Abstract and Applied Analysis

Volume 2015, Article ID 382475, 12 pages

http://dx.doi.org/10.1155/2015/382475

## Approximate Solutions of Delay Differential Equations with Constant and Variable Coefficients by the Enhanced Multistage Homotopy Perturbation Method

^{1}Center for Innovation in Design and Technology, Tecnológico de Monterrey, Campus Monterrey, E. Garza Sada 2501, 64849 Monterrey, NL, Mexico^{2}Department of Mechanical Engineering, University of the Basque Country, Alameda de Urquijo s/n, Bilbao, 48013 Bizkaia, Spain

Received 26 December 2013; Revised 21 July 2014; Accepted 16 August 2014

Academic Editor: Zhichun Yang

Copyright © 2015 D. Olvera et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We expand the application of the enhanced multistage homotopy perturbation method (EMHPM) to solve delay differential equations (DDEs) with constant and variable coefficients. This EMHPM is based on a sequence of subintervals that provide approximate solutions that require less CPU time than those computed from the dde23 MATLAB numerical integration algorithm solutions. To address the accuracy of our proposed approach, we examine the solutions of several DDEs having constant and variable coefficients, finding predictions with a good match relative to the corresponding numerical integration solutions.

#### 1. Introduction

Delayed differential equations (DDEs) are used to describe many physical phenomena of interest in biology, medicine, chemistry, physics, engineering, and economics, among others. Since the introduction of the first delayed models, many publications have appeared as summarizing theorems and homotopy methods of solution that deal with the stability properties of delayed systems (see [1–3] and references cited there in).

For instance, Shakeri and Dehghan introduced an approach to find the solution of delay differential equations by means of the homotopy perturbation technique (HPM) with results that agree well with exact solutions [1]. Wu in [2] used the homotopy analysis method to obtain the approximate solution of a strong nonlinear ENSO delayed oscillator model that provides good agreement when compared to its exact solution under the condition of . Alomari and coworkers in [3] developed an algorithm to obtain approximate analytical solutions for DDEs by using the homotopy analysis method (HAM) and the modified homotopy analysis method (MHAM). They used their derived method to obtain the approximate solution of various linear and nonlinear DDEs with numerical predictions that agree well with the numerical integration solutions, and they also proved that their derived solutions converge to the exact ones. By applying the homotopy perturbation method (HPM), Biazar and Behzad found approximate solutions of neutral differential equations with proportional delays which describe well their corresponding numerical integration solutions [4]. Recently, Anakira and co-workers in [5] extended the applicability of the so called optimal homotopy asymptotic method (OHAM) that does not depend on small or large parameters, to find the approximate analytic solution of DDEs. They used their proposed approach to compare the derived approximate solutions of several DDEs with their exact analytical solutions with predictions that compare well with the exact ones.

On the other hand, Insperger and Stépán in [6] used the semidiscretization method to determine the stability lobes of DDEs that model the dynamics of cutting machine operations. Based on the properties of the Chebyshev polynomials, Butcher and coworkers in [7] developed a methodology to obtain the stability lobes of milling machine operations and they proved that this technique is faster than that of the full and the semidiscretization methods since these solution techniques approximate the original DDEs by a series of ODEs [8].

Here in this paper, we develop a generalized procedure to solve linear and nonlinear DDEs by introducing some modifications to the multistage homotopy perturbation method (MHPM) derived by Hashim and Chowdhury to obtain approximate solutions of ordinary differential equations [9]. The proposed enhanced multistage homotopy perturbation method (EMHPM) is based on a sequence of subintervals that allow us to find more accurate approximated solutions under a numerical-analytical procedure that requires less CPU time when compared to the numerical integration solutions provided by the MATLAB dde23 algorithm written by Shampine and Thompson in [10]. The EMHPM is based on a homotopy function that could be divided into a linear operator and a nonlinear operator to satisfy its assumed initial solution. This split of the homotopy function allows us to modify the nonlinear operator to guarantee, by using the enhanced homotopy perturbation method, the stability of the proposed approximate solutions of nonlinear differential equations [11].

To clarify our proposed method, we briefly review in Section 2 some basic concepts of the homotopy perturbation method, and, then in Section 3, we introduce the EMHPM to solve DDEs. The difference between the HPM and the EMHPM is discussed in Section 4 by addressing the approximate solutions of a nonlinear delayed differential equation with variable coefficients. Finally, the general solution of two DDEs that describe the dynamics of two engineering problems, by using the EMHPM, is discussed in Section 5.

#### 2. Homotopy Perturbation Method

The homotopy perturbation method (HPM) is a coupling of the traditional perturbation method and homotopy in topology which eliminates the limitation of the small parameter assumed in the perturbation methods [12]. Under this approach, a nonlinear problem can be transformed into an infinite number of simple problems without the restriction of having small nonlinear parameter values. This homotopy perturbation method takes the main advantages of traditional perturbation methods together with homotopy analysis [13–15].

To illustrate the basic ideas of the HPM, let us consider the following nonlinear differential equation: with boundary conditions where is a general differential operator, is a boundary operator, is a known analytic function, and is the boundary of the domain .

The operator can generally be divided into two parts: and , where involves the linear terms and the nonlinear ones. Equation (1) therefore can be rewritten as follows: By the homotopy perturbation technique, we construct a homotopy that satisfies where is an embedding parameter and is an initial approximation of (1) which satisfies the boundary conditions (2). Thus, from (4), we have The changing process of from zero to unity is just that of from to . In topology, this is called deformation, and and are called homotopic.

He in [12] uses the embedding parameter as the small parameter and assumed that the solution of (4) can be written as a power series of in the form By setting , He obtained the approximate solution of (1) as Then, this method was applied to obtain the approximate solution of some nonlinear ordinary differential equations valid not only for small, but also for large nonlinear parameter values.

We next will introduce an approach based on homotopy methods, to obtain the solution of DDEs with constant and variable coefficients.

#### 3. The EMHPM Methodology to Solve DDEs

The HPM is an asymptotic method that depends on the auxiliary linear operator form and the initial guess of the initial conditions. Therefore, the convergence of the approximate solution cannot be guaranteed in some cases [16]. Hashim and Chowdhury showed in [9] that the solutions obtained by the standard HPM were not valid for large time span unless more terms are calculated. Thus, they proposed a multistage homotopy perturbation method (MHPM) which treated the HPM algorithm in a sequence of subintervals in an attempt to improve the accuracy of the approximate solutions of linear and nonlinear ordinary differential equations (ODEs).

However, when the MHPM is applied to obtain the approximate solutions of ODEs which contain coefficients as a function of time, this method cannot provide accurate solutions when . In this work, we introduce some modifications to the MHPM and focus on the derivation of approximate solutions of DDEs equations with variable coefficient terms. This new approach is based on the enhanced multistage homotopy perturbation method (EMHPM) introduced in [17] to obtain the solution of nonlinear ordinary differential equations.

The EMHPM is an algorithm which approximates the HPM solution by subintervals, utilizing the following transformation rule: , where satisfies the initial condition , is a shifted time scale used to determine the approximate solution in each subinterval, and represents the approximate solution in the th subinterval. In this case, the initial suggested solution in the th subinterval is given by , where represents the time at the end of the previous subinterval (i.e., the value of the approximate solution at the end of the previous subinterval represents the initial conditions of the next subinterval under consideration).

To apply the homotopy technique to solve delay differential equations, we also assume the following.(1)The linear operator represents , where the assumed approximate solution is set equal to the initial condition ; that is, . To simplify the notation, we let .(2)The transformation on holds in the homotopy -subinterval. Thus, higher order equations are integrated with respect to , while the terms related to the independent variable are assumed to remain constant.Therefore, we may conclude that the order approximate solution, by applying the EMHPM, can be written as where the solution is valid only in the th subinterval . Hence, the solution on the th subinterval can be written as with initial condition , and . Thus, the approximate solution of at the time is given by In summary, the solution for an open-closed interval (] is divided into subintervals that, in general, are not equally spaced: . Thus, the approximated solution of for the span time interval is obtained by coupling the solutions.

#### 4. Approximate Solutions of Some DDEs by Applying the EMHPM

In this section we focus on the solution of DDEs with constant and variable coefficients and examine the applicability of the EMHPM to find the corresponding approximate solutions.

##### 4.1. Delay Differential Equations with Constant Coefficients

First, let us consider the simplest DDE of the form with initial condition . Here, the independent variable is a scalar , the dot stands for differentiation with respect to time , and is the time delay. To evaluate (11) on , the term must represent a known function on . For instance, if , the solution of (11) can be obtained in the interval by assuming an initial function that satisfies the initial condition. By using this solution, it becomes possible to obtain the solution of (11) in the next th interval , , where is an integer number that can be chosen as . With this approach, we can apply the HPM to find the solution of (11) by assuming that the previous delayed function is ; thus the solution for the first interval is given by , valid on . In terms of (4), we now construct the homotopy of (11): We next substitute the first order expansion in (12) and balance the terms with identical power of to obtain the following set of linear differential equations: Integration of (13) yields Hence, the first order solution of (12) is given by Notice that (15) represents the exact solution of (11) on the first interval. By following the same procedure, it is easy to show that the exact solution of (11), for the second and third intervals, is given, respectively, as Figure 1 shows the exact solution of (11) obtained by coupling at each interval the solution obtained by following HPM procedure for .