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This paper presents a Modified Power Series Method (MPSM) for the solution of delay differential equations. Unlike the traditional power series method which is applied to solve only linear differential equations, this new approach is applicable to both linear and nonlinear problems. The method produces a system of algebraic equations which is solved to determine the coefficients in the trial solution. The method provides the solution in form of a rapid convergent series. The obtained results for numerical examples demonstrate the reliability and efficiency of the method.

Although both ordinary differential equations (ODEs) and delay differential equations (DDEs) are used to describe physical phenomena, they are different. While in ODEs the derivatives of unknown functions are dependent on only the current value of the independent variable, in addition to this in DDEs, the derivatives of unknown functions are dependent on the values of the functions at previous time. This implies that the solution of DDEs requires the knowledge of the current state and the state at certain previous times.

Some of the application areas of delay differential equations are population dynamics, infectious disease, physiological and pharmaceutical kinetics, chemical kinetics, models of conveyor belts, urban traffic, heat exchangers, robotics, navigational control of ships and aircrafts, and more general control problems (see [

There are a few classes of nonlinear ODEs for which solutions can be easily found and despite the obvious similarities between ODEs and DDEs, solutions of DDE problems can differ from solutions for ODE problems in several striking and significant way [

In this paper, we consider a modified power series method for solving the delay differential equations of the form.

where n and

According to the MPSM, the Nth degree approximate solution to the DDE (1)-(2) is given by

where

The MPSM is described by the following five-step procedure:

Step 1

Rewrite Equation (1) such that only the nonhomogeneous term is on the right hand side of the equation

Step 2

On the left hand side of the nonhomogeneous differential equation, substitute

and the derivatives of

Step 3

Collect the power of t on the left hand side of the equation resulting from step 2 and set the coefficient of each power of t on the left hand side equal to the corresponding coefficient on the right hand side of the equation.

Step 4

Solve, using either the Newton’s method or forward substitution method, the first

Step 5

Substitute the coefficients

Example 3.1 (see [

Consider the first-order nonlinear DDE

Subject to

Applying the proposed method illustrated in section 2 to this problem for the cases N = 3, 5, 7, 8, 11, we obtain as follows:

Obviously,

which is the exact solution to Example 3.1.

Example 3.2 (see [

Consider the second-order linear DDE

Subject to

Using the proposed method for

Example 3.3 (see [

Consider the third-order nonlinear DDE

Subject to the conditions

with the proposed method illustrated in section 2, we obtain for the cases N = 5, 7, 9, 11 and 13 as follows:

Clearly,

which is the exact solution to Example 3.3.

Example 3.4 (see [

Consider the first-order linear DDE

Subject to

The exact solution is given by

For

which converges to

A simple and straight forward technique based on the power series method has been studied for the solution of delay differential equations. This new approach is implemented without using restrictive assumptions or adding perturbation term and it gives excellent performance compared with existing techniques for solving delay differential equations.