Abstract
In this article, we apply the Daftardar-Gejji and Jafari method (DJM) to solve the multispecies Lotka–Volterra equation. A comparison between the DJM, differential transformation method (DTM), the variational iteration method (VIM), and Adomian decomposition method (ADM) shows that the DJM is a reliable and powerful method for solving nonlinear equations. The efficiency and applicability of this method are confirmed by considering some examples. The proposed procedure provides better results in comparison to some existing methods.
1. Introduction
The area of mathematics called numerical analysis is in charge of coming up with practical methods for calculating answers to difficult computational calculations. The majority of mathematical issues in engineering and science are very challenging, and sometimes, there is no straightforward solution. To make a difficult mathematical problem simpler to solve, measurement is thus crucial. As a contemporary tool for scientists and engineers, numeracy has grown in popularity as a result of the tremendous developments in computing technology. As a consequence, a variety of software packages, including MATLAB, Mathematica, Maple, and others, are being created to solve even the most challenging issues quickly and simply. These programs provide features that make use of conventional numerical techniques, allowing the user to run a single command without entering any parameters and obtain the desired results. The creation, analysis, and application of algorithms for solving numerical problems in continuous mathematics are all made using the numerical analysis approach, which is mostly utilized in mathematics and computer science. These kinds of issues often come up in the actual world when algebra, geometry, and calculus are applied, and they also include continuous variables. These issues arise in all areas of study, including the scientific and social sciences, engineering, health care, and business [1–9]. Numerical analysis introduced realistic mathematical models which have become more prevalent in science and engineering over the last 50 years as a result of the expansion in the power and accessibility of digital computers. We shall learn more about numerical approaches and their analysis here. PDE solutions may be solved using the same numerical techniques used for ODEs. Many difficulties may be solved using the techniques mentioned for handling initial value concerns, for example, see references [6, 10–14].
The Lotka–Volterra equations describe the time history of a biological system [15]. The Lotka–Volterra equations are applied in a number of engineering areas. The one-species Lotka–Volterra equation is used to demonstrate a simple nonlinear control system [16].
The Lotka–Volterra equations were solved by many numerical methods like hybrid deep network [17], Gröbner bases elimination method [18], generalized backstepping control method [19], the differential transformation method (DTM) [20], the Adomian decomposition method (ADM), and variational iteration method (VIM) [21].
The differential transformation method (DTM) was first proposed by Zhou [22] (also check [23, 24]). The DTM is an iterative method that obtains the Taylor series solutions of different kinds of differential equations (see [20, 25–27]). The DTM can be applied directly to different kinds of DEs without requiring linearization, discretization, or perturbation, and it is a very accurate method with less computational work [28].
The Adomian decomposition method (ADM) was introduced by Adomian [29] to solve nonlinear differential equations and physical problems [30–33].
The VIM was first proposed by He [34] (see also [35–37]). The VIM has successfully been used for many ordinary and partial differential equations [21, 38–40].
In 2006, the DJM was first proposed by Daftardar-Gejji and Jafari [41]; the method can solve many nonlinear differential equations and physical problems [42–52]. Recently, the DJM was applied to create a quite new predictor-corrector method [53, 54]. Noor et al. [55–59] used the DJM to create numerical techniques to solve algebraic equations.
In this paper, we apply the DJM to solve the multispecies Lotka–Volterra equation and compare the results obtained with DTM, VIM, ADM, and exact solution to show the simplicity and accuracy of this method. The efficiency and applicability of this method are confirmed by considering some examples. The proposed procedure provides better results in comparison to some existing methods. The DJM method will be implemented in a direct way without any linearization, perturbation, or restrictive assumptions.
2. The Daftardar-Gejji and Jafari Method (DJM)
Here, the DJM ([41]) will be described, which was successfully applied to solve nonlinear DEs of the following form:where is a function given, is the linear operator and is the nonlinear operator. The solution of equation (1) will be as follow:
Suppose
So,
Thus, is decomposed as follows:
So,
Since is linear, then
Then,
Thus,
The term approximate solution is given as follows:
3. Convergence of the DJM
Theorem 1. “For any and for some real and if is in the neighborhood of and , then is absolutely convergent and .”
Proof. Please see reference [46] for full details of the proof.
Theorem 2. “The series is absolutely convergent if is and .”
Proof. Please see reference [46] for full details of the proof.
4. Analysis of Multispecies Lotka–Volterra Equations
In this section, we will study the general Lotka–Volterra system in the form as follows:
To solve equation (11) with the initial condition by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation:
Then, we will apply the DJM as in the previous section.
4.1. One Species
In this section, equation (11) is reduced to one species:where and are constants. With exact solution,
To solve equation (13) with the initial condition by the Daftardar-Gejji and Jafari method (DJM), we write it in the following integral equation:
By applying DJM, we obtain the following:
The four-term solution is as follows:
4.2. Two Species
In this section, equation (11) is reduced to two species:where , , , , , and are constants.
To solve equations (18) and (19) by DJM with initial conditions and , we write it in the following integral equation:
The rest components of the formulas (20) and (20) can be obtained using the computer algebra package Maple.
4.3. Three Species
In this section, equation (11) is reduced to three species:where and are constants.
To solve equations (21)–(23) by DJM with initial conditions , , and , we write it in the following integral equation:
Again, the rest components of the formulas (24)–(26) can be obtained using the computer algebra package Maple.
5. Discussion
We used Maple to code the DJM algorithm. Maple environment variable digits is set to 16 in all calculations done in this paper.
The numerical solutions obtained by using the DJM are compared with the exact solution and those obtained by ADM [21], DTM [20], and VIM [40]. Table 1 shows a comparison between the exact solution, the four iterations DJM with the DTM of order 6, two iterations of VIM, and 3-term ADM in the case , and for ; we can see the method is efficient to solve the one-species Lotka–Volterra equation. In Table 2, we compare four iteration DJM with DTM, VIM, ADM, and the exact solution where , and for ; in this table, we can prove the stability of DJM for large . In Table 3, we perform the numerical comparison when , and . In Table 4, we compare DJM with 4-Iterate VIM and RK4 when , and
6. Conclusions
In this article, the DJM is used for solving the multispecies Lotka–Volterra equation. The Daftardar-Gejji and Jafari method was implemented in a direct way without any linearization, perturbation, or restrictive assumptions. Comparisons with the VIM, DTM, and ADM show that the DJM is a better method for solving nonlinear equations. We proved that DJM is a precise and efficient method to solve the multispecies Lotka–Volterra equation.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.