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A new optimal multistep optimal homotopy asymptotic method to solve nonlinear system of two biological species

  • Zainab Ayati EMAIL logo and Sadegh Pourjafar
Published/Copyright: February 17, 2023
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 12 Issue 1

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.

Keywords: optimal homotopy asymptotic method; multistep optimal homotopy asymptotic method; system of integro-differential equations; biological species

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 q ∈ [ 0 , 1 ] , an auxiliary parameter denoted by h ≠ 0 and so an auxiliary function denoted by H ( X ) ≠ 0 , all together. Marinca and Harrison manipulated Liao’s homotopy analysis method (HAM) by inserting a function of imbedding parameter i.e., H ( p ) instead of all q , h , and H ( X ) .

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.

(1) d n 1 ( t ) d t = n 1 ( t ) K 1 − α 1 n 2 ( t ) − ∫ t − T 0 t f 1 ( t − s ) n 2 ( s ) d s , d n 2 ( t ) d t = n 2 ( t ) − K 2 + α 2 n 1 ( t ) + ∫ t − T 0 t f 2 ( t − s ) n 1 ( s ) d s ,

where K 1 , K 2 , α 1 , α 2 are the known parameters, and n 1 ( 0 ) , n 2 ( 0 ) are the values for initial conditions. n 2 is the number of predators and n 1 is the number of preys. The first equation is known as the prey equation and the second equation is known as the predator equation. k 1 is the index of predator growth, γ 1 is the index of prey growth, γ 2 n 1 n 2 indicates an increase in the population of predators and k 2 n 2 indicates their natural death. In order to get more accuracy, a suitable form of solution has been selected with a determined number of Taylor expansion terms and an acceptable error for each equation, and then multi-step Optimal Homotopy Asymptotic Method (MOHAM) has been used to decrease the error.

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 H ( p ) instead of parameter h , q , and function H ( X ) , used by Liao in the following known relation:

(2) H ˜ [ ϕ ( X ; q ) ; U 0 ( X ) , H ( X ) , h , q ] = ( 1 − q ) { L [ ϕ ( X ; q ) − U 0 ( X ) ] } − q h H ( X ) N [ ϕ ( X ; q ) ] .

where X is a vector of independent variables, q ∈ [ 0 , 1 ] is the embedding parameter, and h is the auxiliary parameter. Also, H ( X ) is an auxiliary function and L is an auxiliary linear operator. These along with the other related definitions and concepts are used in HAM. Besides the governing concepts of Liao’s method and free choice of U o and L , h and H ( X ) also play important roles to obtain the suitable solution.

But in OHAM, a new auxiliary function denoted by H ( x , p ) is used as a power series in an embedding parameter p ∈ [ 0 , 1 ] instead of q , h , and H ( X ) . H ( x , p ) is continuous and non-zero, except for p = 0 , i.e., H ( x , 0 ) = 0 . Let us consider the following equation by OHAM:

(3) A ( u ( x ) ) = g ( x ) , x ∈ Ω , B ( u , ∂ u / ∂ n ) = 0 , x ∈ Г ,

where A , B are the general differential and boundary operators, respectively, g ( x ) is an analytic function, and Γ is the subset of domain Ω . By splitting A ( u ( x ) ) into linear part denoted by L and nonlinear part denoted by N , the above equation can be converted as follows:

(4) L ( u ( x ) ) + N ( u ( x ) ) = g ( x ) .

Finally, we consider a homotopy function such as

(5) h ( φ ( x ; p ) ; p ) : [ 0 , 1 ] × R → R .

(6) ( 1 − p ) [ L ( ϕ ( x ; p ) ) + g ( x ) ] = H ( x , p ) ⋅ [ L ( ϕ ( x ; p ) ) + g ( x ) + N ( ϕ ( x ; p ) ) ] ,

where x ∈ R and p ∈ [ 0 , 1 ] are embedding parameters. Also, H ( x , p ) is an auxiliary non-zero function and ϕ ( x ; p ) is an unknown function with the following property: p = 0 , φ ( x , 0 ) = u 0 ( x ) and p = 1 , φ ( x , 1 ) = u ( x ) . Change occurs continuously and when p varies from 0 to 1, φ ( x , p ) deforms from u 0 ( x ) to u ( x ) [12].

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 n 1 , i , i = 0 , 1 , 2 , for i-th approximation of n 1 and similarly n 2 , j , j = 0 , 1 , 2 , for j-th approximation of n 2 .

According to Eq. (6), we first obtain n 1 , 0 ( t ) and n 2 , 0 ( t ) by linear equations from the concept of homotopy and then we get the other terms of Taylor expansion of n i , i = 1 , 2 , by suitable selection of auxiliary function H ( t , p ; C i ) for each equation as follows:

(7) H ( t ; p , C i ) = p H 1 ( t , C i ) + p 2 H 2 ( t , C i ) + ⋯ .

So, via OHAM we obtain the second term of n i , i = 1 , 2 , by the following equations:

(8) L ( n 1 , 1 ( t , C i ) ) = H 1 , 1 ( t , C i ) N 0 ( n 1 , 0 ( t ) ) , B n 1 , 1 , d n 1 , 1 d t = 0 , t ∈ D ,

(9) L ( n 2 , 1 ( t , C j ) ) = H 2 , 1 ( t , C j ) N 0 ( n 2 , 0 ( t ) ) , B n 2 , 1 , d n 2 , 1 d t = 0 , t ∈ D .

Similarly, we use the other stages as follows:

(10) L ( n 1 , k ( t , C j ) − n 1 , k − 1 ( t , C j ) ) = H 1 , k ( t , C j ) N 0 ( n 1 , 0 ( t ) ) + ∑ i = 1 k − 1 H 1 , i ( t , C j ) [ L ( n 1 , k − i ( t , C j ) ) + N 1 , k − i ( n 1 , 0 ( t ) , n 1 , 1 ( t , C j ) , … , n 1 , k − i ( t , C j ) ) ] , B n 1 , k , d n 1 , k d t = 0 , t ∈ D ,

(11) L ( n 2 , k ( t , C j ) − n 2 , k − 1 ( t , C j ) ) = H 2 , k ( t , C j ) N 0 ( n 2 , 0 ( t ) ) + ∑ i = 1 k − 1 H 2 , i ( t , C j ) [ L ( n 2 , k − i ( t , C j ) ) + N 2 , k − i ( n 1 , 0 ( t ) , n 2 , 1 ( t , C j ) , … , n 2 , k − i ( t , C j ) ) ] , B n 2 , k , d n 2 , k d t = 0 , t ∈ D ,

where B is the initial/boundary condition and C j s , j = 1 , 2 , … are control coefficients for which we have to obtain their values. Consequently, the m-th order approximations of two species by OHAM will be

(12) n i , m ( t , C j ) = n i , 0 ( t ) + ∑ k = 1 m n i , k ( t , C j ) , i = 1 , 2 , m = 1 , 2 , ⋯ .

We obtain the optimal C i s by applying the least square method, using

(13) J ( C 1 , C 2 , … , C m ) = ∫ 0 1 R 2 ( t , C 1 , C 2 , … , C m ) d t ,

and the following system

(14) ∂ J ∂ C 1 = ∂ J ∂ C 2 = ⋯ = ∂ J ∂ C m = 0 ,

where

(15) R ( t , C 1 , C 2 , … , C m ) = L ( n i , m ( t , C 1 , C 2 , … , C m ) ) + N ( n i , m ( t , C 1 , C 2 , … , C m ) ) , i = 1 , 2 , m = 1 , 2 , … .

Finally, by substituting C j , j = 1 , 2 , … in Eq. (12) with a specified iteration, we obtain an approximate numerical solution.

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:

d n 1 ( t ) d t = n 1 ( t ) K 1 − α 1 n 2 ( t ) − ∫ t − T 0 t f 1 ( t − s ) n 2 ( s ) d s , d n 2 ( t ) d t = n 2 ( t ) − K 2 + α 2 n 1 ( t ) + ∫ t − T 0 t f 2 ( t − s ) n 1 ( s ) d s , subject to the following initial conditions and nomenclature values:

n 1 ( 0 ) = 10 , n 2 ( 0 ) = 10 , K 1 = 0.02 , K 2 = 0.01 , α 1 = 0.01 , α 2 = 0.01 , T 0 = 0.1 , t ∈ [ 0 , 2 ] and f 1 ( t − s ) = f 2 ( t − s ) = e − ( t − s ) .

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 [ 0 , 2 ] , we find a special auxiliary function and determined number of terms of polynomial solution. With these data, we use MOHAM in a few steps. The obtained solutions by OHAM and MOHAM are compared at some points and are presented in tables.

4.1 Numerical solution by OHAM

Because of using OHAM, we give auxiliary functions H 1 = ∑ i = 1 n C i p i and H 2 = ∑ i = 1 n C n + i p i and obtain C i s by least square method. To solve the system of C i s , we use the numerical iteration Newton method. To use this method, one can suppose an initial solution named C 0 , then can obtain C k + 1 by C k in the sequence of iterations C k + 1 = C k − J k − 1 A k , where C k = ( C 1 , k , C 2 , k , ⋯ , C n , k ) , and J k named Jacobian matrix, defined as follows:

J k = ∂ f 1 ∂ C 1 , k ∂ f 1 ∂ C 2 , k ⋯ ∂ f 1 ∂ C n , k ∂ f 2 ∂ C 1 , k ∂ f 2 ∂ C 2 , k ⋯ ∂ f 2 ∂ C n , k ⋱ ∂ f n ∂ C 1 , k ∂ f n ∂ C 2 , k ⋯ ∂ f n ∂ C n , k .

The errors of system equations are dependent on the set of C i s , i.e., any set of C i s produces a special set of errors. Because of this, we select a solution that minimizes the sum of errors of system equations and with this C i s , we obtain n 1 and n 2 .

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 H 1 = ∑ i = 1 2 C i p i = C 1 p + C 2 p 2 , H 2 = ∑ j = 3 4 C j p j = C 3 p + C 4 p 2 and apply OHAM with one to seven terms of the series. Also, the solutions and error curves of equations are presented in the text and illustrated in the figures.

So, first we have obtained control coefficients C i = 0 , i = 1 , ⋯ , 4 , by applying initial conditions, defined H 1 , H 2 , and one term of the series. The solution and absolute error functions that have been obtained by inserting the solution in the system are presented as follows: n 1 , 1 = 10 . n 1 , 2 = 10 . R 1 , 1 = 10 .3162581964040 . R 1 , 2 = 10 .4162581964040 .

The curves of solution and errors of equations with one term of series have been plotted as shown in Figure 1.

Figure 1 (a) The curves of n1, n2 obtained by OHAM with one term, without step in the interval [0, 2]. (b) The error curves of the equations by inserting the solution with one term, without step in the interval [0, 2].
Figure 1

(a) The curves of n1, n2 obtained by OHAM with one term, without step in the interval [0, 2]. (b) The error curves of the equations by inserting the solution with one term, without step in the interval [0, 2].

The solution of the system by OHAM with one term results in the maximum error of 10.3162581964040 for prey equation and maximum error of 10.4162581964040 for predator equation in interval [ 0 , 2 ] . Also, the maximum errors do not change in the interval [ 0 , 0.5 ] and the CPU time was 20 .467  s. by Maple 18.

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 H 1 = C 1 p + C 2 p 2 , H 2 = C 3 p + C 4 p 2 have been derived as follows:

C 1 = − 0 .42394468393979110 − 1 , C 2 = 0 , C 3 = 0 .495067903827490 , C 4 = 0 ,

n 1 , 2 ( t ) = 10 − 0 .437352282051577 t , n 2 , 2 ( t ) = 10 − 5 .15675511101965 t ,

R 1 , 2 ( t ) = | 10.1201822434631 − 5.88471300206658 t + 0.237175128787498 t 2 | , R 2 , 2 ( t ) = | 15.5934763216390 − 5.84189249659862 t + 0.237175128787519 t 2 | .

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 10 .1201822434632 and 15 .5934763216390 in the interval [0, 2], respectively. Also, the maximum errors do not differ in the interval [ 0 , 0.5 ] .

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).

Figure 2 (a) The curves of n1 and n2 obtained by OHAM with two terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with two terms, without step in interval [0, 2].
Figure 2

(a) The curves of n1 and n2 obtained by OHAM with two terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with two terms, without step in interval [0, 2].

We have investigated the same process but with three terms of the series and have obtained the following results:

C 1 = − 1 .70554677356656 , C 2 = − 0 .423058872049528, C 3 = − 0 .464274818263567, C 4 = 0 .570433711378159 .

n 3 , 1 ( t ) = 10 − 9.15933573395762 t + 11.1420331472655 t 2 , n 3 , 2 ( t ) = 10 + 1.86719943055459 t − 3.12593934333262 t 2 .

R 3 , 1 ( t ) = | − 15.1800444687121 t − 5.52473124770772 t 3 + 3.66274119608392 t 4 − 6.03252978775368 t 2 − 1.05989046331897 | . R 3 , 2 ( t ) = | 2.39157408129154 t − 5.52473124718345 t 3 + 3.66274119606259 t 4 − 6.32325173422324 t 2 − 9.01207243728546 | .

The CPU time obtaines was 72 .588  s.

With three terms of the series, the maximum error of 41 .8800033922238 for prey and 19 .9989691500608 for predator equations have been obtained in the interval [ 0 , 2 ] and in the sub-interval [ 0 , 0 .5 ] , the maximum errors for prey and predator equations reduce to 10 .6197153605001 and 9 .85876839673482 , respectively. We continue the process with four terms of the series and obtain the following results:

C 1 = − 0 .622400688541859 , C 2 = 0 .182662742935547, C 3 = − 0 .659354586938123 , C 4 = 0 .205217668390281 .

n 4 , 1 ( t ) = 10 − 8.10952932555767 t + 0.973118939280965 t 3 − 0.284061019913224 t 2 , n 4 , 2 ( t ) = 10 + 8.76221541739426 t − 1.03200599446780 t 3 + 0.229509341321545 t 2 ,

R 4 , 1 ( t ) = | − 0.581094737027440 t − 1.79770689792700 + 0.695262955226051 t 3 + 4.42325516525786 t 2 + 0.105611055852865 t 6 − 1.76280911739599 t 4 − 0.68412074614225510 − 1 t 5 | . R 4 , 2 ( t ) = | − 0.506851731552217 t + 0.666998416869458 t 3 + 4.53293489469946 t 2 + 0.105611056662164 t 6 − 1.76313576696487 t 4 − 0.68412068637217510 − 1 t 5 − 2.03237147099184 | .

The CPU time was 41 .746  s.

The maximum error obtained with four terms of the series is 3.33979676054971 and 3 .21859886179310 for prey and predator equations, in the interval [ 0 , 2 ] , but reduced to 1 .81663044019252 and 2 .04644199082422 in the interval [0, 0.5], respectively. Under the determined conditions, but with five terms of the series, the following results have been obtained:

C 1 = − 0 .946705279372507 , C 2 = 0 .0861498282686004, C 3 = − 0 .411900733385533 , C 4 = − 0 .16370425547960010 − 1 ,

n 5 , 1 ( t ) = 10 − 9.83501449286814 t − 0.700201121060218 t 4 + 0.446636860098806 t 2 + 2.42557815465261 t 3 . n 5 , 2 ( t ) = 10 + 9.71603350977239 t + 0.306971915546683 t 4 − 0.917926148085826 t 2 − 1.34077869296380 t 3 ,

R 5 , 1 ( t ) = | − 1.48760227614014 t − 3.53739031130883 t 4 + 0.287504641165046 t 6 + 1.29071465923440 t 5 + 0.22603957155346310 − 1 t 8 − 0.181053360894281 t 7 + 3.17535752664663 t 2 − 0.24121927227619310 − 1 + 0.658526539518602 t 3 | , R 5 , 2 ( t ) = | − 2.04281072755163 t − 3.58180448601210 t 4 − 1.16119802822819 + 0.288413657834328 t 6 + 1.29526379540339 t 5 + 0.22603889855151610 − 1 t 8 − 0.181052041142108 t 7 + 6.92704248906346 t 2 − 1.05959288789440 t 3 | .

The solution of the system by OHAM with five terms result in the maximum error of 0 .687019929650256 for prey and 1 .317324181842 for predator equations in the interval [0, 2]. Also, in the sub-interval [ 0 , 0 .5 ] , the maximum error of prey and predator equations reduce to 0 .199853329620497 and 1 .31732418184292 , respectively. The curves of the solution and error equations have been illustrated in Figure 3(a) and (b).

Figure 3 (a) The curves of n1 and n2 obtained by OHAM with five terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with five terms, without step in interval [0, 2].
Figure 3

(a) The curves of n1 and n2 obtained by OHAM with five terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with five terms, without step in interval [0, 2].

The CPU time was 74 .724  s.

By following the procedure, the required values and equations are obtained with six terms.

C 1 = − 0 .364479654852489 , C 2 = 0 .448894466905287, C 3 = − 0 .284425121894691 , C 4 = 0 .400484523307263 ,

n 6 , 1 ( t ) = − 9.15623466501711 t + 0.502816060936311 t 4 + 2.35351770941634 t 3 + 10 . − 0.468452031382185 t 5 − 1.04607979487628 t 2 . n 6 , 2 ( t ) = 10.1481342795450 t − 0.387113865552237 t 4 − 2.26159312645905 t 3 + 10 . + 0.409547110975836 t 5 + 0.766325299628079 t 2 ,

R 6 , 1 ( t ) = | 0.526161807387484 t − 0.183088635454630 t 8 − 2.20426718948901 t 4 − 0.145665886988149 t 3 + 0.20170731417707010 − 1 t 10 − 1.31876761255119 t 5 + 0.333400524057927 t 7 + 1.30332367126440 t 6 − 0.45448519114240310 − 1 t 9 + 2.73125562757159 t 2 − 0.688130359321417 | ,

R 6 , 2 ( t ) = | 0.40871758384048910 − 1 t − 0.183465454735769 t 8 − 2.72085599192724 t 4 + 0.623593926640201 t 3 + 0.20135963803932710 − 1 t 10 − 1.45854176571276 t 5 + 0.324671881723085 t 7 + 1.32643079089148 t 6 − 0.45324707492017210 − 1 t 9 + 3.46981693399108 t 2 − 0.692766786495488 | .

The CPU time was 212 .738  s.

The solution with six terms of the series have got the maximum error of 9 .25752858556399 for prey and 4 .94535527028125 for predator equations in the interval [ 0 , 2 ] whereas in the sub-interval [ 0 , 0 .5 ] , the maximum error decreases to 0 .688130359321412 and 0 . 6927667864954 for prey and predator equations, respectively.

Finally, the following results have been derived with seven terms of the series,

C 1 = − 0 . 800352671636881 , C 2 = 0 .679720963775363, C 3 = − 0 . 194786484804675 , C 4 = 0 .338118465557743 .

n 1 7 , 1 ( t ) = 10 . − 5.95714727396967 t − 0.132691990644162 t 5 + 0.00497398235120272 t 6 − 0.0796712527517229 t 3 − 0.444383128075060 t 4 + 5.04863867792301 t 2 . n 2 7 , 2 ( t ) = 10 . − 4.93112943379674 t + 0.0361461624459025 t 5 − 0.00116817345815434 t 6 − 0.0867129289545119 t 3 + 0.0691586786215339 t 4 + 0.512098991312087 t 2 ,

R 7 , 1 = | − 1.42115840861149 t + 4.4702162253490510 − 7 t 12 − 0.56987421963832610 − 1 t 5 − 0.87077442392921210 − 2 t 6 + 0.23224809813908710 − 1 t 8 + 0.85206549715798010 − 1 t 7 − 6.43511772956537 t 3 + 0.16567269724103210 − 2 t 9 − 0.321977469217517 t 4 + 0.11915913599655810 − 3 t 11 − 0.59207819421394810 − 3 t 10 + 4.59142588665071 + 9.10133911007724 t 2 | . R 7 , 2 = | 13.0432831293776 t + 0.14614458004359310 − 5 t 12 + 0.86291620523326510 − 1 t 5 − 15.6418293632301 − 0.104110795824955 t 6 + 0.32193224074159010 − 2 t 8 + 0.54730009151480210 − 1 t 7 + 3.41899927798512 t 3 + 0.63479103491410510 − 2 t 9 − 0.243357206321491 t 4 − 0.93592783465433010 − 4 t 11 + 0.16716597788172910 − 2 t 10 − 9.94611738712655 t 2 | .

The CPU time was presented as 528 .594  s.

The case with seven terms has bad situation and the maximum error for prey and predator equations in the interval [ 0 , 2 ] change to 5 .85491204197720 and 15 .6418293632301 , respectively. Specially, in the sub-interval [ 0 , 0 .5 ] , the maximum errors for prey and predator equations increase to 5 .33051035037122 and 15 .6418293632301 .

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 H 1 = C 1 p + C 2 p 2 + C 3 p 3 , H 2 = C 4 p + C 5 p 2 + C 6 p 3 are derived as follows:

C 1 = − 0 . 908298692531751 , C 2 = 0 .161655167489432, C 3 = 0 .0521677545964287 , C 4 = − 0 . 498748546837712, C 5 = − 0 .52226167669950610 − 1 , C 6 = 0 .0527856806175587 .

n 6 , 1 ( t ) = − 9.61583452671766 t − 0.255143066111486 t 4 − 0.165728111514909 t 5 + 10 + 2.53460707332124 t 3 − 0.370811995118856 t 2 . n 6 , 2 ( t ) = 10.3600745862350 t + 0.410263812627818 t 4 + 0.89417427065313910 − 1 t 5 + 10 − 2.61118589752004 t 3 + 0.199829630088061 t 2 ,

R 6 , 1 ( t ) = | − 0.654986154517360 t − 4.73083171596920 t 4 + 0.500713645295256 t 5 − 0.216913197113499 + 2.21759359359414 t 3 + 2.61248948617693 t 2 + 0.15647471773742910 − 2 t 10 + 0.92747678531039910 − 2 t 9 − 0.60025055962493910 − 1 t 8 − 0.167393747662853 t 7 + 1.01126670436202 t 6 | ,

R 6 , 2 ( t ) = | − 0.802142222431769 t − 5.15849915168121 t 4 + 0.492777249925950 t 5 + 2.72327312751350 t 3 − 0.504354377047661 + 3.10061566350234 t 2 + 0.15588778964952910 − 2 t 10 + 0.92670769575903410 − 2 t 9 − 0.59620053286795710 − 1 t 8 − 0.158143614974671 t 7 + 1.01196826219735 t 6 | .

The CPU time obtained was 704.657  s.

The solution of the system by OHAM with six terms and the new auxiliary function results show that the maximum error will be 1 .27258768359967 for prey equation and 0 .9147812136876 for predator equation in the interval [ 0 , 2 ] . And in the sub-interval [ 0 , 0 .5 ] , the maximum error of prey and predator equations will change to 0 .255131456409634 and 0 .5523528852 , respectively. Inserting the obtained solution, the curves of solution and error equations have been illustrated in Figure 4(a) and (b). However, by using MOHAM under the equal conditions, a better result has been obtained, but because of the maximum error 0.699103281349455 for prey and 0.113245187385573 for predator in the interval [0, 2], it can be recognized as unsuitable. We have illustrated the solution and error curves in Figure 5(a) and (b).

Figure 4 (a) The curves of n1 and n2 obtained by OHAM with six terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with six terms, without step in interval [0, 2].
Figure 4

(a) The curves of n1 and n2 obtained by OHAM with six terms, without step in interval [0, 2]. (b) The error curves of system equations by inserting the OHAM solution with six terms, without step in interval [0, 2].

Figure 5 (a) The curves of n1 and, n2 by MOHAM with six terms and step 0.5 in interval [0, 2]. (b) The error curves of system by MOHAM solution with six terms and step 0.5 in interval [0, 2].
Figure 5

(a) The curves of n1 and, n2 by MOHAM with six terms and step 0.5 in interval [0, 2]. (b) The error curves of system by MOHAM solution with six terms and step 0.5 in interval [0, 2].

So, we continue the process with H 1 = C 1 p + C 2 p 2 + C 3 p 3 + C 4 p 4 , H 2 = C 5 p + C 6 p 2 + C 7 p 3 + C 8 p 4 , and initially six terms of expansion by OHAM. The solution of the system by OHAM with six terms results in the maximum error of 1 .37554532150794 for prey and 1 .53164897082781 for predator equation in the interval [ 0 , 2 ] . And in the sub-interval [ 0 , 0 .5 ] , the maximum error of prey and predator equations will change to 0 .671180090226115 and 0 .378985492469005 , respectively. The curves of solution and error equations have been illustrated in Figure 6(a) and (b).

C 1 = − 0 . 721981352182221 , C 2 = 0 .170087909027642, C 3 = 0 .00808105572634990 , C 4 = − 0 .71700975756005410 − 1 , C 5 = − 0 . 722959629895041 , C 6 = 0 .0615223037596436 , C 7 = 0 .137314519001125 , C 8 = − 0 .51595032464992510 − 1 .

n 6 , 1 ( t ) = − 9.81129936207435 t + 0.323344638920389 t 4 − 0.467599061783403 t 2 + 10 . + 2.09714030113537 t 3 − 0.328831069638795 t 5 . n 6 , 2 ( t ) = 10.5257409258193 t − 0.315951984026469 t 4 + 0.200134441722184 t 2 + 10 . − 2.15431687491946 t 3 + 0.329345118129254 t 5 .

R 6 , 1 ( t ) = | − 0.456055577413119 t − 2.77491863871660 t 4 + 4.51048438957523 t 2 − 0.247189548223170 t 3 − 0.925337575396254 t 5 + 0.175744089637865 t 7 + 1.12661649751849 t 6 − 0.131129496131572 t 8 − 0.24297741569921410 − 1 t 9 + 0.11384008247019210 − 1 t 10 − 0.13593130331127110 − 1 | . R 6 , 2 ( t ) = | − 0.789856929264310 t − 2.94375450222681 t 4 + 5.00569051014171 t 2 − 0.125732015599173 t 3 − 0.946968348796092 t 5 + 0.189284991174110 t 7 + 1.13498032192209 t 6 − 0.132330096534341 t 8 − 0.24988451190199310 − 1 t 9 + 0.11375521150423810 − 1 t 10 − 0.347645298474181 | .

Figure 6 (a) The curves of n1 and n2 by OHAM with six terms, without step in the interval [0, 2]. (b) The error curves of system by OHAM solution with six terms, without step in the interval [0, 2].
Figure 6

(a) The curves of n1 and n2 by OHAM with six terms, without step in the interval [0, 2]. (b) The error curves of system by OHAM solution with six terms, without step in the interval [0, 2].

The CPU time was 1489 .872 .

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 0.191023436433359 for prey equation and 0.274730221685456 for predator equation in the interval [ 0 , 2 ] and in the sub-interval [ 0 , 0 .5 ] , the maximum error of prey and predator equations will change to 0 .0774280147649515 and 0 .199802973776686 , respectively. We have illustrated the curves of solution and errors in Figure 7(a) and (b).

C 1 = − 0 . 970635297578064 , C 2 = − 0 .138241462318531 × 10 − 2 , C 3 = − 0 .36565883942957910 − 1 , C 4 = − 0 .17035051161817210 − 1 , C 5 = − 0 . 435542970312829 , C 6 = − 0 .13092495339604510 − 1 , C 7 = − 0 .18087204209165710 − 1 , C 8 = − 0 .14283360959984610 − 1 .

n 7 , 1 ( t ) = − 9.79489533280808 t − 0.546302093246667 t 5 + 0.169821474338011 t 6 + 10 . + 3.98153426365207 t 3 − 0.866396084418935 t 2 − 0.744242058072562 t 4 , n 7 , 2 ( t ) = 10.8691450180496 t + 0.253725432208026 t 5 − 0.76275691627021010 − 1 t 6 + 10 . − 3.34114495837199 t 3 + 0.37546418952799110 − 1 t 2 + 0.753951964540658 t 4 .

R 7 , 1 ( t ) = | − 0.54862859347198510 − 1 t + 0.268046384079861 t 5 + 2.37244887425349 t 6 + 4.21658909113255 t 3 − 0.455273040202060 t 2 − 5.33199712459230 t 4 − 0.275457465823406 t 8 + 0.155194088190006 t 9 − 0.780634310667769 t 7 − 0.92769648681976010 − 2 t 11 − 0.26199891297359410 − 2 t 10 − 0.13713721170279310 − 1 + 0.13621972577584110 − 2 t 12 | . R 7 , 2 ( t ) = | − 1.55877346179692 t + 0.848081889645843 t 5 + 2.39272525562086 t 6 + 3.98883152164637 t 3 + 2.48511132303785 t 2 − 6.98354236610569 t 4 − 0.280400647186822 t 8 + 0.155858813164666 t 9 − 0.772843687664357 t 7 − 0.92769648681958910 − 2 t 11 − 0.26133523962673110 − 2 t 10 + 0.13621972577584510 − 2 t 12 − 0.17890424115606410 − 2 | .

Figure 7 (a) The curves of n1 and n2 by OHAM with seven terms, without step in the interval [0, 2]. (b) The error curves of system by OHAM solution with seven terms, without step in the interval [0, 2].
Figure 7

(a) The curves of n1 and n2 by OHAM with seven terms, without step in the interval [0, 2]. (b) The error curves of system by OHAM solution with seven terms, without step in the interval [0, 2].

The CPU time was 12855 .293 .

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 1.37554532150794 and 1.53164897082781 for six terms and 0.191023436433359 and 0.274730221685456 for seven terms in the interval [ 0 , 2 ] , respectively.

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 H 1 = C 1 p + C 2 p 2 + C 3 p 3 + C 4 p 4 , H 2 = C 5 p + C 6 p 2 + C 7 p 3 + C 8 p 4 , and six terms of expansion by MOHAM. Also, we have used four steps with the length of each step of 0.5 in the interval [ 0 , 2 ] .

The solution of the system by MOHAM with six terms result in the maximum error of 0 .0765758832214998 for prey equation and 0 .0815389676280886 for predator equation in the interval [ 0 , 2 ] . As well as the solution in each sub-interval with maximum error of prey and predator equations have been presented as follows (Figure 8):

n 1 1 ( t ) = − 9.65707300646136 t − 0.963988155199331 t 4 + 0.54588237686402910 − 1 t 5 + 10 + 2.93725223964248 t 3 − 0.53179430055024610 − 1 t 2 , t ∈ [ 0 , 0.5 ] ,

Figure 8 (a) The curves of n1 and n2 by MOHAM with seven terms, without step in the interval [0, 0.5]. (b) The error curves of system by MOHAM solution with seven terms, without step in the interval [0, 0.5].
Figure 8

(a) The curves of n1 and n2 by MOHAM with seven terms, without step in the interval [0, 0.5]. (b) The error curves of system by MOHAM solution with seven terms, without step in the interval [0, 0.5].

The maximum error of n 1 1 ( t ) = 0 .82997704456536510 − 3 .

n 1 2 ( t ) = − 11.96655606 t + 10.35396940 − 0.2786262463686610 − 4 t 5 + 0.53596507810 − 2 t 4 + 0.97945399910 − 1 t 3 + 3.739841190 t 2 , t ∈ [ 0.5 , 1 ] ,

The maximum error of n 1 2 ( t ) = 0 .0765758832214998 .

n 1 3 ( t ) = − 11.21107563 t + 10.33822212 + 0.394974755628339 t 5 − 2.334742779 t 4 + 4.241109907 t 3 + 0.802043346 t 2 , t ∈ [ 1 , 1.5 ] ,

The maximum error of n 1 3 ( t ) = 0 .000816642143110113 .

n 1 4 ( t ) = − 17.11236387 t + 11.85608978 + 0.2813311680 t 4 − 2.666788986 t 3 + 9.869270438 t 2 + 0.31969949250798510 − 3 t 52 , t ∈ [ 1.5 , 2 ] ,

The maximum error of n 1 4 ( t ) = 0 . 27504627078475610 − 3 .

n 2 1 ( t ) = 10.8735773688232 t − 0.447784709274623 t 4 + 0.794895126130086 t 2 + 10 + 1.27752371326862 t 5 − 3.89925672573697 t 3 , t ∈ [ 0 , 0.5 ] ,

The maximum error of n 2 1 ( t ) = 0 .18296727268418710 − 2 .

n 2 2 ( t ) = 13.15253298 t + 9.635028074 + 0.53223188549977910 − 4 t 5 − 4.140437644 t 2 − 0.955905592010 − 2 t 4 − 0.1243833924 t 3 , t ∈ [ 0.5 , 1 ] ,

The maximum error of n 2 2 ( t ) = 0 .0815389676280886 .

n 2 3 ( t ) = 12.51286437 t + 9.605709593 − 0.425103037673921 t 5 − 1.186843360 t 2 + 2.511948624 t 4 − 4.505342012 t 3 , t ∈ [ 1 , 1.5 ] ,

The maximum error of n 2 3 ( t ) = 0 .63199608365107010 − 3 .

n 2 4 ( t ) = 18.98629941 t + 7.936078858 − 0.3245151456 t 4 + 3.017070295 t 3 − 11.09939425 t 2 + 0.79641394022650210 − 3 t 5 , t ∈ [ 1.5 , 2 ] ,

The maximum error of n 2 4 ( t ) = 0 .27233433455061510 − 3 .

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.

Table 1

Comparison of solution and error for prey equation in a few points by OHAM with H 1 = C 1 p + C 2 p 2 + C 3 p 3 + C 4 p 4 , six terms and MOHAM with step length 0.5

t Prey
OHAM Eeq(OHAM) MOHAM Eeq(MOHAM) Eeq(OHAM)- Eeq(MOHAM)
0 10 0 10 0 0
0.5 5 .32271148694077 0 .07742801476495 5 .46678179193861 0 .00006309482132 0 .07736491994363
1 2 .19952016944388 0 .08980515911272 2 .23053171811336 0 .07657588759930 0 .01322927151342
1.5 0 .81410974221815 0 .03160391072340 0 .81965637149020 0 .00081662740260 0 .03078728332080
2 0 .27593355050310 0 .19102343643453 0 .28566097576026 0 .00027516586700 0 .19074827056753
Table 2

Comparison of solution and error for predator equation in a few points by OHAM with H 2 = C 5 p + C 6 p 2 + C 7 p 3 + C 8 p 4 , six terms and MOHAM with step length 0.5

t Predator
OHAM Eeq(OHAM) MOHAM Eeq(MOHAM) Eeq(OHAM)-Eeq(MOHAM)
0 10 0 10 0 0
0.5 15 .0801751038251 0 .04071200295161 15 .1600414469370 0 .00182967266002 0 .03888233029159
1 18 .4969481837521 0 .26273147794353 18 .5132341848685 0 .08153897127920 0 .18119250666433
1.5 19 .9866142565905 0 .12134588905862 19 .9876930141637 0 .00063199530280 0 .12071389375582
2 20 .4601170441125 0 .26553817340842 20 .4809059544872 0 .00027213250700 0 .26526604090142

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 3.33979676054971 , 0 .687019929650256 , and, 9 .25752858556399 respectively, and similarly 3 .21859886179310 , 1 .317324181842 and 4 .94535527028125 for predator equation in the interval [ 0 , 2 ] . With the increase in the number of auxiliary polynomial terms to three, we have obtained the polynomial solutions with six terms and the maximum errors of 1 .27258768359967 and 0 .9147812136876 in the interval [ 0 , 2 ] and 0 .255131456409634 and 0 .5523528852 in the interval [ 0 , 0.5 ] for prey and predator equations, respectively. So, in this situation, we tend to apply MOHAM. But by using MOHAM, the maximum errors are converted to 0.699103281349455 and 0.113245187385573 in the interval [ 0 , 2 ] , which are not suitable.

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 1 .37554532150794 and 1 .53164897082781 and with seven terms 0.191023436433359 and 0.274730221685456 for prey and predator equations in the interval [ 0 , 2 ] , respectively. It was natural that the solution with seven terms was desirable to be applied MOHAM. But, because of the CPU time of solution with seven terms was illustrated as 12855 .293s vs 1489 .872s for the solution with six terms, we decided to use the solution with sex terms for MOHAM. The result was excellent. The maximum errors for prey and predator equations have been 0 .0765758832214998 and 0 .0815389676280886 , respectively. Also, the results in a few points have been illustrated in Tables 1 and 2 which refer to the accuracy of the solution by MOHAM along with the emphasis on how the selection process of the auxiliary polynomials and the series terms.

  1. Funding information: The author states no funding involved.

  2. Author contributions: All author has accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2021-07-21
Revised: 2022-06-18
Accepted: 2022-07-17
Published Online: 2023-02-17

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0230
Keywords for this article
optimal homotopy asymptotic method; multistep optimal homotopy asymptotic method; system of integro-differential equations; biological species
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