- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 490342, 14 pages
Rational Homotopy Perturbation Method
Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, VER, Mexico
Received 28 June 2012; Accepted 16 August 2012
Academic Editor: Turgut Öziş
Copyright © 2012 Héctor Vázquez-Leal. 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.
The solution methods of nonlinear differential equations are very important because most of the physical phenomena are modelled by using such kind of equations. Therefore, this work presents a rational version of homotopy perturbation method (RHPM) as a novel tool with high potential to find approximate solutions for nonlinear differential equations. We present two case studies; for the first example, a comparison between the proposed method and the HPM method is presented; it will show how the RHPM generates highly accurate approximate solutions requiring less iteration, in comparison to results obtained by the HPM method. For the second example, which is a Van der Pol oscillator problem, we compare RHPM, HPM, and VIM, finding out that RHPM method generates the most accurate approximated solution.
Solving nonlinear differential equations is an important issue in sciences because many physical phenomena are modelled using such equations. One of the most powerful methods to approximately solve nonlinear differential equations is the homotopy perturbation method (HPM) [1–28]. The HPM is based on the use of a power series, which transforms the original nonlinear differential equation into a series of linear differential equations. In this paper, we propose a generalization of the aforementioned concept by using a quotient of two power series of homotopy parameter, which will be called rational homotopy perturbation method (RHPM). In the same fashion, like HPM, the use of that quotient of power series transforms the nonlinear differential equation into a series of linear differential equations. We will present two case studies; for the first example, a comparison between the proposed method and the HPM method is presented; it will show how the RHPM generates highly accurate approximate solutions requiring less iteration steps, in comparison to results from HPM method. For the second example, the Van der Pol oscillator problem [3, 29], we compare RHPM, HPM , and variational iteration method (VIM) , resulting that RHPM method generates the most accurate approximated solution.
This paper is organized as follows. In Section 2, we introduce the basic concept of the RHPM method. In Section 3, we present a study of convergence for the proposed method. In Sections 4 and 5, we present the solution of two nonlinear differential equations. In Section 6, numerical simulations and a discussion about the results are provided. Finally, a brief conclusion is given in Section 7.
2. Basic Concept of RHPM
The RHPM and HPM share common foundations. Thus, for both methods, it can be considered that a nonlinear differential equation can be expressed as having as boundary condition where and are a linear and a nonlinear operator, respectively, is a known analytic function, is a boundary operator, is the boundary of domain , and denotes differentiation along the normal drawn outwards from .
Now, a possible homotopy formulation is where is the initial approximation for (2.1) which satisfies the boundary conditions and is known as the perturbation homotopy parameter. Analysing (2.3), can be concluded that
Considering that , it results that the approximate solution for (2.1) is
For the RHPM, we assume that solution for (2.3) can be written as power series quotient of : where are unknown functions to be determined by the RHPM, and are known analytic functions of the independent variable.
For the HPM, the order of the approximation is determined by the highest power of . Nevertheless, for the RHPM the order will be given as , where and are the highest power of employed in the numerator and denominator of (2.7). Here, the number of linear differential equations generated is .
The above limit exists in the case that both limits exist.
3. Convergence of RHPM
In order to analyse the convergence of RHPM, (2.3) is rewritten as
Applying the inverse operator, , to both sides of (3.1), we obtain
Assuming that (see (2.7))
In order to study the convergence of the RHPM, we use the Banach theorem as reported in [1, 2]. Such theorem relates the solution of (2.1) to the fixed point problem of the nonlinear operator . Let us state the theorem as follows.
Theorem 3.1 (Sufficient Condition for Convergence). Suppose that and are Banach spaces and is a contractive nonlinear mapping, that is
Then, according to the banach fixed point theorem, has a unique fixed point ; that is, . Assume that the sequence generated by the RHPM can be written as
and suppose that , where ; then, under these conditions,
Proof. (i) By inductive approach, for we have
Assuming that , as induction hypothesis, then
Using (i), we have
(ii) Because of and , ; that is,
4. Case Study 1
Consider the following nonlinear differential equation having exact solution
4.1. Solution Calculated by RHPM
We establish the following homotopy equations:
Substituting (4.5) into (4.3) and (4.6) into (4.4), regrouping and equating terms having the same -powers, it can be solved for , , , and so on (in order to fulfil initial conditions from ; it follows that , and so on).
Solving (4.7) results in
Choosing the adjustment parameters for (4.10) as and , results in
Now, solving (4.8) results in
Selecting the adjustment parameter as with the procedure reported in [4, 5, 26], (4.13) shows good accuracy for positive values of ; thus, we propose the use of the odd symmetry of exact solution (4.2) to establish a solution with good accuracy throughout the range of
4.2. Solution Obtained by Using HPM
Substituting (4.15) of order 10 and order 2 into (4.3) and (4.4), respectively, regrouping and equalling terms having the same order -powers, it can be solved for , , , and so on (in order to fulfil initial conditions from , it follows that and so on).
By solving (4.2), we obtain
Solving (4.17), we obtain
5. Case Study 2
We suppose that solution for (5.3) has the following rational form:
By solving (5.5), we obtain
6. Numerical Simulation and Discussion
Figure 1 and Table 1 show a comparison between the exact solution (4.2) for the nonlinear differential equation (4.1) and the analytic approximations (4.11), (4.14), (4.19), and (4.22). Considering the odd symmetry from the exact solution and approximations, Table 1 presents the relative error only for positive values of . In the range of , the maximum relative error for (4.11) is 0.0022083, while the maximum error for (4.19) in the same range is (see Figure 1). Besides, the table also shows the relative error for (4.11) at , which is 0.0372487, that is, fifteen orders of magnitude lower than the relative error obtained for (4.19). Also, the RHPM is required to solve (4.7) using just three iterations to obtain (4.11); while in order to obtain (4.19), HPM required to solve the system (4.2) containing ten differential equations. Therefore, for this case study, RHPM reached results having higher precision and wider range requiring less iteration than HPM.
If we perform the Padé [30, 31] approximant of order to the exact solution (4.2), the result is exactly the same to the approximate solution (4.11) calculated by using the RHPM. This result is interesting and deserves deeper study in a future work.
The differential equation (4.1) was solved using homotopy (4.4) in its RHPM and HPM versions, resulting in approximations (4.14) and (4.22), respectively. From Table 1, it is possible to observe that the lowest relative error in the range for (4.14) is 0.000282807, while the minimum relative error for (4.22) is −0.0203519. In fact, there is a difference of one or two orders of magnitude throughout the domain of solutions favouring the RHPM. Furthermore, for , (4.14) has the lower relative error of all approximations with a value of −E-44. In case that a better approximation is required, it would be necessary to perform more iteration for both methods. This shows that using a nonlinear trial equation may generate highly accurate results for both HPM and RHPM.
In Table 2, the relative error for the exact solution (5.2) of the Van der Pol oscillator (5.1), RHPM solution (5.7), and approximations obtained by HPM and VIM reported in  are shown. It can be seen that the approximated solution (5.7) shows the lowest relative error in the range (see Figure 2). Besides, the order solution (5.7) has the lowest number of terms compared to the first-order solutions obtained by HPM  and VIM .
For both case studies, polynomial functions were employed. Nevertheless, is arbitrary and may contain exponentials, trigonometric functions, among others. Likewise, terms play a significant role in the accuracy of the resultant approximation. Therefore, thorough study is required to propose a methodology leading to select functions to obtain more accurate solutions using RHPM method.
In this work, by using two case studies, the RHPM is presented as a novel tool with high potential to solve nonlinear differential equations. Given that HPM and RHPM are closely related, it is highly possible that differential equations solved by HPM can be solved by RHPM in order to find more accurate solutions.
This paper presented the rational homotopy perturbation method as a novel tool with high potential to solve nonlinear differential equations. Also, a comparison between the results of applying the proposed method and HPM was shown. Likewise, for the first example, a comparison between the proposed method and the HPM was presented, showing how the RHPM generates highly accurate approximate solutions using less iteration steps, in comparison to results obtained using the HPM. Besides, a Van der Pol oscillator problem was solved by the proposed method and compared to solutions obtained by HPM and VIM; the result was that the RHPM generated the most accurate approximated solution. Finally, there is a possible connection between the Padé approximant and the RHPM, which will be studied in future works.
The author gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACyT) through Grant CB-2010-01 #157024. The author would like to thank Roberto Castaneda-Sheissa, Uriel Filobello-Nino, Rogelio-Alejandro Callejas-Molina, and Roberto Ruiz-Gomez for their contribution to this project.
- J. Biazar and H. Aminikhah, “Study of convergence of homotopy perturbation method for systems of partial differential equations,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2221–2230, 2009.
- J. Biazar and H. Ghazvini, “Convergence of the homotopy perturbation method for partial differential equations,” Nonlinear Analysis, vol. 10, no. 5, pp. 2633–2640, 2009.
- A. Barari, M. Omidvar, A. R. Ghotbi, and D. D. Ganji, “Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations,” Acta Applicandae Mathematicae, vol. 104, no. 2, pp. 161–171, 2008.
- H. Vazquez-Leal, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, and J. Sanchez-Orea, “High accurate simple approximation of normal distribution related integrals,” Mathematical Problems in Engineering, vol. 2012, Article ID 124029, 22 pages, 2012.
- H. Vazquez-Leal and U. Filobello-Nino, “Modified hpms inspired by homotopy continuation methods,” Mathematical Problems in Engineering, vol. 2012, Article ID 309123, 19 pages, 2012.
- J.-H. He, “Comparison of homotopy perturbation method and homotopy analysis method,” Applied Mathematics and Computation, vol. 156, no. 2, pp. 527–539, 2004.
- J.-H. He, “An elementary introduction to the homotopy perturbation method,” Computers & Mathematics with Applications, vol. 57, no. 3, pp. 410–412, 2009.
- J.-H. He, “Homotopy perturbation technique,” Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257–262, 1999.
- J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,” Applied Mathematics and Computation, vol. 151, no. 1, pp. 287–292, 2004.
- J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003.
- J.-H. He, “A coupling method of a homotopy technique and a perturbation technique for non-linear problems,” International Journal of Non-Linear Mechanics, vol. 35, no. 1, pp. 37–43, 2000.
- J.-H. He, “Application of homotopy perturbation method to nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 26, no. 3, pp. 695–700, 2005.
- J.-H. He, “Homotopy perturbation method for solving boundary value problems,” Physics Letters A, vol. 350, no. 1-2, pp. 87–88, 2006.
- Y. Khan, H. Vazquez-Leal, and Q. Wu, “An efficient iterated method for mathematical biology model,” Neural Computing and Applications. In press.
- Y. Khan, Q. Wu, N. Faraz, A. Yildirim, and M. Madani, “A new fractional analytical approach via a modified riemannliouville derivative,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1340–1346, 2012.
- N. Faraz and Y. Khan, “Analytical solution of electrically conducted rotating flow of a second grade fluid over a shrinking surface,” Ain Shams Engineering Journal, vol. 2, no. 34, pp. 221–226, 201.
- Y. Khan, Q. Wu, N. Faraz, and A. Yildirim, “The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking/stretching sheet,” Computers & Mathematics with Applications, vol. 61, no. 11, pp. 3391–3399, 2011.
- Y. Khan, H. Vázquez-Leal, and L. Hernandez-Martínez, “Removal of noise oscillation term appearingin the nonlinear equation solution,” Journal of Applied Mathematics, vol. 2012, Article ID 387365, 9 pages, 2012.
- H. Aminikhah, “The combined laplace transform and new homotopy perturbation methods for stiff systems of odes,” Applied Mathematical Modelling, vol. 36, no. 8, pp. 3638–3644, 2012.
- F. I. Compean, D. Olvera, F. J. Campa, L. N. Lopez de Lacalle, A. Elias-Zuniga, and C. A. Rodriguez, “Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method,” International Journal of Machine Tools and Manufacture, vol. 57, pp. 27–33, 2012.
- J. Biazar and B. Ghanbari, “The homotopy perturbation method for solving neutral functional-differential equations with proportional delays,” Journal of King Saud University—Science, vol. 24, no. 1, pp. 33–37, 2012.
- A. M. A. El-Sayed, A. Elsaid, I. L. El-Kalla, and D. Hammad, “A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8329–8340, 2012.
- S. T. Mohyud-Din, A. Yildirim, and Mustafa Inc, “Coupling of homotopy perturbation and modified lindstedtpoincar methods for traveling wave solutions of the nonlinear kleingordon equation,” Journal of King Saud University—Science, vol. 24, no. 2, pp. 187–191, 2012.
- J. Biazar and M. Eslami, “A new homotopy perturbation method for solving systems of partial differential equations,” Computers & Mathematics with Applications, vol. 62, no. 1, pp. 225–234, 2011.
- U. Filobello-Nino, H. Vazquez-Leal, Y. Khan et al., “Hpm applied to solve nonlinear circuits: a study case,” Applied Mathematical Sciences, vol. 6, no. 85–88, pp. 4331–4344, 2012.
- U. Filobello-Nino, H. Vazquez-Leal, R. Castaneda-Sheissa et al., “An approximate solution of blasius equation by using hpm method,” Asian Journal of Mathematics and Statistics, vol. 5, pp. 50–59, 2012.
- Y.-G. Wang, W.-H. Lin, and N. Liu, “A homotopy perturbation-based method for large deflection of a cantilever beam under a terminal follower force,” International Journal for Computational Methods in Engineering Science and Mechanics, vol. 13, pp. 197–201, 2012.
- M. Madani, M. Fathizadeh, Y. Khan, and A. Yildirim, “On the coupling of the homotopy perturbation method and Laplace transformation,” Mathematical and Computer Modelling, vol. 53, no. 9-10, pp. 1937–1945, 2011.
- S. H. Behiry, H. Hashish, I. L. El-Kalla, and A. Elsaid, “A new algorithm for the decomposition solution of nonlinear differential equations,” Computers & Mathematics with Applications, vol. 54, no. 4, pp. 459–466, 2007.
- S. Momani, G. H. Erjaee, and M. H. Alnasr, “The modified homotopy perturbation method for solving strongly nonlinear oscillators,” Computers & Mathematics with Applications, vol. 58, no. 11-12, pp. 2209–2220, 2009.
- H. Bararnia, E. Ghasemi, S. Soleimani, R. Abdoul Ghotbi, and D. D. Ganji, “Solution of the falkner-skan wedge flow by hpm-pade’ method,” Advances in Engineering Software, vol. 43, no. 1, pp. 44–52, 2012.