CODEN (USA): IJCRGG, ISSN: 0974-4290, ISSN(Online):2455-9555 Vol.11 No.07, pp 239-246, 2018

1Research in Nonlinear Differential Equations (GEDNOL), Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia 2Research in Statistics and Epidemiology (GIEE), Department of Mathematics, Universidad Tecnológica de Pereira, Pereira, Colombia 3GIEE, Fundación Universitaria del Área Andina, Pereira, Colombia

Abstract : The approximate interpretation of some natural phenomena has led to introduce in certain types of differential equations changes in the temporal variable called delays, which makes these equations and their solutions have a more consistent behavior with reality. These equations, called differential equations with delay require complex methods for their solution and in most cases, only a numerical approximation is achieved. In this article we initially show a theoretical development on the decomposition method applied to ordinary differential equations with delay in which the most important properties were studied. Subsequently, the most relevant Adomian polynomials were tested; some biological models that involve differential equations with delay and integro-differential equations were solved. Finally, the numerical comparison was made with other approximation methods and the convergence of the method in some solutions was analysed. Keywords: Biomathematical Models, Adomian Method.

The Adomian Decomposition Method is of great importance in the resolution of non-linear differential equations with initial value and at the border, with the need to know methods that are generally of

1,2,3

the semi-analytical numerical type using solutionsof the form ቲቛትቜቭዤ ቲሂ. In general, this method

ሂቡቕ

consists of converting a differential equation of real domain into a simpler one (iterative equation) of natural domain. We use the definition and a series of theorems that are applied depending on the characteristic of each of the terms of the differential equation to transform. In essence, this method has its reason for being in the so-called Adomian polynomials and their serial solution, since it is sought to reach an infinite series that represents the approximation of the solution of the equation4,5 .

Pedro Pablo Cárdenas Alzate et al /International Journal of ChemTech Research, 2018,11(07):239-246 DOI= http://dx.doi.org/10.20902/IJCTR.2018.110729

The Adomian decomposition method basically consists of a process of semi-analytic numerical type which is applied to all the members of the equation under study, iterating with the values of the variable k in order to obtain the terms ቲቕሂቲቖሂቲሂሆnecessary to obtain an approximate solution ቲቛትቜalways bearing in mind the existence of a reverse differential operator.

The theory and application6 of differential equations with delay is a very important topic within the dynamic systems, physics and applied mathematics, as well as a particular case of functional differential equations. It deals with models in which the functions at each instant depend not only on a time t, but also on the values for previous instants, which means that the unknown function and its derivatives are evaluated in different arguments. Some important applications are presented in models of population dynamics, heat transfer, biological controls among others. As an interesting example, in the logistic equation it is assumed that the birth rate of a certain organism depends instantaneously on changes in population size, however there are multiple phenomena that affect the process generating a delay. All events of nature require the existence of the time variable to occur or complete a cycle. Some of them take so little time that they do not generate perceptible drawbacks, while others such as those that compromise biological7,8 processes are highly affected in data collection and in scientific measurement. This subtle observation detected long ago was the reason why in 1948 Hutchinson modified his equation introducing a delay in the growth rate.

In general, the main advantage of the method is the fact that it provides an analytical approach, in many cases an exact solution in a rapidly convergent series. The current results both in the theoretical and computational part show that differential equations with delay are a revealing model of the complex dynamics present in nature9,10 .

Experimental

11,12

Given the equation ቲቭብቛቱቜ, with F is a nonlinear operatorwith linear and nonlinear terms, we can write it in the form ቲቦቕቲቦቑቲቭብ. Here, L is the operator for the highest ordered derivative, R is the remainder of the linear term and N is the nonlinear term. Now, applying the inverse operator በቖ we obtain

በቖቲቭበቖብቧበቖቕቲቧበቖቑቲ,

where በቖ is the n-fold definite integration operator from 0 to t.

Therefore, the Adomian method show that the solution ቲቛቱቜcan be written as a series

ቧበቖቕዤ ቧበቖዤ

ቲቛቱቜቭዤ ቲሂቭቲቕ ቲሂ ቄሂ (1)

ሂቡቕ ሂቡቕ ሂቡቕ

The term ቑቛቲቜis defined by Adomian polynomials ቑቛቲቜቭዤ ቄሂ, where ቄሂ are the Adomian Polynomial.

ሂቡቕ

The first Adomian polynomials are:

ቄቕቭባቛቲቕቜሂ ቄቖቭቲቖባቛቖቜቛቲቕቜሂ ቄቭቲባቛቖቜቛቲቕቜቦኔቲቖባቛቜቛቲቕቜ

ንሉ

It is important to note that the ቄዽ depend only on the components ቲቕ to ቲዽ.

Adomian method for differential equations with delay

13,14,15

Consider initially the differential equationwith delay of order n

ቶቛሂቜቛትቜቭባቛትሂቶቛትቜሂቶቛዺቖቛትቜቜሂሆሂቶቛዺሁቛትቜቜቜ, (2)

with ትዱናሂበsubject to initial conditions

ቡቶቛትቜሂቶሒቛትቜሂሆሂቶቛሂበቖቜቛትቜቁቭቛብቖቛትቜሂብቛትቜሂሆሂብሂቛትቜቜ,

where ትዱሂናበ. Here, a is the minimum of the values ዺዽቛትቜቲት (delay functions) for all ትዱናሂበ and ቦዱኔሂቪበ. Here, we assume that ዺዽቛትቜand ቤዽቛትቜarea sufficiently smooth. Then, we have that the Adomian polynomials are

ቡሂ

ኔ

ቄሂቭ ኅባኍዮ ሄዽቶዽኑ

ቫሉ ቡሄሂ ዽቡቕ The calculation of the Adomian polynomials is done as follows. For ቄቕ we have ኔባሒሒቛቶቕቜቶቖ

ቄቕቭባቛቶቕቜሂ ቄቖቭባሒቛቶቕቜቶቖሂ ቄቭ ቦባቃቛቶቕቜቶ

ን

Results Model for growth of tumors in mice

16,17

The growth of ascites Ehrlich typetumors in mice is modeled by the following initial value problem with delay

ልቛለበሩቜ

ቶሒቛቱቜቭቯቶቛቱቧዺቜቡኔቧቁ

{ዝ (3) ቶቛቱቜቭቶቕቛቱቜታናሂ for all ቱዱቧዺሂናበ. Here, ቶቛቱቜ is related to the number of cells (concentration) in the mouse; r is the proportionality (net) rate of reproduction of tumor cells; C is the storage capacity and ዺis the delay showing the duration of a cycle of the multiplicity of cells.

Particular case

Consider next the following particular case of the model (3)

ቶሒቛቱቜቭናህኔቶቛናህኘቱቜቛኔቧቶቛናህኘቱቜቜ

ች

ቶቛናቜቭናህኔ

We can see that the differential equation associated with this problem can be written in such a way that the Adomian method for equations with delay is easy to apply, that is,

ቶሒቛቱቜቭናህኔቶቛናህኘቱቜቧናህኔቶቛናህኘቱቜቶቛናህኘቱቜ

Now, rewriting in terms of the operator we have

ቶቭናህኔቶቛናህኘቱቜቧናህኔቶቛናህኘቱቜቶቛናህኘቱቜ (4) Therefore, from the initial condition ቶቛናቜቭናህኔ and assuming the existence of the inverse operator በቖ we have: In general, we can see that the recursive formula and approximate solution are obtained as follows

በቖቛቶቜቭበቖቛናህኔቶቛናህኘቱቜቧናህኔቶቛናህኘቱቜቶቛናህኘቱቜቜ | (5) | |

from where | ||

ቶቛቱቜቭናህኔቦበቖቛናህኔቶቛናህኘቱቜቧናህኔቶቛናህኘቱቜቶቛናህኘቱቜቜ | (6) |

ቶዿቖቛቱቜቭበቖኌናህኔቶዿቛናህኘቱቜቧናህኔቶዿቛናህኘቱቜቶዿቛናህኘቱቜነ (7) Initially we have that ቶቕቭናህኔ, then iterating for all k ≥ 0 we have: If ቨቭና, then ቶቖቭናህናናኜቱ; if ቨቭኔ, then ቶቭናህናናናኔኛቱ; if ቨቭን, then ቶቘቭቧናህናናኖኖኖኗኔኙቱቘ; if ቨቭኖ, then ቶቭቧናህናናናናናኜንኛኖንኙቱ and so on. Therefore, we obtain

ቶቛቱቜቭቶቕቱቕቦቶቖቱቖቦቶቱቦዏ ቭናህኔቦናህናናኜቱቦናህናናናኔኛቱቧናህናናኖኖኖኗኔኙቱቘቧናህናናናናናኜንኛኖንኙቱቦዏ

Next we present the model18 in which the population of a certain species given by the function ቶቛትቜ where its

growth rate (increasing or decreasing) is denoted by α. Thus, if the population increases, we will then have a

shortage of food as well as space in the habitat. In general, the behavior of this type of phenomena is modeled by the following initial value problem

ቶሒቛትቜቭዺኌዻቧቶቛትቜነቶቛትቜሂ ትታና

{ (8)

ቶቛትቕቜቭቶቕ

Assuming that the growth rate is dependent on the population in previous generations, we obtain the model with delay19 ቶሒቛትቜቭዺቛዻቧቶቛትቧልቜቜቶቛትቜ (9) Now, making the change of variable ቷቛትቜቭዺልቶቛልትቜwe have ቷሒቛትቜቭቶቃቛልትቜቛዺልቜ Therefore, from ቶቛልትቜwe can see that

ቶሒቛልትቜቭዺቡዻቧቶኌልቛትቧኔቜነቁቶቛልትቜሂ

or

ቷሒቛትቜቭቡዺልዻቧዺልቶኌልቛትቧኔቜነቁዺልቶቛልትቜ ቭቛዺልዻቧቷቛትቧኔቜቜቷቛትቜ Finally, returning the change of variable of z by y and ዺልዻby ዻwe obtain ቶሒቛትቜቭቛዻቧቶቛትቧኔቜቜቶቛትቜ (10)

As particular case of (8) we have the following model

ልቛለበቕህቜቜ

ቶሒቛቱቜቭኖህኘቶቛቱቜቡኔቧቁ

{ ቖ (11) ቶቛናቜቭኔኜህናናኔ

Then, to apply the Adomian method we rewrite the differential equation with retardation in the form

ቶሒቛቱቜቭኖህኘቶቛቱቜቧናህኔኛኗንኔኔቶቛቱቜቶቛቱቧናህኚኗቜ

Now, rewriting in terms of the operator we have

ቶቭኖህኘቶቛቱቜቧናህኔኛኗንኔኔቶቛቱቜቶቛቱቧናህኚኗቜ Therefore, of the initial condition ቶቛናቜቭኔኜህናናኔand assuming the existence of the inverse operator በቖ we have

በቖቛቶቜቭበቖቛኖህኘቶቛቱቜቧናህኔኛኗንኔኔቶቛቱቜቶቛቱቧናህኚኗቜቜ (12) from where,

ቶቛቱቜቭኔኜህናናኔቦበቖቛኖህኘቶቛቱቜቧናህኔኛኗንኔኔቶቛቱቜቶቛቱቧናህኚኗቜቜ

In general, we can see that the recursive formula and approximate solution are obtained as follows:

ቶዿቖ ቭበቖቛኖህኘቶዿቛቱቜቧናህኔኛኗንኔኔቶዿቛቱቜቶዿቛቱቧናህኚኗቜቜ

Initially we have that ቶቕቭኔኜህናናኔ, then iterating for all k≥0 we obtain an approximation to the real solution worked by other types of techniques, as evidenced by the attached figure.

Figure 2. Population dynamics.

Now let's consider another particular model20 of the form (8), that is,

ቶሒቛቱቜቭቧኖቶቛቱቧኔቜቛኔቦቶቛቱቜቜ

ች (9)

ቶቛናቜቭናህኔ

As in the previous case, we can write this equation in such a way that the Adomian decomposition method is applicable without any problem, that is, the problem (9) assumes the form

ቶሒቛቱቜቭቧኖቶቛቱቧኔቜቧኖቶቛቱቜቶቛቱቧኔቜ Therefore, of the initial condition ቶቛናቜቭናህኔand assuming the existence of the inverse operator በቖ we have በቖቛቶቜቭበቖቛቧኖቶቛቱቧኔቜቧኖቶቛቱቜቶቛቱቧኔቜቜ, from where

ቶቛቱቜቭናህኔቦበቖቛቧኖቶቛቱቧኔቜቧኖቶቛቱቜቶቛቱቧኔቜቜ

In general, we can see that the recursive formula and approximate solution are obtained as follows: ቶዿቖቛቱቜቭበቖቛቧኖቶዿቛቱቧኔቜቧኖቶዿቛቱቜቶዿቛቱቧኔቜቜ (10)

After iterating the recurrence equation (10), we obtain an excellent approximation with respect to the original evaluated with computational technique (see Figure 3).

Figure 3. Population dynamics.

Conclusion

In this paper we show the most relevant properties of the Adomian decomposition method using its definition. We present the proofs of each of the theorems as well as the properties of the for ordinary differential equations without delay, as well as the properties from the definition of the inverse operator and its linearity.

In the application of the Adomian method to differential equations with delay, we initially used elementary examples that allowed illustrating and comparing its solution with the exact solution found by analytical methods, thus visualizing the behavior regarding the convergence of the solutions. We apply Adomian later for problems with a higher degree of complexity, such as the value problems of the integer-differential type with different types of delay. We generalize the method to be able to apply it to more complex models (models of Biological type) of differential equations with delay, showing again the fast convergence in comparison with its exact solution.

We would like to thank the referee for his valuable suggestions that improved the presentation of this paper and our gratitude to the Department of Mathematics of the Universidad Tecnológica de Pereira, to Fundación Universitaria del Área Andina (Colombia) and the group GEDNOL.

*****