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Journal of Applied Mathematics

Volume 2013 (2013), Article ID 972704, 11 pages

http://dx.doi.org/10.1155/2013/972704

## Fixed-Term Homotopy

^{1}Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Circuito Gonzalo Aguirre Beltrán s/n, 91000 Xalapa, VER, Mexico^{2}Department of Mathematics, Zhejiang University, Hangzhou 310027, China^{3}Department of Electronics, National Institute for Astrophysics, Optics, and Electronics, Luis Enrique Erro No. 1, 72840 Sta. María Tonantzintla, PUE, Mexico

Received 6 September 2012; Revised 6 December 2012; Accepted 20 December 2012

Academic Editor: Chein-Shan Liu

Copyright © 2013 Hector Vazquez-Leal 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

A new tool for the solution of nonlinear differential equations is presented. The Fixed-Term Homotopy (FTH) delivers a high precision representation of the nonlinear differential equation using only a few linear algebraic terms. In addition to this tool, a procedure based on Laplace-Padé to deal with the truncate power series resulting from the FTH method is also proposed. In order to assess the benefits of this proposal, two nonlinear problems are solved and compared against other semianalytic methods. The obtained results show that FTH is a power tool capable of generating highly accurate solutions compared with other methods of literature.

#### 1. Introduction

Many physical phenomena are commonly modelled using nonlinear differential equations, which is a straightforward way to describe the behaviour of their dynamics. Among these methods, the most commonly used is the Homotopy Perturbation Method (HPM) [1–49]. This method is based in the use of a power series of the homotopy parameter, which transforms the original nonlinear differential equation into a series of linear differential equations. In this paper, a generalization of this concept using a product of two power series of the homotopy parameter called Fixed Term Homotopy (FTH) method is proposed. FTH method transforms the nonlinear differential equation into a series of linear differential equations, generating high precision expressions with fewer algebraic terms, reducing the computing cost. Furthermore, in order to deal with the truncate power series obtained with FTH method, the use of Laplace-Pade after-treatment is also proposed. To assess the potential of the proposed methodology, two nonlinear problems, Van Der Pol Oscillator [6, 50] and Troesch’s equation [51–57], will be solved and compared using similar methodologies.

This paper is organized as follows. In Section 2, the fundamental idea of the FTH method is described. Section 3 presents a study of convergence for the proposed method. Section 4 introduces the Laplace-Padé after-treatment. In Sections 5 and 6, the solution procedure of two nonlinear problems is presented. Additionally, a discussion of the obtained results and the finds of this work are summarized in Section 7. Finally, the conclusions are presented in Section 8.

#### 2. Basic Concept of FTH Method

The FTH and HPM methods share common foundations. Both methods consider that a nonlinear differential equation can be expressed as which has as boundary condition where is a general differential operator, is a known analytic function, is a boundary operator, is the boundary of domain , and denotes differentation along the normal drawn outwards from [58]. In general, the operator can be divided into two operators and , which are the corresponding linear and nonlinear operators, respectively. Hence, (1) can be rewritten as

Now, a possible homotopy formulation is given by the expression where is the trial function (initial approximation) for (3) which satises the boundary conditions, and is known as the perturbation homotopy parameter. From analyzing (4), it can be concluded that

For , the homotopy map (4) is reduced to the problem (5) that possesses a trivial solution . Moreover, for , the homotopy map (4) is transformed into the original nonlinear problem (6) that possesses the sought solution.

For the HPM method [9–12], we assume that the solution for (4) can be written as a power series of , such that

Considering that , it results that the approximate solution for (1) is

The series (8) is convergent on most cases [9, 12].

In [59], a homotopy which uses the auxiliary term was reported. Then, modifying that version, it results in the following proposed homotopy: where is an arbitrary function.

When or , the auxiliary term is set to zero. Then, does not affect or change the initial solution when or the sought solution at . Moreover, a properly selection of can be useful to improve convergence of the homotopy. Now, for the FTH method, (7) can be rewritten as the product of two power series, such that where are unknown functions to be determined by the FTH method, and are arbitrarily chosen fixed terms.

After substituting (10) into (9), and equating terms with the same order of , we obtain a set of linear equations that lead us to calculate . The limit of (10), when , provides an approximate solution for (3) in the form of

The upper limit exists in the event that both limits exists so and fixed term exist.

#### 3. Convergence of FTH Method

To analyze the convergence of FTH, (9) is rewritten as

Applying the inverse operator to both sides of (14), we obtain

Assuming that (see (10)) and substituting (16) into the right-hand side of (15), we obtain

The exact solution of (3) is obtained when of (17), resulting in

For the convergence analysis of the FTH method, we used the Banach Theorem as reported in [1, 2, 5, 60]. Such theorem relates the solution of (3) and the fixed point problem of the nonlinear operator .

Theorem 1 (Sufficient Condition of Convergence). *Suppose that and are Banach spaces and is a contractive nonlinear mapping, then
*

According to Banach Fixed Point Theorem, has a unique fixed point such that . Assume that the sequence generated by the FTH method can be written as

If one assumes that , where . Then, under these conditions:(i),(ii).

*Proof. *(i) By inductive approach, we have for

Assuming as induction hypothesis that , then

Using (i), we have

(ii) Because of and , , that is,

#### 4. Laplace-Padé after Treatment for FTH Series

The coupling of Laplace transform and Padé approximant [61] is used in order to recover part of the lost information due to the truncated power series [60, 62–70]. The process can be recast as follows.(1)First, Laplace transformation is applied to power series obtained by FTH method. (2)Next, is substituted by in the resulting equation. (3)After that, we convert the transformed series into a meromorphic function by forming its Padé approximant of order . and are arbitrarily chosen, but they should be of smaller value than the order of the power series. In this step, the Padé approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. (4)Then, is substituted by .(5)Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution.

We will denominate to this process as the Laplace-Padé fixed term homotopy (LPFTH) method.

#### 5. Van Der Pol Oscillator

Consider the Van der Pol Oscillator problem [6, 50] which have the exact solution

In order to find an approximate solution for (25) by means of LPFTH, we obtain the Taylor series of the trigonometric terms, resulting is where the expansion order is 7.

From (9), we establish the following homotopy equation where is an adjustment parameter due to the auxiliary term .

From (10), we assume that solution for (28) has the following form: where is an adjustment parameter of the fixed term of the homotopy map.

Substituting (29) into (28), and rearranging the terms of the same order of ,

By solving (30), we obtain

Substituting (31) into (29), and calculating the limit when , we obtain the second order approximated solution

Then, we select the adjustment parameters as: , ; where the parameters are calculated using the NonlinearFit command from Maple Release 13 [5, 32–34]. Moreover, NonlinearFit command finds values of the approximate model parameters such that the sum of the squared -residuals is minimized.

In order to guarantee the validity of the approximate solution (32) for large , the series solution is transformed by the Laplace-Padé after-treatment. First, Laplace transformation is applied to (32) and then is written in place of in the equation. Then, the Padé approximant is applied and is written in place of . Finally, by using the inverse Laplace transformation, we obtain the modified approximate solution

#### 6. Troesch’s Problem

The Troesch’s equation is a boundary value problem (BVP) that arises in the investigation of confinement of a plasma column by a radiation pressure [71] and also in the theory of gas porous electrodes [72, 73]. The problem is expressed as where prime denotes differentiation with respect to and is known as Troesch’s parameter.

Straightforward application of FTH to solve (34) is not possible due to the hyperbolic term of dependent variable. Nevertheless, the polynomial type nonlinearities are easier to handle by the FTH method. Therefore, in order to apply FTH successfully, we convert the hyperbolic-type nonlinearity in Troesch’s problem into a polynomial type nonlinearity, using the variable transformation reported in [51, 52]

After using (35), we obtain the following transformed problem: where conditions are obtained by using variable transformation (35).

Substituting original boundary conditions and into (35) results in

From (9) and (36), we can formulate the following homotopy [7–9]: where is the homotopy parameter and is an adjustment parameter due to the auxiliary term .

From (10), we assume that solution for (38) has the following form: where are the adjustment parameters due to the fixed term of the homotopy map.

Substituting (39) into (38) and equating identical powers of terms, we obtain where .

We solve (40) by using Maple software, resulting in

Substituting (41) into (39), and calculating the limit when , we obtain the second order approximated solution of (36)

Finally, from (35) and (42), the proposed solution of Troesch’s problem is

If we consider , then we choose the adjustment parameter as: , , and , by using the procedure explained for the Van der Pol Oscillator problem.

#### 7. Numerical Simulation and Discussion

A comparison of the exact solution (26) of the Van der Pol Oscillator against the approximated solutions obtained by LPFTH (33), HPM [6], and VIM [6] is shown in Table 1 and Figure 1. In the comparison can be seen that the LPFTH method exhibited the higher accuracy for a large period of time reproducing successfully the oscillatory behaviour of the exact solution (26), therefore, being a suitable alternative to solve nonlinear oscillators.

The Troesch’s BVP problem is a benchmark equation for numerical [74, 75] and semi-analytical methods [51–57] which has been solved by FTH to obtain the approximated solution (43). Table 2 shows a comparison of the results obtained with other semi-analytical methods as: homotopy perturbation method (HPM) [52, 54, 55], decomposition method approximation (DMA) [53, 54], homotopy analysis method (HAM) [56], and Laplace transform decomposition method (LTDM) [57]. The comparison shows that the average absolute relative error (A.A.R.E.) of (43) is lower than most of the reported results and similar to LDTM [57].

For both cases of study, polynomial functions were used for fixed and auxiliary terms as they generate the best results. Nonetheless, functions as exponential, trigonometric or hyperbolic may enlarge the domain of convergence. Thereby, a methodology that allows the selection of fixed and auxiliary terms like these to obtain more accurate solutions is a window of opportunity to be exploited.

Since FTH does not rely in the linearization of the input equations, a perturbation parameter nor assumption of weak nonlinearity, the solution generated may be general and more realistic than the method of simplifying the physical problems.

#### 8. Conclusions

In this work, the Fixed Term Homotopy and the Laplace-Padé Fixed Term Homotopy methods are presented as novel tools to solve nonlinear ordinary differential equations. The proposed methods were tested using two nonlinear problems: a second order nonlinear oscillator and a high nonlinear boundary value problem. From a comparison against several semi-analytical methods from the literature, FTH and LPFTH probed to be power tools that generate highly accurate easy manageable expressions. Furthermore, the homotopy given in (9) may be replaced or modified by a formulation inspired by some other homotopy method proposed in literature [76–96], which can lead to improvement of convergence of proposed methods. Additionally, being FTH a modified version of HPM can be assumed that the differential equations solved by HPM should be also solvable by FTH. In that fashion, further research can be focused on applying FTH method in the solution of nonlinear partial differential equations, nonlinear fractional differential equations, among others.

#### Acknowledgments

The authors gratefully acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACyT) through Grant CB-2010-01 no. 157024. The authors would like to thank Roberto Castaneda-Sheissa, Rogelio-Alejandro Callejas-Molina, and Roberto Ruiz-Gomez for their contribution to this project.

#### References

- 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and H. Ghazvini, “Convergence of the homotopy perturbation method for partial differential equations,”
*Nonlinear Analysis. Real World Applications*, vol. 10, no. 5, pp. 2633–2640, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Ağırseven and T. Öziş, “An analytical study for Fisher type equations by using homotopy perturbation method,”
*Computers & Mathematics with Applications*, vol. 60, no. 3, pp. 602–609, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Öziş and D. Ağırseven, “He's homotopy perturbation method for solving heat-like and wave-like equations with variable coefficients,”
*Physics Letters A*, vol. 372, no. 38, pp. 5944–5950, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Vazquez-Leal, “Rational homotopy perturbation method,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 490342, 14 pages, 2012. View at Publisher · View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Comparison of homotopy perturbation method and homotopy analysis method,”
*Applied Mathematics and Computation*, vol. 156, no. 2, pp. 527–539, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “An elementary introduction to the homotopy perturbation method,”
*Computers & Mathematics with Applications*, vol. 57, no. 3, pp. 410–412, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Homotopy perturbation technique,”
*Computer Methods in Applied Mechanics and Engineering*, vol. 178, no. 3-4, pp. 257–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “The homotopy perturbation method nonlinear oscillators with discontinuities,”
*Applied Mathematics and Computation*, vol. 151, no. 1, pp. 287–292, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,”
*Applied Mathematics and Computation*, vol. 135, no. 1, pp. 73–79, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - J. H. He, “Application of homotopy perturbation method to nonlinear wave equations,”
*Chaos, Solitons and Fractals*, vol. 26, no. 3, pp. 695–700, 2005. View at Publisher · View at Google Scholar · View at Scopus - J.-H. He, “Homotopy perturbation method for solving boundary value problems,”
*Physics Letters A*, vol. 350, no. 1-2, pp. 87–88, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Khan, H. Vazquez-Leal, and Q. Wu, “An efficient iterated method for mathematical biology model,”
*Neural Computing and Applications*. In press. - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and B. Ghanbari, “The homotopy perturbation method for solving neutral functional-differential equations with proportional delays,”
*Journal of King Saud University*, vol. 24, no. 1, pp. 33–37, 2012. View at Google Scholar - D. D. Ganji, H. Tari, and M. B. Jooybari, “Variational iteration method and homotopy perturbation method for nonlinear evolution equations,”
*Computers & Mathematics with Applications*, vol. 54, no. 7-8, pp. 1018–1027, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Sheikholeslami, H. R. Ashorynejad, D. D. Ganji, and A. Yildirim, “Homotopy perturbation method for three-dimensional problem of condensation film on inclined rotating disk,”
*Scientia Iranica*, vol. 19, no. 3, pp. 437–442, 2012. View at Google Scholar - D. D. Ganji and A. Sadighi, “Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations,”
*Journal of Computational and Applied Mathematics*, vol. 207, no. 1, pp. 24–34, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. D. Ganji, “The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer,”
*Physics Letters A*, vol. 355, no. 4-5, pp. 337–341, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - M. Rafei, D. D. Ganji, and H. Daniali, “Solution of the epidemic model by homotopy perturbation method,”
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 1056–1062, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. D. Ganji and A. Rajabi, “Assessment of homotopy-perturbation and perturbation methods in heat radiation equations,”
*International Communications in Heat and Mass Transfer*, vol. 33, no. 3, pp. 391–400, 2006. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Google Scholar - 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, 2011. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Fathizadeh, M. Madani, Y. Khan, N. Faraz, A. Yildirim, and S. Tutkun, “An effective modification of the homotopy perturbation method for mhd viscous flow over a stretching sheet,”
*Journal of King Saud University*. In press. - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Vázquez-Leal, R. Castaneda-Sheissa, U. Filobello-Nino, A. Sarmiento-Reyes, and J. Sánchez-Orea, “High accurate simple approximation of normal distribution related integrals,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 124029, 22 pages, 2012. View at Publisher · View at Google Scholar - H. Vazquez-Leal, U. Filobello-Nino, R. Castaneda-Sheissa, L. Hernandez-Martinez, and A. Sarmiento-Reyes, “Modified hpms inspired by homotopy continuation methods,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 309123, 19 pages, 2012. View at Publisher · View at Google Scholar - 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. View at Google Scholar - 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. View at Google Scholar · View at MathSciNet - J. Biazar, F. Badpeima, and F. Azimi, “Application of the homotopy perturbation method to Zakharov-Kuznetsov equations,”
*Computers & Mathematics with Applications*, vol. 58, no. 11-12, pp. 2391–2394, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and H. Ghazvini, “Homotopy perturbation method for solving hyperbolic partial differential equations,”
*Computers & Mathematics with Applications*, vol. 56, no. 2, pp. 453–458, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and H. Ghazvini, “Exact solutions for non-linear Schrödinger equations by He's homotopy perturbation method,”
*Physics Letters A*, vol. 366, no. 1-2, pp. 79–84, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar, H. Ghazvini, and M. Eslami, “He's homotopy perturbation method for systems of integro-differential equations,”
*Chaos, Solitons & Fractals*, vol. 39, no. 3, pp. 1253–1258, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar, M. Eslami, and H. Aminikhah, “Application of homotopy perturbation method for systems of volterra integral equations of the first kind,”
*Chaos, Solitons and Fractals*, vol. 42, no. 5, pp. 3020–3026, 2009. View at Google Scholar - J. Biazar and H. Ghazvini, “He's homotopy perturbation method for solving systems of Volterra integral equations of the second kind,”
*Chaos, Solitons & Fractals*, vol. 39, no. 2, pp. 770–777, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Biazar and H. Ghazvini, “Numerical solution for special non-linear Fredholm integral equation by HPM,”
*Applied Mathematics and Computation*, vol. 195, no. 2, pp. 681–687, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Jalaal, D. D. Ganji, and G. Ahmadi, “Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media,”
*Advanced Powder Technology*, vol. 21, no. 3, pp. 298–304, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Jalaal and D. D. Ganji, “On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids,”
*Advanced Powder Technology*, vol. 22, no. 1, pp. 58–67, 2011. View at Publisher · View at Google Scholar · View at Scopus - Z. Ziabakhsh and G. Domairry, “Solution of the laminar viscous flow in a semi-porous channel in the presence of a uniform magnetic field by using the homotopy analysis method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 4, pp. 1284–1294, 2009. View at Publisher · View at Google Scholar · View at Scopus - D. D. Ganji, G. A. Afrouzi, and R. A. Talarposhti, “Application of variational iteration method and homotopy-perturbation method for nonlinear heat diffusion and heat transfer equations,”
*Physics Letters A*, vol. 368, no. 6, pp. 450–457, 2007. View at Publisher · View at Google Scholar · View at Scopus - M. Rafei, H. Daniali, and D. D. Ganji, “Variational interation method for solving the epidemic model and the prey and predator problem,”
*Applied Mathematics and Computation*, vol. 186, no. 2, pp. 1701–1709, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - G. Domairry, A. Mohsenzadeh, and M. Famouri, “The application of homotopy analysis method to solve nonlinear differential equation governing Jeffery-Hamel flow,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 1, pp. 85–95, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Sadighi and D. D. Ganji, “Analytic treatment of linear and nonlinear Schrödinger equations: a study with homotopy-perturbation and Adomian decomposition methods,”
*Physics Letters A*, vol. 372, no. 4, pp. 465–469, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-H. Chang, “A variational iteration method for solving Troesch's problem,”
*Journal of Computational and Applied Mathematics*, vol. 234, no. 10, pp. 3043–3047, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Vazquez-Leal, Y. Khan, G. Fernandez-Anaya et al., “A general solution for troesch's problem,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 208375, 14 pages, 2012. View at Publisher · View at Google Scholar - E. Deeba, S. A. Khuri, and S. Xie, “An algorithm for solving boundary value problems,”
*Journal of Computational Physics*, vol. 159, no. 2, pp. 125–138, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Feng, L. Mei, and G. He, “An efficient algorithm for solving Troesch's problem,”
*Applied Mathematics and Computation*, vol. 189, no. 1, pp. 500–507, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. H. Mirmoradi, I. Hosseinpour, S. Ghanbarpour, and A. Barari, “Application of an approximate analytical method to nonlinear Troesch's problem,”
*Applied Mathematical Sciences*, vol. 3, no. 29-32, pp. 1579–1585, 2009. View at Google Scholar · View at MathSciNet - H. N. Hassan and M. A. El-Tawil, “An efficient analytic approach for solving two-point nonlinear boundary value problems by homotopy analysis method,”
*Mathematical Methods in the Applied Sciences*, vol. 34, no. 8, pp. 977–989, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. A. Khuri, “A numerical algorithm for solving Troesch's problem,”
*International Journal of Computer Mathematics*, vol. 80, no. 4, pp. 493–498, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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, no. 3, pp. 197–201, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - J.-H. He, “Homotopy perturbation method with an auxiliary term,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 857612, 7 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Vazquez-Leal, A. Sarmiento-Reyes, Y. Khan, U. Filobello-Nino, and A. Diaz-Sanchez, “Rational biparameter homotopy perturbation method and laplace-padé coupled version,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 923975, 21 pages, 2012. View at Publisher · View at Google Scholar - Y. C. Jiao, Y. Yamamoto, C. Dang, and Y. Hao, “An aftertreatment technique for improving the accuracy of Adomian's decomposition method,”
*Computers & Mathematics with Applications*, vol. 43, no. 6-7, pp. 783–798, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. H. Sweilam and M. M. Khader, “Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method,”
*Computers & Mathematics with Applications*, vol. 58, no. 11-12, pp. 2134–2141, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Khan and N. Faraz, “Application of modified Laplace decomposition method for solving boundary layer equation,”
*Journal of King Saud University*, vol. 23, no. 1, pp. 115–119, 2011. View at Publisher · View at Google Scholar · View at Scopus - D. Bahuguna, A. Ujlayan, and D. N. Pandey, “A comparative study of numerical methods for solving an integro-differential equation,”
*Computers & Mathematics with Applications*, vol. 57, no. 9, pp. 1485–1493, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Momani and V. S. Ertürk, “Solutions of non-linear oscillators by the modified differential transform method,”
*Computers & Mathematics with Applications*, vol. 55, no. 4, pp. 833–842, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - A. Gökdoğan, M. Merdan, and A. Yildirim, “The modified algorithm for the differential transform method to solution of Genesio systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 1, pp. 45–51, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - P.-Y. Tsai and C.-K. Chen, “An approximate analytic solution of the nonlinear Riccati differential equation,”
*Journal of the Franklin Institute. Engineering and Applied Mathematics*, vol. 347, no. 10, pp. 1850–1862, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. E. Ebaid, “A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 16, no. 1, pp. 528–536, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Merdan, A. Gökdoğan, and A. Yildirim, “On the numerical solution of the model for HIV infection of CD4
^{+}Tcells,”*Computers & Mathematics with Applications*, vol. 62, no. 1, pp. 118–123, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. S. Weibel, “On the confinement of a plasma by magnetostatic fields,”
*Physics of Fluids*, vol. 2, no. 1, pp. 52–56, 1959. View at Google Scholar · View at Scopus - D. Gidaspow and B. S. Baker, “A model for discharge of storage batteries,”
*Journal of the Electrochemical Society*, vol. 120, no. 8, pp. 1005–1010, 1973. View at Google Scholar · View at Scopus - V. S. Markin, A. A. Chernenko, Y. A. Chizmadehev, and Y. G. Chirkov, “Aspects of the theory of gas porous electrodes,” in
*Fuel Cells: Their Electrochemical Kinetics*, pp. 22–33, Consultants Bureau, New York, NY, USA, 1966. View at Google Scholar - U. Erdogan and T. Ozis, “A smart nonstandard finite difference scheme for second order nonlinear boundary value problems,”
*Journal of Computational Physics*, vol. 230, no. 17, pp. 6464–6474, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Lin, J. A. Enszer, and M. A. Stadtherr, “Enclosing all solutions of two-point boundary value problems for ODEs,”
*Computers and Chemical Engineering*, vol. 32, no. 8, pp. 1714–1725, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Vazquez-Leal, R. Castaneda-Sheissa, A. Yildirim et al., “Biparameter homotopy-based direct current simulation of multistable circuits,”
*British Journal of Mathematics & Computer Science*, vol. 2, no. 3, pp. 137–150, 2012. View at Google Scholar - K. Reif, K. Weinzierl, A. Zell, and R. Unbehauen, “A homotopy approach for nonlinear control synthesis,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 9, pp. 1311–1318, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Aslam and A. K. Sunol, “Homotopy continuation based prediction of azeotropy in binary and multicomponent mixtures through equations of state,”
*Physical Chemistry Chemical Physics*, vol. 6, no. 9, pp. 2320–2326, 2004. View at Publisher · View at Google Scholar · View at Scopus - F. Kubler and K. Schmedders, “Computing equilibria in stochastic finance economies,”
*Computational Economics*, vol. 15, no. 1-2, pp. 145–172, 2000. View at Google Scholar - R. C. Melville and L. Trajković, “Artificial parameter homotopy methods for the dc operating point problem,”
*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*, vol. 12, no. 6, pp. 861–877, 1997. View at Google Scholar - K. Yamamura, T. Sekiguchi, and Y. Inoue, “A fixed-point homotopy method for solving modified nodal equations,”
*IEEE Transactions on Circuits and Systems*, vol. 46, no. 6, pp. 654–665, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. Geoghegan, J. C. Lagarias, and R. C. Melville, “Threading homotopies and dc operating points of nonlinear circuits,”
*SIAM Journal on Optimization*, vol. 9, no. 1, pp. 159–178, 1999. View at Publisher · View at Google Scholar · View at MathSciNet - J. Lee and H.-D. Chiang, “Convergent regions of the Newton homotopy method for nonlinear systems: theory and computational applications,”
*IEEE Transactions on Circuits and Systems I*, vol. 48, no. 1, pp. 51–66, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Ushida, Y. Yamagami, Y. Nishio, I. Kinouchi, and Y. Inoue, “An efficient algorithm for finding multiple DC solutions based on the SPICE-oriented Newton homotopy method,”
*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*, vol. 21, no. 3, pp. 337–348, 2002. View at Publisher · View at Google Scholar · View at Scopus - H. Vazquez-Leal, L. Hernaandez-Martinez, and A. Sarmiento-Reyes, “Double-bounded homotopy for analysing nonlinear resistive circuits,”
*International Symposium on Circuits and Systems*, vol. 4, pp. 3203–3206, 2005. View at Google Scholar - M. Van Barel, Kh. D. Ikramov, and A. A. Chesnokov, “A continuation method for solving symmetric toeplitz systems,”
*Computational Mathematics and Mathematical Physics*, vol. 48, no. 12, pp. 2126–2139, 2008. View at Google Scholar - H. Vázquez-Leal, L. Hernández-Martínez, A. Sarmiento-Reyes, and R. Castañeda-Sheissa, “Numerical continuation scheme for tracing the double bounded homotopy for analysing nonlinear circuits,” in
*Proceedings of the International Conference on Communications, Circuits and Systems*, pp. 1122–1126, May 2005. View at Scopus - M. M. Green, “An efficient continuation method for use in globally convergent dc circuit simulation,” in
*Proceedings of URSI International Symposium on Signals, Systems, and Electronics (ISSSE '95)*, pp. 497–500, October 1995. - L. B. Goldgeisser and M. M. Green, “A method for automatically finding multiple operating points in nonlinear circuits,”
*IEEE Transactions on Circuits and Systems*, vol. 52, no. 4, pp. 776–784, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - D. M. Wolf and S. R. Sanders, “Multiparameter homotopy methods for finding dc operating points of nonlinear circuits,”
*IEEE Transactions on Circuits and Systems*, vol. 43, no. 10, pp. 824–838, 1996. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Yamamoto, “A variable dimension fixed point algorithm and the orientation of simplices,”
*Mathematical Programming*, vol. 30, no. 3, pp. 301–312, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Vázquez-Leal, L. Hernández-Martinez, A. Sarmiento-Reyes, and R. S. Murphy-Arteaga, “Improving multi-parameter homotopy via symbolic analysis techniques for circuit simulation,” in
*Proceedings of the European Conference on Circuit Theory and Design*, vol. 2, pp. 402–405, 2003. - J. S. Roychowdhury and R. C. Melville, “Homotopy techniques for obtaining a DC solution of large-scale MOS circuits,” in
*Proceedings of the 33rd Annual Design Automation Conference*, pp. 286–291, June 1996. View at Scopus - J. Roychowdhury and R. Melville, “Delivering global DC convergence for large mixed-signal circuits via homotopy/continuation methods,”
*IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems*, vol. 25, no. 1, pp. 66–78, 2006. View at Publisher · View at Google Scholar · View at Scopus - Y. Inoue, S. Kusanobu, K. Yamamura, and T. Takahashi, “Newton-fixed-point homotopy method for finding dc operatingpoints of nonlinear circuits,” in
*Proceedings of the International Technical Conference on Circuits/Systems, Computer and Communications (ITC-CSCC '01)*, vol. 1, pp. 370–373, Tokushima, Japan, July 2001. - H. Vazquez-leal, L. Hernandez-Martinez, A. Sarmiento-Reyes, R. Castaneda-Sheissa, and A. Gallardo-Del-Angel, “Homotopy method with a formal stop criterion applied to circuit simulation,”
*IEICE Electronics Express*, vol. 8, no. 21, pp. 1808–1815, 2011. View at Google Scholar