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.
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 [
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 [
On the other hand, Insperger and Stépán in [
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 [
To clarify our proposed method, we briefly review in Section
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 [
To illustrate the basic ideas of the HPM, let us consider the following nonlinear differential equation:
The operator
He in [
We next will introduce an approach based on homotopy methods, to obtain the solution of DDEs with constant and variable coefficients.
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 [
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
The EMHPM is an algorithm which approximates the HPM solution by subintervals, utilizing the following transformation rule:
To apply the homotopy technique to solve delay differential equations, we also assume the following. The linear operator The transformation
Therefore, we may conclude that the
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.
First, let us consider the simplest DDE of the form
Exact solution of (
It is easy to show that the solution of (
We next show how the EMHPM approach can be applied to obtain the approximate solution of nonlinear delay differential equation with variable coefficients. In this case, we obtain the approximate solutions of a DDE of the form
Figure
EMHPM and dde23 solution of (
To further assess the applicability of our proposed EMHPM approach to high order delay differential equations, we will next describe a methodology to obtain the approximate solutions of well-known high order delay differential equations by generalizing our EMHPM approach.
Let us consider an
Schematic of the zeroth order polynomial used to fit the approximate EMHPM solution.
Schematic EMHPM solution using first polynomial to approximate delay subinterval.
In the next section, we will apply our EMHPM procedure to obtain the solution of two second order delay differential equations: (a) the damped Mathieu equation with time delay, and (b) the well-known delay differential equation that describes the dynamics in one degree-of-freedom milling machine operations.
In order to assess the accuracy of our EMHPM approach, we first obtain the solution of the damped Mathieu differential equation with time delay that combines the effect of parametric excitation and damping. This equation is described by the following equation:
By following the EMHPM procedure, we first write (
Numerical solutions of the damped Mathieu equation with time delay by dde23, the zeroth EMHPM, and the first EMHPM with
It can be seen from Figure
Computer time needed to solve the damped Mathieu equation with time delay. The
dde23 |
EMHPM | |||
---|---|---|---|---|
|
|
Zeroth |
First | |
19 | 15 | 5 | 4 | 5 |
20 | 4 | 6 | 6 | |
40 | 3 | 12 | 12 | |
60 | 2 | 17 | 18 | |
60 | 5 | 18 | 19 | |
60 | 10 | 20 | 21 |
Figure
Estimated relative error values between the numerical solution dde23 and the EMHPM approximate solutions. Here we use for the EMHPM the values of
We next use our EMHPM procedure to obtain the solution of the single degree-of-freedom milling operation. We use the simplified form based on [
By following the EMHPM procedure, we can write (
EMHPM approximate solutions of (
EMHPM approximate solutions of (
Figure
CPU time comparison among the approximate solutions of (
Solution method | Time [ms] |
---|---|
dde23 | 139 |
Zeroth order EMHPM ( |
77 |
First order EMHPM ( |
87 |
Estimated relative error values between dde23 and the EMHPM approximate solutions. Here we have used the system parameter values of
We have developed a new algorithm based on the homotopy perturbation method to solve delay differential equations. The proposed EMHPM approach is based on a sequence of subintervals that approximate the solution of delayed differential equations by using the transformation rule
The authors declare that they have no conflict of interests with any mentioned entities in the paper.
This work was funded by Tecnológico de Monterrey, Campus Monterrey, through the Research Group in Nanomaterials for Medical Devices and the Research Group in Advanced Manufacturing. Additional support was provided from Consejo Nacional de Ciencia y Tecnología (Conacyt), Mexico.