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This paper concentrates on the differential transform method (DTM) to solve some delay differential equations (DDEs). Based on the method of steps for DDEs and using the computer algebra system Mathematica, we successfully apply DTM to find the analytic solution to some DDEs, including a neural delay differential equation. The results confirm the feasibility and efficiency of DTM.

The differential transform method (DTM) is a semi analytical-numerical technique depending on Taylor series for solving integral-differential equations (IDEs). The method was first introduced by Pukhov [

Delay differential equations (DDEs) arise in many applied fields, such as control technology, communication networks, and biological population management, and hence they have attracted considerable attention. There are many papers devoted to the problem of approximate solution of DDEs [

and

with

It should be pointed out that the solution to DDEs (2) maybe be non-unique (see Section 2 in [

where

In this paper, we will apply DTM to find the analytic solution to DDEs (3) with the help of the computer algebra system Mathematica. Thus, in some sense, our work can be viewed as a supplement to [

The basic theory of differential transform can be found in [

Consider a function

represented by one series whose center is located at

where

Differential inverse transformation of

From (4) and (5), it is easy to see that the concept of the differential transformation is derived from the Taylor series expansion. By our assumption,

In this study, we use the lower case letters to represent the original functions and upper case letters to stand for the transformed functions (T-functions). The fundamental mathematical operations performed by differential transform method are listed in

There are many methods to deal with the delay differential Equation (3). For example, linear multistep (LM) methods, Runge-Kutta (RK) methods, waveform relaxation (WR) methods, etc. However, the basic idea to solve the DDE (3) is to solve the following system of ODEs step by step:

Sample Size | MLE | LSE | ||||||
---|---|---|---|---|---|---|---|---|

0.3 | 0.1 | 100 | 2.8 | 0.33 | 0.150 | 3.925 | 0.239 | 0.159 |

200 | 2.701 | 0.315 | 0.156 | 5.117 | 0.233 | 0.157 | ||

300 | 2.979 | 0.312 | 0.147 | 5.917 | 0.222 | 0.155 | ||

0.25 | 0.15 | 100 | 2.809 | 0.271 | 0.173 | 3.860 | 0.234 | 0.188 |

200 | 2.93 | 0.263 | 0.176 | 3.808 | 0.254 | 0.184 | ||

300 | 3.146 | 0.262 | 0.171 | 4.232 | 0.251 | 0.182 | ||

0.2 | 0.15 | 100 | 3.44 | 0.208 | 0.161 | 4.136 | 0.212 | 0.169 |

200 | 3.403 | 0.208 | 0.159 | 4.72 | 0.225 | 0.166 | ||

300 | 3.261 | 0.208 | 0.158 | 5.111 | 0.242 | 0.164 |

with

principle, be continued as far as desired. It is called, quite naturally, the method of steps [

Using the basic idea of the method of steps, first, we apply the DTM to find the solution to the following ODEs:

with

Suppose the approximate solution is given by

If

Applying the DTM to the differential equation above again, we will obtain the following solution

Of course, we should go on if

until for some

Remark 1 If

Remark 2 If

Remark 3 If we want to improve the accuracy of the approximate solution in each interval, we can combine the above method with the multi-step method given by [

Remark 4 In fact, the DTM based on the method of steps can also be applied to solve the following neutral delay differential equations

In this section, four examples are given to show the performance of the DTM based on the method of steps. First, we want to solve the following simple but classical DDE to further illustrate the process of DTM.

Example 4.1 Consider the DDE [

First, since

It is easy to get

Thus we have the analytic solution

Second, we should continue to solve the following DDE:

or equivalently,

with initial condition

From (8), we have the following differential transform

It is easy to get

Thus we have the analytic solution

Now, if we want to obtain the solution in the interval [2, 3], we should deal with the following DDE:

or equivalently,

with the initial condition

Then we have the following differential transform

and get

Thus the analytic solution defined on [2, 3] is given by

The DTM can be proceed till the desire solution is obtained.

Example 4.2 Consider the nonlinear DDE of third-order [

Since

Thus, applying DTM to the equation above, we obtain

The initial conditions lead to

Noting that

Remark 5 In [

It’s worth pointing out that, using the method given in [

solution. In fact, Example 4.2 shows that

also a solution to (10), then

initial function

to” get the approximate solution of

Example 4.3 Consider a single delay equation with a stiffness parameter [

with

Similarly, we should solve the following DDE limited in the interval

Thus applying DTM to this equation, we obtain

The initial values lead to

Then, we obtain the solution to (12):

This is the analytic solution to (12). Particularly, if

As the last example, we apply the DTM based on the method of steps to solve a neutral delay differential equations.

Example 4.4 Consider the neutral delay differential equation

with the initial function

According to the idea of the method of steps, DDE (13) becomes

Applying DTM to this equation, we have

From the initial function, we get

Then, the solution to (13) is

Although the theory of differential transform method is not complete yet, it has been successfully applied to solve ordinary differential equations, partial differential equations, integral-differential equations, differential- algebraic equations and etc. In this paper, we apply DTM based on the method of steps to solve some delay differential equations, including neutral delay differential equations, successfully. Numerical experiments show that DTM is feasible and efficient for them. We believe that the operations of DTM presented in this paper also can be used to solve some partial delay differential equations (PDDEs), which is worth while studying in the future work.

This work is supported by the National Natural Science Foundation of China, contract/grant number 11371198 and 11401296, Jiangsu Provincial Natural Science Foundation of China, contact/grant no. BK20141008, Natural Science Fund for colleges and universities in Jiangsu Province contact/grant no. 14KJB110007, Jiangsu Provin- cial Key Laboratory for Numerical Simulation of Large Scale Complex Systems contract/grant number 201305 and 201401.