- Research
- Open Access

# GHM method for obtaining rationalsolutions of nonlinear differential equations

- Hector Vazquez-Leal
^{1}Email author and - Arturo Sarmiento-Reyes
^{2}

**4**:241

https://doi.org/10.1186/s40064-015-1011-x

© Vazquez-Leal and Sarmiento-Reyes. 2015

**Received:**29 December 2014**Accepted:**3 May 2015**Published:**4 June 2015

## Abstract

In this paper, we propose the application of the general homotopy method (GHM) to obtain rational solutions of nonlinear differential equations. It delivers a high precision representation of the nonlinear differential equation using a few linear algebraic terms. In order to assess the benefits of this proposal, three nonlinear problems are solved and compared against other semi-analytic methods or numerical methods. The obtained results show that GHM is a powerful tool, capable to generate highly accurate rational solutions.

**AMS subject classification** 34L30

## Keywords

- General homotopy method
- Nonlinear differential equations
- Epidemic model
- Heat radiation
- Boundary value

## Introduction

Solving nonlinear differential equations is an important issue in sciences because many physical phenomena are modelled using such equations (Vazquez-Leal and Sarmiento-Reyes 2015). One of the most powerful methods to approximately solve nonlinear differential equations is the homotopy perturbation method (HPM) (Aminikhah 2012; Barari et al. 2008; Biazar and Eslami 2011; Biazar and Ghanbari 2012; Compean et al. 2012; El-Sayed et al. 2012; Faraz and Khan 2011; Fathizadeh et al. 2011; Filobello-Nino et al. 2012a,b; He 2004, 2009; Khan et al. 2013, 2011a,b; Mohyud-Din et al. 2012; Vazquez-Leal et al. 2012a; Wang et al. 2012). Recently, HPM method was generalized by introducing the Rational Homotopy Perturbation method (RHPM) (Vazquez-Leal 2012; Vazquez-Leal et al. 2012b), multiparameter and nonlinearities distribution HPM (Vazquez-Leal et al. 2012c), fixed-term homotopy (Vazquez-Leal et al. 2013), and the generalized homotopy method (GHM) (Vazquez-Leal 2014). Using as inspiration the RHPM method, we propose a rational expression as a particular case of application of the GHM method. In RHPM method, we consider that the approximate solution of a differential equation can be represented by the quotient of two power series of the homotopy parameter; that quotient of power series transforms the nonlinear differential equation into a series of linear differential equations. Therefore, we propose a rational GHM version of the RHPM method with the advantage of automatically obtaining the numerator and denominator of the rational solution. The main characteristic of rational version of GHM is that obtains a Taylor series of the quotient in terms of the homotopy parameter. The resulting power series is used in the same fashion like the RHPM or HPM methods, transforming a nonlinear differential equation into a series of linear differential equations. Once solved the system of differential equations, we use the results to reconstruct the original rational expression which increase the accuracy of the approximations. To assess the potential of the proposed methodology, three nonlinear problems will be solved and compared using similar methodologies or numerical methods: a nonlinear boundary valued problem (BVP) (Li and Liao 2005), a heat radiation initial valued problem (IVP) (Ganji and Rajabi 2006), and an epidemic model containing several variables (Guerrero et al. 2011).

This paper is organized as follows. In Section ‘Basic concept of GHM method’, we introduce the basic concept of the rational version of GHM method. In Section ‘Case studies’, we show the solution of three nonlinear differential equations of different kind. Numerical simulations and a discussion about the results are provided in Section ‘Numerical simulation and discussion’. Finally, a brief conclusion is given in Section ‘Conclusions’.

## Basic concept of GHM method

where *L* and *N* are a linear and a non-linear operator, respectively; *f*(*r*) is a known analytic function, *B* is a boundary operator, *Γ* is the boundary of domain *Ω*, and *∂*
*u*/*∂*
*η* denotes differentiation along the normal drawn outwards from *Ω* (Wang et al. 2012).

where *u*
_{0} is the initial approximation for (1) which satisfies the boundary conditions and *p* is known as the homotopy parameter. When *p*=0, (3) is reduced to a trivial equation easy to solve, and when *p*=1, (3) is reduced to the original nonlinear differential Eq. 1 (Barari et al. 2008; He 2004,2009; Khan et al. 2013; Vazquez-Leal et al. 2012a).

*p*

where *v*
_{1},*v*
_{2},… are unknown functions to be determined by the RHPM method and *w*
_{1},*w*
_{2},… are known (arbitrary) functions of the independent variable.

*M*represents the order of the approximation, and

where the maximum order of the power of *p* employed is considered as the order of the approximation.

In RHPM, we obtain only the unknown coefficients of the numerator because the denominator is proposed by user. However, the improvement in this work, is that GHM obtains coefficients directly for numerator and denominator. In (5), *W* represents the order of the numerator, and *M* the order of the denominator considering that the lowest power of the denominator is *W*+1. In (6), the numerator is composed by even powers and the denominator by odd powers. Here, it is important to remark that the powers of *p* of the rational expressions (5) or (6), are in fact not repeated in numerator and denominator, otherwise, the GHM will not work properly. Therefore, this is a restriction of GHM for rational expressions.

where *v*
_{0},*v*
_{1},… are unknown functions to be determined by the GHM method and *g*
_{
i
} are functions obtained by the application of Taylor series method.

*p*-powers and equating its coefficients to zero. The resulting system of linear differential equations is solved to obtain

*v*

_{0},

*v*

_{1},…. Next, substituting

*v*

_{0},

*v*

_{1},… into (5) or (6) and calculating the limit, when

*p*→1, provides an approximate solution for (1) in the form of

Usually, a low order approximation is enough to obtain a highly accurate result as depicted in the next section. A study of convergence of GHM method was reported in (Vazquez-Leal 2014).

## Case studies

In the present section, we will solve three case studies to show the utility of the GHM method to solve nonlinear differential equations.

### Nonlinear boundary value problem

As it is known, Gelfand’s equation (Li and Liao 2005) (also known as Bratu’s problem in 1D) models the chaotic dynamics in combustible gas thermal ignition. Therefore, it is important to search for accurate solutions for this equation.

where prime denotes differentiation with respect to *t* and *κ* is known as Gelfand’s parameter.

*p*, we obtain

*κ*=1, we solve (15), resulting

### Heat radiation equation

where prime denotes differentiation with respect to *τ* and *ε*
_{1},*ε*
_{2} are parameters of the equation.

*p*, we obtain

### Model for evolution of smoking habit in Spain

where dots denote differentiation with respect to *t*.

The sub-populations included in the model are: *n* is the proportion of the total population who has never smoked, *s* is the proportion of people who smoke less than 20 cigarettes per day, *c* is the proportion of individuals who smoke more than 20 cigarettes per day, and *e* is the proportion of ex-smokers.

Parameter *μ* denotes birth rate in Spain; *β* denotes the transmission rate due to social pressure to adopt smoking habit; *ρ* express the rate at which ex-smokers return to smoking; *α* is the rate at which an excessive smoker becomes a normal smoker by decreasing the number of cigarettes per day; *γ* is the rate at which normal smokers become excessive smokers by increasing the number of cigarettes per day; *λ* denotes the rate at which normal smokers stop smoking, and *δ* is the rate at which excessive smokers stop smoking.

for any instant of time.

We set the values of the parameters as reported in (Guerrero et al. 2013) for Spain: *μ*=0.01 years ^{−1}, *ρ*=0.0425 years ^{−1}, *β*=0.0381 years ^{−1}, *α*=0.1244 years ^{−1}, *γ*=0.1175 years ^{−1}, *λ*=0.0498 years ^{−1} and *δ*=0.0498 years ^{−1}. Moreover, the initial conditions are chosen as: *n*(0)=0.5045, *s*(0)=0.2059, *c*(0)=0.1559, and *e*(0)=0.1337, as reported in (Guerrero et al. 2013).

*t*and \(\hbar \) is a control parameter. Initial approximations are

where the order of the approximation is choose as 2*Q*=12.

*Q*+1 is calculated, resulting

*v*

_{ i,j }(

*i*,

*j*=1,2,3,…,2

*Q*) are functions yet to be determined. Substituting (30) into (27) and rearranging the coefficients of

*p*-power, we have

*v*

_{ i,j }(

*t*) (

*i*,

*j*=1,2,3,…), we must construct and solve the following system of equations, considering the initial conditions of

*v*

_{ i,j }(0)=0 for

*i*,

*j*=1,2,3,…

*v*

_{1,3},

*v*

_{2,3},

*v*

_{3,3},

*v*

_{4,3}, and the succeeding terms; nevertheless, because they are too cumbersome, we skip them and will be used only in the final results. Now, from (29), we obtain a 12-th order approximation; then, considering

*p*→1, yields the approximate solution for (25) as

where *n*(*t*)=*V*
_{1}, *s*(*t*)=*V*
_{2}, *c*(*t*)=*V*
_{3}, and *n*(*t*)=*V*
_{4}.

Now, we need to determine the value of the parameter \(\hbar \) to obtain the best fit for the exact solution (25).

*E*

_{ m }), defined as

where *K*=500, *Δ*
*t*=0.1; *n*
_{
r
}(·), *s*
_{
r
}(·), *c*
_{
r
}(·), and *e*
_{
r
}(·) are the numerical values obtained using the Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) (Enright et al. 1986; Fehlberg 1970) solution (built-in function of Maple software). We considered an absolute error of 10^{−12} for the setup.

*E*

_{ m }is the residual error due to the difference between the GHM solution and the exact solution within the interval 0≤

*t*≤50 years. Therefore, Fig. 1 shows the minimum mean square error that corresponds, approximately, to \(\hbar =0.265\). Hence, we obtain

## Numerical simulation and discussion

For all case studies, we used built-in numerical routines from Maple 13 for comparison purposes. For the BVP problem, it was utilized the scheme based on trapezoid combined with Richardson extrapolation. For the IVP problems, it was used the Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant (RKF45) (Enright et al. 1986; Fehlberg 1970). For both types of algorithms, it was used a tolerance of absolute error (A.E.) of 10^{−12}.

*s*(

*t*)). A control of convergence \(\hbar \) is employed to increase accuracy of the solution; it is done in similar fashion to the control of convergence for HAM method (Guerrero et al. 2013; Li and Liao 2005).

Thus, GHM method can potentially generate higher accurate solutions in comparison than the well established HAM method. In this example, we used the rational series (6), instead of (5), to show the flexibility of the GHM method. In fact, as long as the Taylor series of the proposed rational series keeps the general form (7), we can propose other combinations of *p*-powers to obtain more accurate solutions. In order to extend the convergence of GHM method, this method may be combined with others methods like those reported for HPM or HAM: the nonlinearities distribution homotopy perturbation method (NDHPM) (Vazquez-Leal et al. 2012c), the variational homotopy perturbation method (Matinfar et al. 2011; Noor and Mohyud-Din 2008), Padé approximants (Bararnia et al. 2012; Guerrero et al. 2013; Raftari and Yildirim 2011; Torabi and Yaghoobi 2011), Laplace-Padé after-treatment (Bahuguna et al. 2009; Ebaid 2011; Gökdoğgan et al. 2012; Jiao et al. 2002; Khan and Faraz 2011; Merdan et al. 2011; Momani and Ertürk 2008; Merdan et al. 2009; Sweilam et al. 2009; Tsai and Chen 2010), fixed-term homotopy (Vazquez-Leal et al. 2013), among others.

GHM and RHPM methods can generate highly accurate rational solutions. Nonetheless, on one side, RHPM (Vazquez-Leal 2012; Vazquez-Leal et al. 2012b) requires the proposal of an arbitrary power series for the divisor. Therefore, the RHPM solution procedure calculates only the numerator. This feature implies the requirement of adjustment parameters that should be recalculated for each specific value of the parameters of the nonlinear problem under study. On the other side, the rational version of GHM method obtains automatically the solutions for numerator and denominator. Therefore, this characteristic converts the GHM method into a more attractive tool due to its ability to generate general solutions.

The case studies where chosen in order to test the ability of GHM for the solution of problems with different type of nonlinearities and boundary conditions. For instance, the first case study exhibits an exponential nonlinearity and boundary valued conditions. Next, second case study is an initial condition problem with a four order power nonlinearity. Finally, in the last case study, we show that GHM can be applied to solve a system of nonlinear differential equations with initial conditions. In the present work, we choose arbitrary order approximations to depict the basic procedure of GHM for rational solutions, resulting highly accurate solutions (see Figs. 1, 3, and 5). In the same fashion as HPM, increasing the order of the GHM approximations will increase the accuracy. However, it is important to highlight that future work is required in order to propose a systematic procedure to choose the order of the GHM rational approximations.

In this manuscript, GHM is presented as a novel tool to find rational solutions of different nonlinear differential equations. For instance, we can observe that (24) is expressed in terms of the division of the sum of exponential terms (see (23)) and the coefficients (*ε*
_{1} and *ε*
_{2}) of (18); this type of approximation is indeed very difficult to obtain (or impossible) with HPM, PM, HAM, among other approximative methods. Thereupon, further research is required to explore all the potential benefits of this proposal.

## Conclusions

This work introduced a rational version of the generalized homotopy method (GHM) as a useful tool with high potential to solve nonlinear differential equations. We were able to obtain accurate and handy rational solutions for different types of problems: a nonlinear BVP problem, a highly nonlinear IVP problem, and an epidemic model. The high precision of the GHM solutions is due to the generated rational expressions that can potentially fit a wider scope of non-linearities. Also, a comparison between the results of applying the proposed method and PM/HPM/HAM was shown; concluding that GHM method provided more accurate approximations. Finally, further research can be focused on the application of rational version of GHM method for the solution of nonlinear differential algebraic equations, nonlinear fractional differential equations, nonlinear partial differential equations, among others.

## Declarations

### Acknowledgements

We gratefully acknowledge the financial support from 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.

## Authors’ Affiliations

## References

- Aminikhah, H (2012) The combined laplace transform and new homotopy perturbation methods for stiff systems of odes. Appl Math Model 36(8): 3638–3644.View ArticleGoogle Scholar
- Bahuguna, D, Ujlayan A, Pandey DN (2009) A comparative study of numerical methods for solving an integro-differential equation. Comput Math Appl 57(9): 1485–1493.View ArticleGoogle Scholar
- Barari, A, Omidvar M, Ghotbi A, Ganji D (2008) Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations. Acta Applicandae Mathematicae 104: 161–171.View ArticleGoogle Scholar
- Bararnia, H, Ghasemi E, Soleimani S, Ghotbi AR, Ganji DD (2012) Solution of the falkner-skan wedge flow by hpm-pade’ method. Adv Eng Softw 43(1): 44–52.View ArticleGoogle Scholar
- Biazar, J, Eslami M (2011) A new homotopy perturbation method for solving systems of partial differential equations. Comput Math Appl 62(1): 225–234.View ArticleGoogle Scholar
- Biazar, J, Ghanbari B (2012) The homotopy perturbation method for solving neutral functional-differential equations with proportional delays. J King Saud University - Sci 24(1): 33–37.View ArticleGoogle Scholar
- Compean, FI, Olvera D, Campa FJ, de Lacalle LNL, Elias-Zuniga A, Rodriguez CA (2012) Characterization and stability analysis of a multivariable milling tool by the enhanced multistage homotopy perturbation method. Int J Mach Tools Manuf 57(0): 27–33.View ArticleGoogle Scholar
- Ebaid, AE (2011) A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities. Commun Nonlinear Sci Numeric Simul 16(1): 528–536.View ArticleGoogle Scholar
- El-Sayed, AMA, Elsaid A, El-Kalla IL, Hammad D (2012) A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains. Appl Math Comput 218(17): 8329–8340.View ArticleGoogle Scholar
- Enright, W, Jackson K, Norsett SA, Thomsen P (1986) Interpolants for runge-kutta formulas. ACM TOMS 12: 193–218.View ArticleGoogle Scholar
- Faraz, N, Khan Y (2011) Analytical solution of electrically conducted rotating flow of a second grade fluid over a shrinking surface. Ain Shams Eng J 2(3-4): 221–226.View ArticleGoogle Scholar
- Fathizadeh, M, Madani M, Khan Y, Faraz N, Yildirim A, Tutkun S (2011) An effective modification of the homotopy perturbation method for mhd viscous flow over a stretching sheet. J King Saud University - Sci. doi:10.1016/j.jksus.2011.08.003.
- Fehlberg, E (1970) Klassische runge-kutta-formeln vierter und niedrigerer ordnung mit schrittweiten-kontrolle und ihre anwendung auf waermeleitungsprobleme. Computing 6: 61–71.View ArticleGoogle Scholar
- Filobello-Nino, U, Vazquez-Leal H, Khan Y, Castaneda-Sheissa R, Yildirim A, Hernandez-Martinez L, Sanchez-Orea J, Castaneda-Sheissa R, Bernal FR (2012a) Hpm applied to solve nonlinear circuits: a study case. Appl Math Sci 6(85-88): 4331–4344.Google Scholar
- Filobello-Nino, U, Vazquez-Leal H, Castaneda-Sheissa R, Yildirim A, Hernandez-Martinez L, Pereyra-Diaz D, Perez-Sesma A, Hoyos-Reyes C (2012b) An approximate solution of blasius equation by using hpm method. Asian J Math Stat 5: 50–59.Google Scholar
- Ganji, DD, Rajabi A (2006) Assessment of homotopy-perturbation and perturbation methods in heat radiation equations. Intl Commun Heat Mass Transfer 33(3): 391–400.View ArticleGoogle Scholar
- Gökdoğan, A, Merdan M, Yildirim A (2012) The modified algorithm for the differential transform method to solution of genesio systems. Commun Nonlinear Sci Numeric Simul 17(1): 45–51.View ArticleGoogle Scholar
- Guerrero, F, Santonja FJ, Villanueva RJ (2011) Analysing the effect of spanish smoke-free legislation of year 2006: A new method to quantify its impact using a dynamic model. Int J Drug Policy 22: 247–251.View ArticleGoogle Scholar
- Guerrero, F, Santonja F, Villanueva R (2013) Solving a model for the evolution of smoking habit in spain with homotopy analysis method. Nonlinear Anal: Real World Appl 14(1): 549–558.View ArticleGoogle Scholar
- He, JH (2004) Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 156(2): 527–539.View ArticleGoogle Scholar
- He, J H (2009) An elementary introduction to the homotopy perturbation method. Comput Math Appl 57(3): 410–412.View ArticleGoogle Scholar
- Jiao, YC, Yamamoto Y, Dang C, Hao Y (2002) An aftertreatment technique for improving the accuracy of adomian’s decomposition method. Comput Math Appl 43(6-7): 783–798.View ArticleGoogle Scholar
- Khan, Y, Vazquez-Leal H, Wu Q (2013) An efficient iterated method for mathematical biology model. Neural Comput Appl 23(3-4): 677–682.View ArticleGoogle Scholar
- Khan, Y, Faraz N (2011) Application of modified laplace decomposition method for solving boundary layer equation. J King Saud University - Sci 23(1): 115–119.View ArticleGoogle Scholar
- Khan, Y, Wu Q, Faraz N, Yildirim A, Madani M (2011a) A new fractional analytical approach via a modified riemann-liouville derivative. Appl Math Lett. doi:10.1016/j.aml.2011.11.041.
- Khan, Y, Wu Q, Faraz N, Yildirim A (2011b) The effects of variable viscosity and thermal conductivity on a thin film flow over a shrinking-stretching sheet. Comput Math Appl 61(11): 3391–3399.Google Scholar
- Li, S, Liao SJ (2005) An analytic approach to solve multiple solutions of a strongly nonlinear problem. Appl Math Comput 169(2): 854–865.View ArticleGoogle Scholar
- Matinfar, M, Mahdavi M, Raeisy Z (2011) Exact and numerical solution of lineard’s equation by the variational homotopy perturbation method. J Inf Comput Sci 6(1): 73–80.Google Scholar
- Merdan, M, Gökdoğan A, Yildirim A (2011) On the numerical solution of the model for hiv infection of cd4
^{+}t cells. Comput Math Appl 62: 118–123.View ArticleGoogle Scholar - Momani, S, Ertürk VS (2008) Solutions of non-linear oscillators by the modified differential transform method. Comput Math Appl 55(4): 833–842.View ArticleGoogle Scholar
- Momani, S, Erjaee GH, Alnasr MH (2009) The modified homotopy perturbation method for solving strongly nonlinear oscillators. Comput Math Appl 58(11-12): 2209–2220.View ArticleGoogle Scholar
- Mohyud-Din, ST, Yildirim A, Inc M (2012) Coupling of homotopy perturbation and modified lindstedt-poincaré methods for traveling wave solutions of the nonlinear klein-gordon equation. J King Saud University - Sci 24(2): 187–191.View ArticleGoogle Scholar
- Noor, MA, Mohyud-Din ST (2008) Variational homotopy perturbation method for solving higher dimensional initial boundary value problems. Math Probl Eng 2008(Article ID 696734): 11. doi:10.1155/2008/696734.
- Raftari, B, Yildirim A (2011) Series solution of a nonlinear ode arising in magnetohydrodynamic by hpm-padé technique. Comput Math Appl 61(6): 1676–1681.View ArticleGoogle Scholar
- Sweilam, NH, Khader MM (2009) Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method. Comput Math Appl 58(11 - 12): 2134–2141.View ArticleGoogle Scholar
- Torabi, M, Yaghoobi H (2011) Novel solution for acceleration motion of a vertically falling spherical particle by hpm-padé approximant. Adv Powder Technol 22(5): 674–677.View ArticleGoogle Scholar
- Tsai, PY, Chen CK (2010) An approximate analytic solution of the nonlinear riccati differential equation. J Franklin Inst 347(10): 1850–1862.View ArticleGoogle Scholar
- Vazquez-Leal, H (2012) Rational homotopy perturbation method. J Appl Math14. doi:10.1155/2012/490342.
- Vazquez-Leal, H, Castañeda-Sheissa R, Filobello-Niño U, Sarmiento-Reyes A, Sánchez-Orea J (2012a) High accurate simple approximation of normal distribution related integrals. Math Probl Eng 2012: 22.Google Scholar
- Vazquez-Leal, H, Sarmiento-Reyes A, Khan Y, Filobello-Nino U, Diaz-Sanchez A (2012b) Rational biparameter homotopy perturbation method and laplace-padé coupled version. J Appl Math 2012: 21. doi:10.1155/2012/923975.Google Scholar
- Vazquez-Leal, H, Filobello-Nino U, Castaneda-Sheissa R, Hernandez-Martinez L, Sarmiento-Reyes A (2012c) Modified hpms inspired by homotopy continuation methods. Math Probl Eng 2012: 19.Google Scholar
- Vazquez-Leal, H, Khan Y, Filobello-Nino U, Sarmiento-Reyes A, Diaz-Sanchez A, Cisneros-Sinencio LF (2013) Fixed-term homotopy. J Appl Math: 1–11. doi:10.1155/2013/972704.
- Vazquez-Leal, H (2014) Generalized homotopy method for solving nonlinear differential equations. Comput Appl Math 33(1): 275–288.View ArticleGoogle Scholar
- Vazquez-Leal, H, Guerrero F (2014) Application of series method with padé and laplace-padé resummation methods to solve a model for the evolution of smoking habit in spain. Comput Appl Math 33(1): 181–192. doi:10.1007/s40314-013-0054-2.
- Vazquez-Leal, H, Sarmiento-Reyes A (2015) Power series extender method for the solution of nonlinear differential equations. Math Probl Eng 15(7). doi:10.1155/2015/717404.
- Wang, YG, Lin WH, Liu N (2012) A homotopy perturbation-based method for large deflection of a cantilever beam under a terminal follower force. Intl J Comput Methods Eng Sci Mech 13: 197–201.View ArticleGoogle Scholar

## Copyright

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.