# The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus

^{1}School of Mathematics and Statistics, Huanghuai University, Zhumadian, China^{2}Department of Mathematics, Government College University, Faisalabad, Pakistan^{3}School of Mathematics and Statistics, Nanyang Normal University, Nanyang, China

Fractional differential equations can model various complex problems in physics and engineering, but there is no universal method to solve fractional models precisely. This paper offers a new hope for this purpose by coupling the homotopy perturbation method with Aboodh transform. The new hybrid technique leads to a simple approach to finding an approximate solution, which converges fast to the exact one with less computing effort. An example of the fractional casting-mold system is given to elucidate the hope for fractional calculus, and this paper serves as a model for other fractional differential equations.

## 1 Introduction

Fractional calculus has triggered much interest in both physics and mathematics [1, 2]. Traditional differential equations cannot accurately represent many physical problems, and the fractional partner can provide deeper insight into these complex physical phenomena with ease. In general, this newly developed field is for studying real-world applications in the fractal space, so most literature labeled it as the fractal–fractional calculus [3–5] or the local fractional calculus on the Cantor set [6]. A continuum medium, e.g., water or air, becomes a fractal space (porous medium) when we observe it on a molecule’s scale. Any phenomena arising in molecules’ perturbation have to be modeled by the fractal–fractional model [7]. As an example, we consider a nanoparticle’s motion in the air, which is stochastic and difficult to be modeled by the traditional differential equation; however, if the air is considered as a fractal space on a molecule’s scale, its motion is determinate and can be modeled by the fractal–fractional model. So, we need two scales for a porous medium; one is large enough so that the continuum assumption works, and the other is small enough so that the porosity can be measured, as pointed out by Ji-Huan He that “seeing with a single scale is always unbelieving” [8]. Another example is the motion of the Moon, which is naturally periodic; however, if we measure its motion at an extremely far distance, its motion becomes stochastic, and the Heisenberg-like uncertainty principle works for the Moon [9]. He and Qian showed that the fractal diffusion process in water depends on the fractal dimensions [10], and other scientists also discussed the fractional advection–reaction–diffusion process [11] and the fractal diffusion–reaction process [12]. A cocoon’s air/moisture permeability and its thermal property can best be revealed by the fractal–fractional model [13, 14], and the fractal micro-electromechanical systems show even more amazing properties [15–18].

Fractional calculus is a good and reliable tool for scientists and engineers but a mixed blessing for practical applications because an intractable problem arises; that is, fractional models are extremely difficult to be solved. Researchers have been racing to test various analytical methods which were originally proposed to solve traditional differential equations. Though there are many famous analytical methods in the literature, for example, the homotopy perturbation method [19–23] and its various modifications [24–26], the decomposition method [27], the variational iteration method [28–30], the exp-function method [31], and the differential transform method [32], so far, there is not a universal approach to solving exactly fractional differential equations, and this paper offers a new hope for this purpose by coupling the homotopy perturbation method [33, 34] and the Aboodh transform [35].

The homotopy perturbation method (HPM) was first proposed by Chinese mathematician Prof. Ji-Huan He in the later 1990s [33]; it is mathematically simple and physically insightful. The method is equally suitable for linear or non-linear, homogeneous or inhomogeneous, and initial and/or boundary value problems. The solution is expressed in an infinite series and typically converges to the exact solution. The HPM is now considered a matured tool for almost all kinds of problems, and many researchers have used this method for an accurate insight into the solution properties of a complex problem [36–38].

The Aboodh transform (AT) was proposed by Aboodh [35] and derived from the classical Fourier integral. This transform is now considered a simple technique for solving linear differential equations but is unable to solve non-linear ones. By coupling AT with the HPM, one has the capability to solve linear and non-linear problems, and a lot of literature works have been witnessed to utilize this coupling for solving various types of problems. Using AT–HPM, Manimegalai et al. [39] solved strongly non-linear oscillators with great success. Jani and Singh [40] found it had obvious advantages over the decomposition method, Yasmin [41] revealed the dynamic behavior of the fractional convection–reaction–diffusion process, and Jani and Singh [42] extended it to the soliton theory.

Though much work was achieved, in this study, we will show that AT–HPM is a universal tool for fractional calculus. As an example, we consider the time-fractional casting-mold system which is used in manufacturing various medical equipment, ranging from injections to the COVID-19 tool-kit [43]. The significant findings reveal that AT–HPM is an accurate and effective approach that reduces the computational work with fast convergence ratio.

## 2 Aboodh transform-based homotopy perturbation technique

This section is divided into two sections. In the first section, the methodology will be proposed, and the convergence of the suggested technique will be discussed in the second section.

### 2.1 Methodology

In this section, we give a brief introduction to the Aboodh transform [35] and homotopy perturbation method [33, 34].

If

where

The Aboodh transform of the partial derivative of time can be obtained using the following formula:

where

where

where

By employing the differential characteristic of Aboodh transform, we can express Eq. 3 as

and after using the initial conditions, we have

or

where

where

where

By substituting Eqs 7, 8 in Eq. 6, we get

Comparing the coefficients of like powers of

We can obtain the best approximation for the solution as

### 2.2 Convergence analysis

To show that the series solution of the system in Eq. 14 converges to the solution of Eq. 3, we are to prove the sufficient condition of the convergence, and the following theorem will help us.

Theorem: We assume that *X* and *Y* are Banach spaces and

Then, according to Banach’s fixed point theorem, *M* has a unique fixed point

and considering that

(i)

(ii)

Proof: (i) By the principle of mathematical induction, for

Assuming

By employing the definition of

(ii) As

Hence, the given statement is proved.

## 3 Numerical examples

In this section, three examples are presented to illustrate the idea explained in Section 2. First, we will study the method for a homogeneous linear system of PDEs. Second, the analytical solution will be obtained for an inhomogeneous linear system of PDEs. Finally, the inhomogeneous non-linear system of PDEs will be examined.

### 3.1 The system of homogeneous linear PDEs

We consider the following linear system:

with initial conditions

By employing the Aboodh transform method, we have

Using the initial conditions given in Eq. 16, we reach

or

The Aboodh transform-based homotopy perturbation method considers a series solution given by

By using the aforestated equation, the system of equations in Eq. 19 gets the form

By comparing like powers of

Hence, the series solution by using Eq. 14 can be expressed as

or in a closed form as

which is the exact solution of Eq. 15.

### 3.2 The system of inhomogeneous linear PDEs

Suppose the following inhomogeneous linear system of PDEs:

with initial conditions

Applying the Aboodh transform on each side of the equations in Eq. 24 and then putting on the given initial conditions, we obtain

or

By using the Aboodh transform-based homotopy perturbation method, the series solution is expressed by

The system of equations in Eq. 27 gets the following form after employing the aforestated equation:

By comparing the coefficient of like powers of

Therefore, the solution in the form of an infinite series by using Eq. 14 can be expressed as

or in its convergent form as

which is the exact solution of Eq. 24.

### 3.3 The system of inhomogeneous non-linear PDEs

Suppose the following inhomogeneous non-linear system of PDEs:

with initial conditions

Employing the Aboodh transform on each side of the equations in Eq. 36 and then applying the given initial conditions give

Taking the inverse Aboodh transform on each side, we obtain

According to the Aboodh transform-based homotopy perturbation method, the solution functions

where the non-linear terms

By comparing the coefficient of like powers of

Therefore, the solution in the form of an infinite series by using Eq. 14 can be expressed as

or in its convergent form as

which is the exact solution of Eq. 36.

## 4 Time-fractional casting-mold system

Now, we turn back to a time-fractional casting-mold system which models the temperature distribution in the casting and molding processes. For this, two heat conduction equations are used with initial and Dirichlet boundary conditions [45]. The mathematical model is depicted as follows:

where

It is necessary to point out that Eq. 48 was originally studied in [45], where the series solution was presented and no closed-form solution was formulated. Our aim here is to overcome the main shortcomings in [45] and to offer a totally new hope for numerical approximation. To this end, applying the Aboodh transform in the aforementioned system, we have

Now, by inverse Aboodh transformation, we obtain

which can further be written as

According to the standard HPM [33, 34], the solution

By substituting Eq. 52 in Eq. 51, the solution can be written as

Equating coefficients of powers of

The approximate solution can be obtained as

### 4.1 Example

We consider the system expressed in Eq. 48 for the case

By employing Eq. 57, the solution can be written as

The expressions are similar to those obtained by the fractional complex transform [46–49]. In the closed form, we obtain

where

### 4.2 Results and discussion

This section is devoted to test the applicability and validity of the suggested technique for the time-fractional casting-mold system over the series-based solution of the same model.

Figures 1, 2 present the errors of the series solutions obtained by the HPM [45] for the fractal dimension *β* = 1. It is observed that for all the parameters and for both casting and molding processes, the errors grow exponentially for the case of a series solution [45] and can be reduced by adding more terms in the solution. On the other hand, the suggested solution has the exact solution, and there is no chance of error even for a larger range of t. Therefore, based on these findings, we can say that the proposed technique is more effective than the previous method [45].

## 5 Conclusion

The Aboodh transform-based homotopy perturbation method is successfully employed to solve traditional differential equations and fractional differential equations successfully. This approach has been shown to have the potential to solve both linear and non-linear problems. For a linear system, the exact solution is predicted, while for a non-linear system, with the help of He’s polynomials, a series solution is obtained, which converges fast to the exact one. So, the method pushes the progress of non-linear science and will make a “big change” to increase the number of practical applications, and this paper serves as a model for other applications.

## Data availability statement

The original contributions presented in the study are included in the article/Supplementary Materials; further inquiries can be directed to the corresponding author.

## Author contributions

Conceptualization: HT and NA; methodology: NA and YY; validation: NA and YY; writing—original draft preparation: HT and YY; writing—review and editing: HT, NA, and YY; supervision: HT and YY; and funding acquisition: YY. All authors read and agreed to the published version of the manuscript.

## Funding

The study was supported by the Natural Science Foundation of Henan Province (No. 222300420507); National Natural Science Foundation of China (No. 12171193), Key Scientific Research Project of High Education Institutions of Henan Province (No. 23A110019), Science and Technology Research Projects of Henan Province (No. 182102110292), Basic and Frontier Technology Research Project of Henan Province (Nos. 12300410398 and 132300410084), and Zhumadian Key Laboratory of Statistical Computing and Data Modeling [No. (2022)12].

## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Publisher’s note

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## Nomenclature

Keywords: homotopy perturbation method, Aboodh transform, He’s polynomials, fractional differential equation, two-scale fractal theory

Citation: Tao H, Anjum N and Yang Y-J (2023) The Aboodh transformation-based homotopy perturbation method: new hope for fractional calculus. *Front. Phys.* 11:1168795. doi: 10.3389/fphy.2023.1168795

Received: 18 February 2023; Accepted: 03 April 2023;

Published: 27 April 2023.

Edited by:

Hamid M. Sedighi, Shahid Chamran University of Ahvaz, IranReviewed by:

Szabolcs Fischer, Széchenyi István University, HungarySantanu Saha Ray, National Institute of Technology Rourkela, India

Magaji Adamu Yunbunga, Abubakar Tafawa Balewa University, Nigeria

Copyright © 2023 Tao, Anjum and Yang. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Naveed Anjum, xsnaveed@yahoo.com