Approximate solution of a model describing biological species living together by Taylor collocation method

Headnote

Abstract: In this paper, a numerical method is presented to obtain approximate solutions for the system of nonlinear delay integrodifferential equations derived from considering biological species living together. This method is essentially based on the truncated Taylor series and its matrix representations with collocation points. Also, to illustrate the pertinent features of the method examples are presented and results are compared to the Adomian decomposition method, the variational iteration method, pseudospectral Legendre method. All numerical computations have been performed on the computer algebraic system Maple 15.

Keywords: System of nonlinear integro-differential equations, Taylor polynomials and series, Biological species ,Collocation points.

(ProQuest: ... denotes formulae omitted.)

1 Introduction

In science and engineering some important problems can usually be modeled to a system of integral and integro-differential equations (IDE). IDEs and their systems play significant role in biology, ecology, medicine, physics such as population growth, one dimensional viscoelasticity, electro magnetic theory and reactor dynamics [1]-[6]. Since few of these equations can be solved explicitly, it is needed to develop the numerical techniques to solve IDEs and their systems [7,8]. This paper is concerned with the dynamic of two interacting species which was first modeled by Volterra [9]. It is considered two separate species with numbers y1(t) and y2(t) at time t where first species increases and second one decreases. If they are put together, assuming that the second species will feed on the first, there will be increase in the rate of the second species dy2 dt which depends not only on the present population y1(t) but also on all previous values of the first species. When a steady-state condition or equilibrium is reached between these two species, it is described by the following system of nonlinear delay Volterra integro-differential equations:

... (1)

where ... b with initial conditions

y1(0) =a1; y2(0) =a2 (2)

where h1 and -h2 are coefficients of increase and decrease of the first and the second species, respectively. The parameters f1; f2;g1 and g2 are given functions while y1 and y2 are unknown functions and T0 ? R is assumed to be the finite heredity duration of both species [10].

Up to now, Adomian decomposition method (ADM) [11], rational Chebyshev tau method (CTM)[12], variational iteration method (VIM), pseudospectal Legendre method (PLM) [13] and Legendre multiwavelet method (LVM) [14] have been used to solve the model (1)-(2).

The main purpose of this study is to solve equations (1)-(2) using the Taylor matrix method. Since the beginning of 1994, Taylor, Chebyshev, Legendre, Laguerre, Hermite and Bessel collocation and matrix methods have been used by Sezer et al.[15]-[25] to solve differential, difference, integral , integro-differential, delay differential equations and their systems. In this article, by modifying and developing matrix and collocation methods studied in [15,17,20], we will find the approximate solutions of the system (1)-(2) in the truncated Taylor series form

... (3)

where yi;n; (n = 0;1; : : : ;N; i = 1;2) are unknown coefficients to be determined.

2 Fundamental relations

Let us consider the system of nonlinear Volterra integro-differential equations in the form (1) and find matrix representations of each term in the system. First we convert the solution defined by (3) and its first derivative, for n = 0;1; : : : ;N to the following matrix forms:

... (4)

... (5)

where

...

On the other hand, we can write the matrix forms of the expressions y1(t)y2(t), y1(t)y2(t ) and y1(t )y2(t) as, respectively,

... (6)

... (7)

and

... (8)

where

...

Now, we convert the kernel functions

...

to the matrix forms, by means of following procedure.

The functions K1 = f1 and K2 = f2 can be expressed by the truncated Taylor series, respectively,

... (9)

... (10)

where

...

The expressions (9) and (10) can be written in the matrix forms

... (11)

... (12)

where

...

So that Ki = [ ki m;n ] ; m;n = 0;1; : : : ;N; i = 1;2 are the Taylor coefficients matrices of functions Ki(t;t ) = fi(t -t ); i = 1;2 at point (0;0):

For example, in case N = 2, the matrix K1 becomes

...

where

...

3 Fundamental matrix equations for the system

We now ready to construct the fundamental matrix equations for the system of nonlinear delay integro-differential equations (1). For this purpose, substituting the matrix relations (4)-(8), (11) and (12) into system (1) and simplifying, we obtain the two matrix equations.

...

and

...

or the system

...

where

...

Therefore, we can write the matrix representation of the system (13) in the form

...

where

...

and

Consequently, the fundamental matrix equations of the system (14) can be written in the following compact form

...

where

...

4 Taylor matrix-collocation method

In this section, by substituting the collocation points defined by

...

into the fundamental matrix equation (15), we obtain the new system

D(ts)Y+A(ts)Y = G(ts); s = 0;1; : : : ;N (16)

and therefore, the new fundamental matrix equation

DY* +AY = G (17)

where

...

By using equation(4), we obtain the matrix forms of the conditions as

T(0)Y1 =a1 (18)

and

T(0)Y2 =a2: (19)

By replacing the row matrices (18) and (19) by two rows of the matrix equation (17), we have the matrix equation based on the conditions. Thus, the fundamental matrix equation of the system (1) under initial conditions (2) corresponds to a system of 2(N +1) nonlinear algebraic equations with the unknown coefficients y1;n and y2;n; (n = 0;1; : : : ;N): Finally, the unknown coefficients are computed by solving this system and they are substituted in equation (3). Hence, the Taylor polynomial solutions

... (20)

can be obtained.

5 Accuracy of solutions

Accuracy of the approximate solutions are checked by substituting this solutions into the system (1); that is, for t ? [0;1],

... (21)

We expect that Ei;N(t) = 0 on collocation points. The closer yi(t) ~= yi;N(t) the closer Ei;N(t) ~= 0 for t ? [0;1].

6 Numerical examples

In this section, some numerical examples are given to illustrate the accuracy and efficiency of the presented method (PM). The values of exact solutions yi(t); i = 1;2; and the absolute error functions ei;N(t) = |yi(t)-yi;N(t)| ; i = 1;2; are presented at selected points of the given interval. Results are shown with tables and figures. All of them were performed on the computer using a program written in Maple 15.

Example 1. As the first example, consider the system (1)-(2) with ... and ... . The exact solutions of this system are in the following form y1(t) = t, y2(t) = 1: By applying the presented method in Section 4 for N = 2, we have the solutions ... 1 which are the exact solutions.

Example 2. [11], [13], [14] Now we consider the system (1)-(2) with ..., and ... 1. The exact solutions of this system are in the following form

...

We obtain the approximate solutions by Taylor polynomials of the problem for N = 5;6;7: In Tables 1-2 and Figure 1, the absolute errors obtained by the present method for N = 5 are compared with the results obtained by variational iteration method [13], Adomian decomposition method and pseudospectral Legendre method [11]. It is seen from this tables and figures that the present method is closer to exact solution than the other methods. Additionally in Tables 3-4 and Figure 2, the absolute errors for the present method are shown for different values of N. These datas show that as N increases, than the absolute errors decrease more rapidly.

Example 3. [13] In this example, we solve the system (1)-(2) with ... and ... are the exact solutions of this system.

Using the procedure in section 4, we calculate the approximate solutions y1;N(t) and y2;N(t) for N = 7;8;9: The exact solutions, absolute errors obtained by other methods and the present method are given in Tables 5-6 and Figure 3. On the other hand, in Tables 7-8 and Figure 4, the absolute errors for the present method are shown for different values of N.

Additionally, in Table 9, the accuracy of solutions are stated. These results show that present method is closer to exact solution than the other methods and if N increases, than the absolute errors decrease more rapidly.

7 Conclusion

In this study, a new Taylor matrix-collocation method is proposed for numerical solutions of a model describing biological species living together. It is observed from Figures and Tables that the method is a simple and powerful tool to obtain the approximate solution. When the numerical experiments are analyzed and the results are compared, it is seen that, the present method is quite effective. Additionally, if is increased, it can be seen that approximate solutions obtained by the mentioned method are closed to the exact solutions. One of the considerable advantage of the method is founding the approximate solutions very easily by using the computer program written in Maple 15. Shorter computation time and lower operation count results in a reduction of cumulative truncation errors and improvement of overall accuracy. In addition, the method can also be extended to other models in the future.

References

References

[1] Kot, M., Elements of Mathematical Ecology, Cambridge University Press, (2001).

[2] Pougaza, D. B., The Lotka integral equation as a stable population model, Postgraduate Essay, African Institute for Mathematical Sciences (AIMS), (2007).

[3] Kopeikin I.D., V.P. Shishkin, Integral form of the general solution of equations of steady-state thermoelasticity, Journal of Appl. Math. Mech. (PMM U.S.S.R.), 48(1), (1984), 117-119.

[4] Lotka, A. J., On an integral equation in population analysis, Ann. Math. Stat., 10, (1939), 144-161.

[5] Bloom, F., Asymptotic bounds for solutions to a system of damped integro-differential of electromagnetic theory, J Math Anal Appl., 73, (1980), 524-542.

[6] Abdou, M.A., Fredholm-Volterra integral equation of the first kind and contact problem, Appl. Math. and Comput., 125,(2002), 177-193.

[7] Baker, C.T.H., A perspective on the numerical treatment of Volterra equation, J Comput Appl Math.,125, (2000), 217-249.

[8] Linz, P., Analytical and numerical methods for Volterra equations, Philadelphia (PA): SIAM, (1985).

[9] Volterra, V., Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Memorie del R. Comitato talassografico italiano, (1927).

[10] Jerri, A.J., Introduction to integral equations with applications, John Wiley and Sons Inc., (1999).

[11] Babolian, E., Biazar, J., Solving the problem of biological species living together by Adomian decomposition method, Appl. Math. and Comput., 129, (2002), 339-343.

[12] Parand, K., Razzaghi M., Rational Chebyshev tau method for solving Volterra's population model, Appl. Math. and Comput., 149, (2004), 893-900.

[13] Shakeri, F., Dehghan M., Solution of a model describing biological species living together using the variational iteration method, Mathematical and Computer Modelling, 48, (2008), 685-699.

[14] Yousefi, S. A., Numerical Solution of a Model Describing Biological Species Living Together by Using Legendre Multiwavelet Method, International Journal of Nonlinear Science, 11, (2011), 109-113.

[15] Sezer, M., Taylor polynomial solutions of Volterra integral equations, Int. J. Math. Educ. Sci. Technol., 25(5), (1994), 625-633.

[16] Akyüz A., Sezer M., Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients, Appl. Math. Comput., 144, (2003), 237-247.

[17] Gülsu M, Sezer M, Taylor collocation method for solution of systems of high-order linear Fredholm-Volterra integro-differential equations, Int. J. Comput. Math., 83(4), (2006), 429-448.

[18] Kurt N, Sezer M, Polynomial solution of high-order linear Fredholm integro-differential equations with constant coefficients, Journal of Franklin Institude, 345(8), (2008), 839-850.

[19] Yalçinbas S., Sezer M., Sorkun H., Legendre polynomial solutions of high-order linear Fredholm integro-differential equations, Appl. Math. Comput., 210, (2009), 334-349.

[20] Bülbül B., Gülsu M., Sezer M., A new Taylor collocation method for non-linear Fredholm-Volterra integro-differential equations, Journal of Numerical Methods Partial Differential Equations, 26, (2010), 1006-1020.

[21] Gülsu M., Gürbüz B., Öztürk Y., Sezer M., Laguerre polynomial approach for solving linear delay difference equations, Appl. Math. Comput., 217, (2011), 6765-6776.

[22] Yalçinbas S., Aynigül M., Sezer M., A collocation method using Hermite polynomials for approximate solution of pantograph equations, Journal of the Franklin Institute, 348, (2011), 1128-1139.

[23] Isik O.R., Sezer M., Güney Z., Bernstein series solution of a class of linear integro-differential equations with weakly singular kernel, Appl. Math. Comput., 217(16), (2011), 7009-7020.

[24] Yüzbas 'y S ., S ahin N., Sezer M., A collocation approach for solving modeling the pollution of a system of lakes, Mathematical and Computer Modelling, 55(3-4), (2012), 330-341.

[25] Gülsu M., Öztürk Y., Numerical approach for the solution of hypersingular integro-differential equations, Appl. Math. Comput., 230, (2014), 701-710.