform, which is defined as follows: Let $\mathit{V}\left(\varrho \right)$ have the Aboodh transformation $Z(k,s)$ and the ZZ transformation $G\left(s\right)$ [24]

$$ZZ\left\{\begin{array}{c}ABC\hfill \\ 0\hfill \end{array}{D}_{\varrho }^{\upsilon }\mathit{V}\left(\varrho \right)\right\}=\left[\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{\frac{s^{a+2}}{k^{\upsilon +2}}G\left(\frac{s}{k}\right)-\frac{s^{\upsilon }}{k^{\upsilon }}f\left(0\right)}{\frac{s^{\upsilon }}{k^{\upsilon }}+\frac{\upsilon }{1-\upsilon }}\right]$$

(11)

Theorem 9. 

Assuming that the ZZ transform of $\mathit{V}\left(\varrho \right)$ is denoted by $G\left(s\right)$, and the Aboodh transform of $\mathit{V}\left(\varrho \right)$ is denoted by $Z(k,s)$, we can define the ZZ-transform of the AB Riemann-Liouville operator as follows: [24]

$$ZZ\left\{\begin{array}{c}ABR\hfill \\ 0\hfill \end{array}{D}_{\varrho }^{\upsilon }f\left(\varrho \right)\right\}=\left[\frac{B\left(\upsilon \right)}{1-\upsilon }\frac{\frac{s^{\upsilon +2}}{k^{\upsilon +2}}G\left(\frac{s}{k}\right)}{\frac{s^{\mu }}{k^{\mu }}+\frac{\upsilon }{1-\upsilon }}\right]$$

(12)

3. Methodology

The general methodology of q-HATM for fractional Kolmogorov IVP is

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

(13)

with initial condition

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

(14)

where ${}^{ABC}{D}_{\varsigma }^{\upsilon }\mathcal{U}(\varrho ,\varsigma )$ symbolises the AB derivative of $\mathcal{U}(\varrho ,\varsigma )$.

On using the ZT on Equation (28), we have, after simplification,

$${\mathcal{Z}}_{\varsigma }\left[\mathcal{U}(\varrho ,\varsigma )\right]-\frac{f\left(\varrho \right)}{s}+\frac{1}{\mathfrak{N}\left[\upsilon \right]}\left(1-\upsilon +\frac{\upsilon }{s^{\upsilon }}\right){\mathcal{Z}}_{\varsigma }\left[{\mathrm{N}}_{\varrho }\left[\mathcal{U}(\varrho ,\varsigma )\right]\right].$$

(15)

The definition of the non-linear operator is as follows:

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

(16)

The function $\varphi (\varrho ,\varsigma ;q)$ is a real-valued function that depends on $\varrho$, $\varsigma$, and $q\in [0,\frac{1}{n}]$. We will now introduce a homotopy in the following manner:

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

(17)

The parameter ℏ is auxiliary, and $\mathcal{Z}$ represents ZT. The embedding parameter $q\in [0,\frac{1}{n}]$ with $n\ge 1$. The statements below are valid for $q=0$ and $q=\frac{1}{B}$.

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

(18)

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 define $\varphi (\varrho ,\varsigma ;q)$ with respect to q by applying the Taylor theorems to achieve

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

(19)

where

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

(20)

The series given in Equation (40) converges when $q=\frac{1}{n}$, provided that appropriate values of ${\mathcal{U}}_0(\varrho ,\xi ,\varsigma )$, n, and ℏ are chosen.

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

(21)

To derive the derivative of Equation (40) concerning the embedding parameter q, we take its derivative and subsequently let $q=0.$ Then, by dividing the outcome by $m!$, we can obtain the desired result.

$${\mathcal{Z}}_{\varsigma }[\mathcal{U}(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\Re }_m(\to U_{m-1}),$$

(22)

where the vector are given as

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

(23)

On applying inverse ZT to Equation (22), one can obtain

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

(24)

where

$$\begin{array}{c}\hfill {\Re }_m(\to U_{m-1})={\mathcal{Z}}_{\varsigma }\left[{\mathcal{U}}_{m-1}(\varrho ,\varsigma )\right]-\left(1-\frac{k_m}{n}\right)\left(\frac{f\left(\varrho \right)}{s}\right)+\frac{1}{\mathfrak{N}\left[\upsilon \right]}\left(1-\upsilon +\frac{\upsilon }{s^{\upsilon }}\right){\mathcal{Z}}_{\varsigma }\left[{\mathrm{N}}_{\varrho }\mathcal{U}(\varrho ,\varsigma )\right],\end{array}$$

(25)

and

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

(26)

Using Equations (24) and (25), one can obtain the series of ${\mathcal{U}}_m(\upsilon ,\varsigma )$. Lastly, the series q-HATM solution is defined as

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

(27)

4. Numerical Problems

Problem 1. 

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

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

(28)

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

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

(29)

Applying the Laplace transform on Equation (28) and using initial conditions, we obtain

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

(30)

The non-linear operator is defined as

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

(31)

The mth order deformation equation, defined by the assistance of the suggested method, is as follows:

$$\mathcal{Z}[{\mathcal{U}}_m(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\mathcal{R}}_m(\to U_{m-1})$$

(32)

where ${\mathcal{R}}_m(\to U)$ is

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

(33)

Applying inverse ZT on Equation (40), we obtain

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

(34)

Solving the above equations, we obtain

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

(35)

Problem 2. 

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

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

(36)

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

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

(37)

Applying the Laplace transform on Equation (36) and using initial conditions, we obtain

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

(38)

The non-linear operator is defined as

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

(39)

The mth order deformation equation is defined by the assistance of the suggested method as follows:

$$\mathcal{Z}[{\mathcal{U}}_m(\varrho ,\varsigma )-k_m{\mathcal{U}}_{m-1}(\varrho ,\varsigma )]=\hslash {\mathcal{R}}_m(\to U_{m-1})$$

(40)

where  ${\mathcal{R}}_m(\to U)$  is

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

(41)

Applying inverse ZT to Equation (40), we obtain

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

(42)

Solving the above equations, we obtain

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

(43)

In Figure 1, two dimensional graph of analytical solutions of Problem 1 at various values of $\upsilon$. In Figure 2, 3D plots of approximate solutions to Problem 1 at various values of $\upsilon$. In Figure 3, 3D plots of approximate solution at $\upsilon =1$ and the exact solutions for Problem 2. In Figure 4, 2D plots of the exact and approximate solutions for Problem 2 at various values of $\upsilon$. In Figure 5, three dimensional plots of approximate solutions for Problem 2 at various values of $\upsilon$. In Figure 6, 3D plots of approximate solution at $\upsilon =1$ and exact solutions for Problem 2. In

Table 1. Absolute error of the exact and fourth term approximate solutions for Problem 1 with various $\varsigma$ and $\varrho$ for $\upsilon =1$.

$\mathit{\varsigma }$	$\mathit{\varrho }$	${\mathit{Approximate}}_{\mathit{\upsilon }=1}$	Exact	AE
0.1	0	1.105166667	1.105170918	4.251 $\times {10}^{-6}$
	0.2	1.326200000	1.326205102	5.102 $\times {10}^{-6}$
	0.4	1.547233333	1.547239285	5.952 $\times {10}^{-6}$
	0.6	1.547239285	1.768273469	6.802 $\times {10}^{-6}$
	0.8	1.989300000	1.989307652	7.652 $\times {10}^{-6}$
	1	2.210333333	2.210341836	8.503 $\times {10}^{-6}$
0.2	0	1.221333333	1.221402758	6.9425 $\times {10}^{-5}$
	0.2	1.465600000	1.465683310	8.3310 $\times {10}^{-5}$
	0.4	1.709866667	1.709963861	9.7194 $\times {10}^{-5}$
	0.6	1.954133333	1.954244413	1.11080 $\times {10}^{-4}$
	0.8	2.198400000	2.198524964	1.24964 $\times {10}^{-4}$
	1	2.442666667	2.442805516	1.38849 $\times {10}^{-4}$

and

Table 2. Absolute error of the exact and fourth term approximate solutions for Problem 2 with various $\varsigma$ and $\varrho$ for $\upsilon =1$.

$\mathit{\varsigma }$	$\mathit{\varrho }$	${\mathit{Approximate}}_{\mathit{\upsilon }=1}$	Exact	AE
0.1	0	10.56035556	10.56035508	4.8 $\times {10}^{-7}$
	0.2	10.64002126	10.64002221	9.5 $\times {10}^{-7}$
	0.4	10.66666431	10.66666667	2.36 $\times {10}^{-6}$
	0.6	10.64001846	10.64002221	3.75 $\times {10}^{-6}$
	0.8	10.56034998	10.56035508	5.10 $\times {10}^{-6}$
	1	10.42845487	10.42846127	6.40 $\times {10}^{-6}$
0.2	0	10.24568889	10.24565863	3.026 $\times {10}^{-5}$
	0.2	10.42844612	10.42846127	1.515 $\times {10}^{-5}$
	0.4	10.42846127	10.56035508	6.042 $\times {10}^{-5}$
	0.6	10.63991712	10.64002221	1.0509 $\times {10}^{-4}$
	0.8	10.66651797	10.66666667	1.4870 $\times {10}^{-4}$
	1	10.63983139	10.64002221	1.9082 $\times {10}^{-4}$

, absolute error of the exact and fourth term approximate solutions for Problem 1 and 2 with various $\varsigma$ and $\varrho$ for $\upsilon =1$.

5. Conclusions

In conclusion, the fractional-order Kolmogorov and Rosenau–Hyman models have been shown to be solvable using the q-HATM method. This is a significant breakthrough in the field of mathematical modeling, as it provides researchers with a powerful tool for accurately predicting complex systems with fractional-order dynamics. The q-HATM method has proven to be effective in solving complex equations involving fractional derivatives, and its successful application to the Kolmogorov and Rosenau–Hyman models is a testament to its robustness and versatility. With further development and refinement, the q-HATM method holds great promise for solving a wide range of problems in science and engineering. Overall, the successful solution of the fractional-order Kolmogorov and Rosenau–Hyman models using the q-HATM method represents a significant milestone in the field of fractional calculus, and highlights the potential of this field to provide new insights into the behavior of complex systems.