Abstract
In the present work, the natural transform iterative method (NTIM) is implemented to solve the biological population model (BPM) of fractional order. The method is tested for three nonlinear examples. The NTIM is a combination of a new iterative method and natural transform. We see that the solution pattern converges to the exact solution in a few iterations. The method handles an extensive range of differential equations of both fractional and integer order. The fractional order derivative is considered in Caputo’s sense. For mathematical computation, Mathematica 10 is used.
1. Introduction
The globe and our everyday lives have been revolutionized by modern technologies. Technology is being used in a wide range of engineering applications, including aerodynamics, fluid dynamics, medical sciences, and finance. The essence of technology is influenced and designed by mathematical modeling. The modeling might take the form of mathematical models that can be described using differential equations. These differential equations may have been used to represent the transmission of electromagnetic waves, which is at the root of many present technologies. A variety of applications ranging from wireless communications to radar, medical imaging, and remote sensing have played a great role in our life [1, 2]. Mathematics and biosciences have also numerous practical applications related to real life [3, 4]. Several diseases can be modeled through mathematical calculations and can be controlled by collecting data and making precise analysis [5]. There is a strong and interesting relationship between biology and mathematics utilizing differential equations. The noninteger order differential equations are termed as fractional order differential equations (FDEs) [32–35]. The branch of mathematics dealing with FDEs is known as fractional calculus [6]. Depending on the nature of the problem, differential equations can be linear or nonlinear. Simple analytical methods may be used to analyze linear differential equations, but investigators have developed several ways for solving nonlinear differential equations as their exact solutions are not always feasible. The importance of the FDEs can be discussed in many fields of sciences [7, 23–27]. Many operators for fractional derivatives have been given by several researchers. The most famous is Caputo’s fractional derivative operator [8]. Li et al. introduced the fractional order integral operator for handling differential equations [9]. Recently, many transformations have been used to solve fractional order differential equations. Some of them are the Laplace transform, Sumudu transform, Elzaki transform, etc. [10–12, 28–31]. In this work, we will deal with the natural transform iterative method (NTIM), a combination of the natural transform and the new iterative method (NIM). The proposed techniques have been recently applied by Nawaz et al. for solving noninteger order differential equations [13]. Many other researchers have applied NIM and natural transform for handling the FDEs [14–17]. In this article, we will consider fractional biological population model (FBPM) as [18]
In Equation (1), is the population density and is the supply of population due to births and deaths. The are the real numbers and is the initial condition. The detailed solution of Equation (1) can be found in [19, 20]. FBPM is a mathematical model of biology which will be thoroughly investigated in this paper. FBPM aids in the understanding of the dynamical procedure of population changes in biological population models, as well as providing useful predictions.
The remaining paper is structured as follows: Preliminary definitions from fractional calculus are contained in Section 2. The notion of NTIM is introduced in Section 3. The NTIM is used to solve three FBPM in Section 4. In Section 5, some results have been discussed. Lastly, a concrete conclusion is given.
2. Preliminaries
Definition 1. Riemann-Liouville (R-L) fractional integral is defined as where is the gamma function.
Definition 2. Caputo’s time-fractional derivative operator of order is defined as
Definition 3. Natural transform of is defined as [21] where and are the transform variables.
Definition 4. The inverse of natural transform of is defined as where and the integral are taken in the complex plane along .
Definition 5. If the th derivative of is , then its natural transform is given as
Theorem 6. If the natural transform of and are and respectively, defined on set A, then where is convolution the functions and .
3. Natural Transform Iterative Method (NTIM) [13]
Consider FDE of the form where and The linear operator, nonlinear operator, and the source term are , , and , respectively. The initial condition is given as By applying the natural transform to Equation (8), we have Using the natural transform differentiation property, Equation (10) can be written as By rearranging Equation (11), we have For the NTIM solution, is expanded as and , the nonlinear term, is defined as Using Equation (13) and Equation (14) in Equation (12), we obtain Using the recursive relation,
Now by taking the inverse natural transform of Equation (16), we have
Then by adding the components, the approximate solution of Equations (8) and (9) by NITM is given as
Convergence of NTIM is as a convergence of NIM and is proved by Bhalekar and Daftardar-Gejji [22].
4. Applications of NTIM
In this section, we apply the natural transform iterative method NTIM for handling the three nonlinear cases of FBPM. The method is applied directly to the problems without any discretization by using the given initial conditions. Then, the comparison is made with the help of plots and numerical tables with the existing methods which shows the effectiveness of the proposed method [18].
Problem 7. Consider the population model as [18] where together with initial conditions and the exact solution is Taking natural transformation of Equation (19), we have Applying the natural transform differentiation property to Equation (22), we get Taking the inverse natural transform of Equation (23), we have Using the idea of NTIM and the recursive relation of Equation (16), we obtained the solution components as By using the software package, the solution components are obtained as Combining the components, the 4th order approximate solution is given as Equation (27) can be simplified as For , Equation (28) converges as which converges to the exact solution given by Equation (21).
Problem 8. Consider the biological population model as [18] where together with initial conditions and the exact solution is Taking natural transform of Equation (30), we have Using the natural transform differentiation property, we obtain Taking the inverse natural transform, we have Using the recursive relation, the solution components can be obtained as By using the software package, the solution components are obtained as Combining the components, the 4th order approximate solution is given as Equation (38) can be simplified as For , Equation (39) converges as which yields the exact solution given by Equation (32).
Problem 9. Consider the biological population model as [18] where subject to the initial condition and the exact solution is Using the same procedure as for Problems 7 and 8, we obtain the solution as Combining the components, the 3rd order approximate solution is given as Equation (27) can be simplified as For , Equation (29) converges as which converges to the exact solution given by Equation (43).
5. Results and Discussions
The biological population model of fractional order has been investigated in the present work. We observe that the solution pattern for Examples 1-3 converges very rapidly to the exact solution in a few iterations. The results have been compared through graphs and tables which confirms the convergence of NTIM. Figure 1 is the NTIM approximate solution, and Figure 2 is the exact solution for of Example 1. The 6th order approximate solution for different fractional values of β for Example 1 is depicted in Figure 3. Furthermore, Figures 4 and 5 show the 6th order NTIM solution and the exact solution, respectively, for Example 2 by mean of 3D plots. The comparison for different fractional values of and exact solution is made in Figure 6 for Example 2. Similarly Figures 7 and 8 show, respectively, the 6th order NTIM solution and exact solution for Example 3. Figure 9 is the comparison of fractional values of and exact solution for Example 3. In Table 1–3, the approximate solution has been compared in tabular form for , , , and , with the exact solution. The value converts the FDE to the classical PDE. The absolute error by the proposed NTIM in Tables 1–3 has been compared with the absolute error obtained by the modified generalized Taylor fractional series method (MGTFSM) for Examples 1-3. It is concluded from the results of Examples 1-3 that as the fractional value of reaches 1, the NTIM approximate solution meets with the exact solution. We also observe that NTIM yields an excellent approximate solution.
6. Conclusion
Three nonlinear problems of fractional order biological population model have been investigated by the natural transform iterative method in the current study. The method is applied to nonlinear problems without any discretization. We found that NTIM converges very rapidly to the exact solution. The advantage of the method is that it is free of any large or small parameter assumptions or to find any constant at the end of the solution. The obtained results of the FBPM have been compared through 3D and 2D plots, and also, the numerical values have been compared in tabular form for different values of . The comparison between absolute errors of the NTIM approximate solution and modified generalized Taylor fractional series method solution is done with the help of tables. In each case, NTIM reveals an efficient approximate solution as compared with other methods in the literature.
Data Availability
No data were generated or analyzed during the study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups under grant number R.G.P.2/14/43.