Research Article | Open Access
Advanced Analytical Treatment of Fractional Logistic Equations Based on Residual Error Functions
In this article, an analytical reliable treatment based on the concept of residual error functions is employed to address the series solution of the differential logistic system in the fractional sense. The proposed technique is a combination of the generalized Taylor series and minimizing the residual error function. The solution methodology depends on the generation of a fractional expansion in an effective convergence formula, as well as on the optimization of truncated errors, , through the use of repeated Caputo derivatives without any restrictive assumptions of system nature. To achieve this, some logistic patterns are tested to demonstrate the reliability and applicability of the suggested approach. Numerical comparison depicts that the proposed technique has high accuracy and less computational effect and is more efficient.
Scientists were interested in studying differential equations in the fractional sense as flexible mathematical frameworks for modeling, measuring, and describing genetic structures, memory and material transfer, and multiple processes that have recently become an increasingly stimulating area of engineering and science applications, including but not limited to viscosity, fluid mechanics, optimal control, oscillation, signal processing, anomalous diffusion, electromagnetic, and fractal geometry [1–6]. When comparing with the integer order issues, the memory and hereditary properties of several substances are well and fully described by noninteger order issues. Anyhow, numerous attempts to provide numerical solutions to such equations exist in the literature, often due to the difficulty in finding analytical solutions accurately. Therefore, different numerical and approximation methods were introduced to handle those fractional systems [7–10].
In this research direction, the application of fractional residual power series method is employed based on residual error concept to find the analytic-numeric solutions of the fractional logistic model:along with the initial conditionwhere , indicates Caputo fractional derivatives, while indicates smooth solution to be obtained. In this regard, suppose that the logistic models (1) and (2) have a unique analytical solution for However, when , the differential equation (1) will be called the standard logistic model in the following form , which has the exact solution Further, P.-F. Verhulst was the first who presented the standard logistic model. Indeed, there are two basic kinds of the logistic models: continuous and discrete models. The continuous cases are explained by ordinary differential equations of the first order that are called “standard logistic,” while the discrete case representing a simple recurrence formula detects the properties of chaotic in specific domains. Furthermore, logistic solutions provide a consistent classification of the rates of population growth and their rapid and startling development, which does not include reducing food supplies, basic needs, or disease outbreaks. For solution behavior, the logistic system curve increases exponentially starting from the factor of multiplication up to the limit of saturation, that is, the maximum carrying capacity. Consequently, where represents the size of population growth in terms of , indicates the maximum population growth rate, while indicates carrying capacity. On the other hand, the continuous logistic system has a steady population growth rate in the form of with initial population data . In addition, the theory of existence and uniqueness of the continuous logistic system in a fractional sense has been presented in the literature; for more details, see [11–15].
In this research work, a reliable numerical treatment, the fractional residual power series method (FRPSM), is suggested for solving a sort of fractional logistic system. The FRPSM is an easy and reliable tool to obtain the values of unknown coefficients of desired fractional series solution for different types of linear and nonlinear FDEs without discretization, perturbation, and linearization by solving sequence of algebraic system [16–23]. The FRPS technique is primarily applied using the residual error concept and the repetition of Caputo derivatives to obtain the appropriate series solution by choosing a fit initial data, whereas the gained series solution and all fractional derivatives are valid for all mesh points of the domain of interest. To view the characteristics and advantages of numerical methods developed in dealing with various physical and engineering phenomena in the fractional sense, we refer to [24–31].
The rest of the current study is outlined in five sections. Characterization and primary results of the theory of fractional calculus are given as well as representation of fractional series solution is also provided in Section 2. In Section 3, the main procedures of the proposed algorithm are discussed to construct the required series solution. In Section 4, several applications are considered to confirm the performance and reliability of the present FRPSM. The conclusion is briefly presented finally.
2. Primary Mathematical Concepts
This section is devoted to concepts and results concerning the Caputo fractional derivatives and generalized power series representations. Throughout this research, the order of fractional derivatives is a nonnegative real constant.
Definition 1. . The Riemann–Liouville fractional integral operator with order is given by
Definition 2. . The operator of fractional derivative with order for is given bywhich is called the Caputo fractional operator.
The operator satisfies the following properties:(i), (ii), , , and is equal to zero otherwise(iii) for , that is, is linear operatorMoreover, for , and , , as well as for , we have .
Remark 1. . For an arbitrary function the Caputo fractional derivative can be computed by the following formula:It is worth mentioning that the fractional calculus has a nonlocal property, so solving fractional differential equations is a challenge, especially for numerical calculations. This property indeed is the main reason why fractional calculus is more popular and good tool for modeling reality. However, Taylor expansion in the fractional sense does not give an approximation of the function at a point because of nonlocality. Anyhow, the local fractional Taylor formula has been successfully generalized and applied in science and engineering problems based on the theory of fractal geometry; for more details, we refer to [32–34].
Definition 3. . The fractional power series “FPS” about for is given by
Theorem 1. . According to the FPS , there are the following possibilities:(1)The series converges only for whenever is equal to zero(2)The series converges whenever is equal to infinity(3)The series converges at for and diverges for where indicates the radius of convergence of the power series.
Lemma 1. Assume that , and Then,
Proof. From the properties of Caputo fractional operator, one can deduce thatOn the other hand, for , we infer that
Theorem 2. . Assuming that has the following power series expansion about :If and , , then coefficient is given by where (-times).
3. The FRPS Approach
This section is dedicated to presenting the procedures needed to implement the FRPS algorithm to solve the continuous logistic equation in the fractional sense by expanding FPS and utilizing repeated fractional differentiation of the truncated residual functions. To perform this, suppose that the fractional logistic equations (1) and (2) have the solution form about :
This analysis aims at extending the application of fractional Taylor series framework to get an accurate analytic series solution of fractional systems (1) and (2). Thus, if we use the initial data given by (2), , as initial truncated series of , so the FPS solution of equation (1) can be written by
Therefore, the truncated series solution of is given by
According the FRPS approach, we define the residual function, , for the proposed logistic model as follows:whereas the residual function, can be defined by
In this point, we noted that for all , which leads to , for all . Consequently, the following fractional relations assist us to determine the unknown coefficients, , , of equation (13):
To show the iteration concept of the FRPS technique to find out , put in residual function of equation (14) at to get that
Thus, by using the fact that , it yields
In the same way, to find out , put the second truncated series in such that
Consequently, by applying the fact in equation (21), one can get . The second FRPS approximation is
Similarly, substituting into the residual function such that , calculating the fractional derivative of , and finally solving the following obtained result by the third coefficient is determined such that
So, the third FRPS approximation is given by
Further, when the same routine is repeated as above up to the arbitrary order , the coefficients , can be obtained.
Theorem 3. Assuming that satisfies the conditions of (11) with such that . Then,where and , for some . Here, is the -term truncation of and is the remainder error function.
Proof. Clearly,Using Lemma 1, it is obvious thatThis, in turn, implies thatOn the other hand, we get thatHence, one can deduce the stated result.
Remark 2. If on then the upper bound of can be obtained by
4. Illustrative Examples
In this section, numerical applications of the fractional logistic differential equations are presented and quantified at some mesh points. Numerical outcomes highlight the globality of the proposed algorithm in obtaining string solutions consistently and also show that the approximate values are highly acceptable in terms of stability and accuracy. In the calculation, all symbolic and digital calculations are performed using Mathematica software package.
Example 1. Consider the fractional logistic differential equationwith the initial conditionThe exact solution of IVP (31) and (32) at is given byAccording the FRPS algorithm, the FPS solution to (31) has the formNow, define the residual error for equation (31) byand the residual function byFollowing the FRPS algorithm to find out the coefficients , of equation (34). Let the first truncated PS approximation has the formFrom equation (36) at , we haveand depending on the fact on equation (16), , we have . So, the first approximation isFor , the second truncated PS approximation has the formand the second residual function isNow, applying on both sides of equation (41) such thatUsing the results of equation (16) at , , we have . So, the second approximation isFor , the third truncated PS approximation has the formand the third residual function isNow, applying on both sides of (46) such thatUsing and then continuing in this process, one can get thatand so on.
Consequently, few terms of RPS solution areThe numerical results of the 5th FRP solution are given in Table 1 with in the interval with step size . Figure 1 shows a comparison between the behavior of the exact solution and the approximate solution at , while in Table 2, numerical comparison is given between the proposed method and the optimal homotopy asymptotic method (OHAM)  at . In Figure 2, the behavior of the 5th FPRS approximation is presented with different values of , where and with step size of , whereas in Table 3, we review the numerical comparison between the FRPS solutions and the OHAM  when with step size of . Anyhow, Table 4 shows the representation of the 6th approximate solution with different values of such that . From these results, it can be observed that the behavior of the approximate solutions for different values of is in good agreement with each other that depends on the fractional order .