Abstract
In this paper, we propose an innovative approach to determine the approximate solution of the coupled time-fractional Keller-Segel (K-S) model. We use the fractional complex transform (FCT) to switch the model into its differential partner, and then, the homotopy perturbation method (HPM) is introduced to tackle its nonlinear elements using He’s polynomials. This two-scale theory helps to define the physical meaning of the FCT for the solution of the K-S model. Some examples are illustrated to show that the proposed scheme presents the significant results. The considerable findings show that this strategy does not require any assumptions and also reduces the massive computations without imposing any constraints. This technique is also suitable in functional studies of fractal calculus due to its powerful and robust support for nonlinear problems.
1. Introduction
Fractional differential equations (FDEs) are the generalizations of classical differential equations with integer orders. It is worth reporting that some mathematical models of integer-order derivatives particularly nonlinear models do not work adequately for most of the cases [1–3]. This is because integer order derivatives are limited operators and are inappropriate for infinite variance whereas fractional order derivatives are worldwide to take account of the domination of the neighborhood. In recent years, nonlinear FDEs in mathematical physics are competing against a principal role in miscellaneous domains, such as biological science, applied science, signal processing, control theory, finance, and fractal dynamics [4–7].
In 1970, Keller and Segel introduced a hypothesis to express the combination system of cellular slime mold by chemical fascination. The K-S model has broadly been practiced for chemotaxis terms due to its competency to capture the key facts and its impulsive nature. The significance of chemotaxis has achieved much attraction due to its crucial function in a broad variety of biological occurrences [8, 9]. In this article, we examine the coupled time-fractional K-S model of the form subject to the initial solutions, we get where and represent the bacterial density and the concentration of chemical substance as a function of and , respectively, and denotes the chemotactic term and shows that the cells are sensitive to the chemicals and are attracted by them [10]. The sensitivity function is a smooth function which describes the cell’s perception and response to the chemical stimulus while and are positive constants. If , the time-fractional K-S model (1) leads to a simple nonlinear differential equation that has been studied extensively whereas taken as Caputo’s sense [11] and is He’s fractional derivative defined [12, 13]
Many actions in physics and engineering can be precisely characterized by utilizing differential equations with various sorts of fractional derivatives. The finding of the approximate and exact solution of FDEs is a very crucial challenge. There have been a lot of developments to solve FDEs in nonlinear dynamics. FDEs are used extensively because they do not have exact solutions, and thus, approximate and numerical solutions are needed. The homotopy perturbation method (HPM) [14] is one of the most famous approaches to achieve the series solutions of linear and nonlinear differential equations of arbitrary orders. Later, various methods have been developed to show that HPM a is very efficient and powerful tool for finding the approximate solution to FDEs [15–18]. In order to get the solution of the K-S model, many powerful and efficient techniques have been suggested to obtain the analytical solutions such as Laplace homotopy perturbation method [19], iterative method [20], homotopy perturbation Sumudu transform [21], and natural homotopy transform method [22] with a logic sensitivity function and small diffusivity. Some partial differential equations with fractional order are not easy to solve, and then, their approximate solution can be evaluated. The two-scale approach converts the fractional order to a simple partial differential equation which is now easy to solve by the homotopy perturbation method.
This study presents the idea of a two-scale method to obtain the solution of the fractional K-S model in Caputo sense. The FCT converts the model into its differential partner, and then, HPM is introduced to bring down the nonlinear terms in algebraic series. The quality of the current method is appropriate to provide the analytical results to the given examples. This study is summarized as follows: In Section (2), we recall some basic definitions of fractional calculus. We present the idea of the homotopy perturbation method and the two-scale approach in Sections (3) and (4), respectively. Some numerical examples are provided to demonstrate the performance of this approach in Section (5) and the discussion of results in Section (6). The conclusion is given in Section (7).
2. Preliminary Concepts
Definition 1. The Riemann-Liouville fractional integral operator of order of a function , , is defined as [23]
Definition 2. The Caputo fractional derivative of in the Caputo sense is given [23]
Lemma 3. If , , , , then The fractional derivatives are considered in Caputo sense which allows the conditions to deal with the expression of the problems.
3. Basic Idea of the Homotopy Perturbation Method
In this segment, we explain the fundamental concept of HPM. Let the following nonlinear equation [24] with boundary conditions where is a general function with boundary operator , is analytic function, and is the boundary of the domain . The operator can normally be separated into two operators with as a linear and being a nonlinear operator. Thus, Equation (7) can be accompanied as follows:
Let us consider that confirms or where is said to be a homotopy parameter and is an initial approximation of Equation (7). According to HPM, we can take as a small element, and suppose that the solution of Equation (11) can be written as a power series of :
Considering , the approximate solution of Equation (7) is obtained as follows:
Using Equations (11) and (12), we can identify the similar powers of to obtain the following series solution form: where depending upon called He’s polynomials can be computed by adopting the following rule:
The system of nonlinear equations in (14) is evidently simple to calculate, and thus, the components of HPM can be identified easily which leads to the series solutions very rapidly.
4. Fractional Complex Transform
The dimension and scale are highly important elements due to its impressive outcomes and properties of the configuration through the modeling of a problem. FCT is a systematic technique that turns FDEs into its differential parts in a steady period and is described as [25–27] where is the slighter scale and is the greater scale. The time fractional K-S model reacts discontinuously on a slighter scale, particularly at the highest point whereas it anticipates a plane solitary wave on the greater scale. Thus, Equation (11) is considered two-scale transform [28–30]. The outcomes of any study problem depend on the scale. For an observable scale, the fluid is consistent; therefore, Newton’s laws can be applied; however, they are illegitimate at the molecular scale. If the motion is free of time, then Newton’s law is acceptable; otherwise, it can be revoked.
4.1. Convergence Theorem
Let and be the Banach spaces and be a contraction nonlinear mapping. If the sequence generated by HPM such as then the following conditions must be true: (1)(2) is always in the neighborhood of meaning (3)
Proof. (1)We prove condition (1) by induction on , and according to the Banach fixed point theorem, has a fixed point meaning ; therefore,since is a contraction mapping. Assume that is an induction hypothesis, then (2)The first concern is to demonstrate that , and this is achieved by induction on . So, for with the initial condition. Assume that for is an induction hypothesis, thenNow, for all , using condition (1), we have (3)Using condition (2) and the fact that yields that ; therefore,Thus, converges.
5. Numerical Examples
In this segment, we implement a two-scale method to achieve the approximate solution of the K-S model in one dimension. Results disclose that this approach is an extremely efficient and powerful aid for solving FDEs.
5.1. Example 1
Consider the K-S model in one dimension given as with the initial conditions
Considering that the sensitivity function ; thus, the chemotactic term is zero, i.e., and using
Thus, the coupled K-S model of Equation (23) becomes
We can use HPM with He’s polynomials on the system of Equation (26), and we get
With the help of Equation (24), we can get the following iterations:
In the same way, other of the elements can be identified. So, the series solution of Equation (23) with the help of Equation (25) is as follows:
Only some of terms are evaluated while the other terms can be obtained using the iterative formula. As a result, the solution of the system of Equation (23) is as follows:
5.2. Example 2
with the initial conditions
Considering that the sensitivity function ; thus, the chemotactic term becomes and using
Thus, the coupled K-S model of Equation (44) becomes
Now, using the two-scale approach, with the help of Equation (45), we can get the following iterations directly
In the same way, other of the elements can be identified. So, the series solution of Equation (44) with the help of Equation (46) is as follows:
6. Results and Discussion
In this segment, we demonstrate the validity and the accuracy of the two-scale approach through the 3D graphical representations. We also present the graphical models and physical behaviors of the time-fractional K-S model. Mathematica program 11.0.1. is used to calculate the iterations and the graphical representations. Figures 1–4 show the surface graphs of the K-S model for and , respectively, at different values of with and . On behalf of graphical illustrations, we adopt . The graphical illustrations have validated the convergence of fractional-order solutions in the direction of integer-order solutions. We calculate the iteration only up to 3 terms, and the series of the solution converges to the exact solution very rapidly. Some more iterations can be evaluated for more accuracy of the approximate solutions. It is noted that the obtained solutions are similar which legitimize the reliability of the proposed strategies:
(a) Surface solution of when
(b) Surface solution of when
(c) Surface solution of when
(d) Surface solution of when
(a) Surface solution of when
(b) Surface solution of when
(c) Surface solution of when
(d) Surface solution of when
(a) Surface solution of when
(b) Surface solution of when
(c) Surface solution of when
(d) Surface solution of when
(a) Surface solution of when
(b) Surface solution of when
(c) Surface solution of when
(d) Surface solution of when
7. Conclusion
In this study, we have successfully applied a hybrid strategy where FCT has coupled with HPM to investigate the approximate solution of the nonlinear time fractional K-S model. The current association is not just helpful for fractional-order differential equations but also other differential equations with some variants. The main advantage of FCT is that it deals with the nonlinear problems straightforward to customize FDEs into their differential parts. We performed two numerical illustrations of the fractional K-S model to examine the reliability of the suggested approach. The results indicate that the two-scale approach is a more effective and powerful strategy in determining the analytical solutions of nonlinear differential equations. Thus, we conclude that our proposed scheme is suitable and can be considered for the other nonlinear fractional partial differential equations with fractal derivatives in future study.
Data Availability
All the data are available within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Foundation of Yibin University, China (Grant no. 2019QD07).