Abstract
Recently solving integro-differential equations have been the focus of attention among many researchers in the field of mathematic and engineering. The aim of current study is to apply the well-known optimal homotopy asymptotic method (OHAM) on a specific and famous model of these equations. It is illustrated that auxiliary functions and the number of Taylor series terms affect the accuracy of the solution. Hence, at first a solution has been found with an acceptable error by OHAM. Then, it has been continued to attain a better solution by Multistep optimal homotopy asymptotic method. All these processes had improved the precision of the solution. Auxiliary polynomials of two, three, and four degrees and different numbers of Taylor series term have been investigated to solve a nonlinear system derived by two biological species living together. Ultimately, appropriate results with auxiliary polynomials of degree four and Taylor series with six terms have been obtained precisely. In addition, the error values decrease significantly compared to the other cases.
1 Introduction
It is clear that solving more accurately differential equation, integro-differential equations, or their systems are very invaluable. The necessity of this accuracy is more perceptible as we apply them in practical cases. For solving mathematical models for science problems or natural issues raised in daily living, various analytical or numerical methods have been applied such as Variation iteration method [1,2,3], Homotopy perturbation method [4,5,6,7,8,16,17], Homotopy analytical method [9,10,11], Optimal homotopy asymptotic method (OHAM) [12,13,14,15], and so on [18,19,20,21,22,23,24,25].
OHAM has been used in this study to solve a biological system. Generally, in homotopy methods, an embedding parameter would have been used and by this parameter the equation deforms continuously. Liao formed a special homotopy by using an embedding parameter denoted by
One of the properties of OHAM is its simplicity compared with HAM. But, by using the Taylor series and paying attention to its convergence, it is important to find the number of the first few terms of an approximate polynomial as a solution. Meanwhile, the neighborhood radius and auxiliary functions proposed in the OHAM have significantly influenced in finding the suitable solutions. Therefore, at first an appropriate approximated solution has been obtained with an auxiliary polynomial. Then, by continuously changing the number of terms of auxiliary polynomials and solution polynomials, we have gotten a suitable solution with less error for the equations. Also, in order to minimize the error, we have reduced the neighborhood radius to a satisfactory level.
In the current research, OHAM have been applied to solve the following system of equations known as predator–prey equation, which is often used to describe the dynamics of biological systems, in which two species interact, one as a predator and the other as a prey.
where
Solving such mathematical models in recent years has been considered by researchers in various fields of science, including mathematics, physics and engineering, and so on, such as microorganisms, Biorheological fluid model, cilia-driven fluid model, and so on [18,19,20,21].
2 Description of OHAM
OHAM has been presented by Marinca and Herisanu [12]. The idea is derived from Liao’s HAM. With a clever choice, they have used a special function of
where
But in OHAM, a new auxiliary function denoted by
where
Finally, we consider a homotopy function such as
where
3 Solution of the proposed problem via OHAM and MOHAM
3.1 Solution via OHAM
In this study, the system of Eq. (1) is an integro-differential system of two species living together and affecting each other. We use the symbol
According to Eq. (6), we first obtain
So, via OHAM we obtain the second term of
Similarly, we use the other stages as follows:
where B is the initial/boundary condition and
We obtain the optimal
and the following system
where
Finally, by substituting
3.2 Solution via MOHAM
After obtaining a suitable solution with acceptable maximum error, the proper auxiliary functions and the number of terms of solution polynomials are determined. Now for more accuracy, we use an expansion with a fewer neighborhood radius. So, we divide the domain interval into sub-intervals and continue to solve the problem. To solve in each sub-interval, we have to consider the continuity of solution in nods. Therefore, we replace the value of solution in the end of each sub-interval, for the initial value of the next sub-interval. Using MOHAM enables us to minimize the amount of error in each sub-interval.
4 Numerical example
We consider the system of integro-differential equations of prey and predator as follows:
First, we obtain the approximate solution by OHAM with various auxiliary functions and different numbers of expansion terms in the interval. Then, with an appropriate selection that results in minimizing of maximum equation errors in the interval
4.1 Numerical solution by OHAM
Because of using OHAM, we give auxiliary functions
The errors of system equations are dependent on the set of
Also, the solution will be changed by increasing the number of expansion terms in OHAM and evidently, the error of equations will be changed by inserting the different solutions in equations. So, we must select an appropriate finite expansion of solution with an acceptable error in a suitable sub-interval.
To reach this goal, we suppose
So, first we have obtained control coefficients
The curves of solution and errors of equations with one term of series have been plotted as shown in Figure 1.
The solution of the system by OHAM with one term results in the maximum error of
The curves of solution and the value of equations by inserting the obtained solution (remained or error of equations) have been illustrated in Figure 1(a) and (b).
The solution of the prey and predator system by OHAM with two terms of the series and
The CPU time presented was 19.905 s.
Inserting the solution of the system using OHAM with two terms of series, the maximum errors of prey and predator equations have been
The CPU time was 20.593s.
The curves of solution and errors of equations with two terms have been illustrated in Figure 2(a) and (b).
We have investigated the same process but with three terms of the series and have obtained the following results:
The CPU time obtaines was
With three terms of the series, the maximum error of
The CPU time was
The maximum error obtained with four terms of the series is
The solution of the system by OHAM with five terms result in the maximum error of
The CPU time was
By following the procedure, the required values and equations are obtained with six terms.
The CPU time was
The solution with six terms of the series have got the maximum error of
Finally, the following results have been derived with seven terms of the series,
The CPU time was presented as
The case with seven terms has bad situation and the maximum error for prey and predator equations in the interval
Apparently, it has been illustrated that the solution with defined auxiliary functions H1, H2, and five terms of the series is better than the before cases. Also, the obtained solutions with 6 and 7 terms of series have not been better than the other cases in the whole interval, as well as the CPU time has increased very much. It was expected to achieve a decrease in the error of equations with increase in the terms of the series, but this has not occurred. Meanwhile, increasing the CPU time raises the consumption costs. So, we decided to use the new auxiliary functions with six terms of the series. The solution of the presented system by OHAM with six terms of the series and
The CPU time obtained was
The solution of the system by OHAM with six terms and the new auxiliary function results show that the maximum error will be
So, we continue the process with
The CPU time was
Because of unacceptable error, we continue the process with seven terms of expansion. The solution of the system by OHAM with seven terms results in the maximum error of
The CPU time was
As noted above, with auxiliary polynomials of degree four, and the solutions containing six and seven terms of series, the error values are reduced. The maximum errors obtained for prey and predator equations were
As illustrated, the solution with seven terms is much better than the six terms, but the CPU time for the seven terms is about ten times that of the six terms. In this regard, it is a good idea to go to the multi-step approach by choosing these auxiliary functions and 6 terms of the series.
4.2 Numerical solution by MOHAM
We have obtained the solution of the proposed system with auxiliary functions
The solution of the system by MOHAM with six terms result in the maximum error of
The maximum error of
The maximum error of
The maximum error of
The maximum error of
The maximum error of
The maximum error of
The maximum error of
The maximum error of
We have presented the solution and error equations in each sub-interval and have calculated the values of solution and error values in a few points of domain by OHAM and MOHAM with auxiliary polynomials of four degrees and solutions with six terms, then we have compared those values. The results are presented in Tables 1 and 2.
t | Prey | ||||
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OHAM | Eeq(OHAM) | MOHAM | Eeq(MOHAM) | Eeq(OHAM)- Eeq(MOHAM) | |
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t | Predator | ||||
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OHAM | Eeq(OHAM) | MOHAM | Eeq(MOHAM) | Eeq(OHAM)-Eeq(MOHAM) | |
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5 Conclusion
Although the simplicity of OHAM affects more in practical applications, we have to reduce the error values for solutions. So, to obtain a satisfactory solution by OHAM, we have first changed the number of terms of the series and auxiliary polynomials with unknown control coefficients. In each case, the control coefficients have been calculated at least by 30 iterations of Newton method and consequently the solution of the system has been obtained with a lesser error. In action, and by the given example in this study, we began with auxiliary polynomials of degree two to obtain the solution of prey and predator systems.
With the increase in the number of series terms, we were satisfied with an approximate solution of degree five, because the solutions with less than and more than this degree have had more maximum errors, such that prey equation errors with four, five, and six terms have been obtained as
We followed the solution by auxiliary polynomials with four terms, as well as the solution polynomials with six and seven terms. The obtained results were such that the maximum errors with six terms have been
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Funding information: The author states no funding involved.
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Author contributions: All author has accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: The authors state no conflict of interest.
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