We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter
With the recent progress in nonlinear problems research, there has been an increasing interest in analytical techniques to solve the corresponding nonlinear equations. However, most of the current methods based their solution methodology on the assumption of small nonlinearities that limit its potential usage in the solution of physical and engineering applications.
Recent applications of some asymptotic methods such as the variational iteration, the homotopy perturbation, the energy balance, the parameterexpansion, the variational approach, the improved amplitude frequency formulation, the maxmin approach, the Hamiltonian approach, and the homotopy analysis, to name a few, have been used to obtain approximate solutions of highly nonlinear problems in which the traditional perturbation methods have some limitations [
To obtain approximate analytical solutions of strongly nonlinear differential equations by using the homotopy approach, the methods of homotopy perturbation, homotopy analysis, Adomian decomposition (ADM), and the variational iteration (VIM) are commonly used in the literature. The ADM is an iterative method which provides analytical approximate solutions in the form of an infinite power series for nonlinear equations without linearization [
The HPM is an asymptotic method with limited convergence away from the equation initial conditions [
Based on the findings of these previous research works, the aim of this work focuses on developing a seminumericalanalytic technique that generalizes the MHPM in an attempt to obtain accurate approximate solutions of nonlinear differential equations. To assess the accuracy of this new approach, its numerical predictions will be compared with respect to those of the MHPM techniques introduced in [
In order to illustrate the basic idea behind the HPM, let us consider the following nonlinear differential equation:
The
Hosein Nia et al. proposed to modify the HPM by introducing a stable linear operator. This modified method was called the
In order to improve the convergence of the HPM, Odibat proposed the expansion of the independent variable by using Taylor series [
Comparison between the
Numerical comparisons between the RungeKutta method (
The proposed
In order to establish a homotopy algorithm that will allow us to obtain approximate solutions of nonlinear differential equations, we next assume the following.
The linear operator
Since the homotopy is defined in the
In order to demonstrate the effectiveness of the proposed EMHPM, we next derive the approximate solution of (
Figure
Comparison of the
Before we examine the application of the EMHPM to derive approximate solutions of nonlinear differential equation, we will next address some convergence issues related to our proposed approach.
Recently Turkyilmazoglu introduced a convergence scheme for the homotopy series [
First, we focus on the solution of (
Figure
Absolute error plotted against the control parameter
We next consider the second order nonlinear ordinary differential equation used in [
Influence of the subinterval size
Figure
Absolute error plotted against the control parameter
To further evaluate the accuracy of our proposed EMHPM approach, in the next section, we will derive the approximate solution of some nonlinear differential equations that arise in several engineering applications.
In this section, we will explore the accuracy of the EMHPM approach when this is applied to obtain the approximate solution of the HelmholtzDuffing equation that has quadratic and cubic nonlinearities. This equation is used to describe the nonlinear response of some materials in mechanical engineering applications [
In order to evaluate the accuracy of the EMHPM approach, the derived approximate solution of (
In this case, we suppose that the system parameter values are given as
Figure
Computation time and absolute error comparisons.
EMHPM 
 



Absolute error  Computation time (ms)  Error tolerance  Absolute error  Computation time (ms) 
0.05  7  0.1604  104 

83.9929  59 
0.10  9  0.1321  74 

36.5743  72 
0.15  16  0.1304  78 

1.7213  88 
0.20  22  0.1874  98 

0.0499  107 
Solution of (
In this case, we solve (
Solution of (
Next, we will use our EMHPM approach to obtain the timeamplitude approximate solutions of an irrational nonlinear spring oscillator and of an elastomagnetic passive suspension system.
In nonlinear dynamic systems, there are cases when the amplitudefrequency response curves have jumps due to the nonlinearities of the system. In the hard spring model with cubic nonlinearities, the amplitudefrequency response curves exhibit more than one possible dynamics response [
Nonlinear spring oscillator, right: elastomagnetic passive suspension.
To find the approximate solution of (
Solution of the nonlinear spring oscillator with
The dynamic response of an elastomagnetic passive suspension system of one degree of freedom that is showed in Figure
Solution of (
In all cases discussed here, we can use different system parameter values and still show that the derived seminumericalanalytic technique that generalizes the MHPM provides good results when compared to the numerical integration solution of the corresponding equations of motion.
In this work, we have modified the HPM and introduced a new numericalanalytic approach that is based on a linear operator defined as
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 Chair in Nanomaterials for Medical Devices and the Research Chair in Intelligent Machines. Additional support was provided from Consejo Nacional de Ciencia y Tecnología (Conacyt), México.