## Articulo

• Similares en SciELO

## versión On-line ISSN 2007-9621versión impresa ISSN 0188-6266

### Acta univ vol.29  México  2019  Epub 01-Dic-2019

#### https://doi.org/10.15174/au.2019.2065

Artículos

An easy and computable approximation for Troeschs problem by using the Laplace Transform-Homotopy Perturbation Method

Una solución aproximada y práctica al problema de Troesch utilizando el Método de Perturbación Homotópica con Transformada de Laplace

1Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, Veracruz, 9100, México.

2 Centro de Investigación en Micro y Nanotecnología, Universidad Veracruzana.

3National Institute for Astrophysics, Optics and Electronics.

4Facultad de Ingeniería en Electrónica y Comunicaciones, Universidad Veracruzana.

Abstract

This work introduces the Laplace Transform-Homotopy Perturbation Method (LT-HPM) in order to provide an approximate solution for Troeschs problem. After comparing figures between exact and approximate solutions, as well as the average absolute relative error (AARE) of the approximate solutions of this research, with others reported in the literature, it can be said that the proposed solutions are accurate and handy. In conclusion, LT-HPM is a potentially useful tool.

Keywords: Laplace Transform; Homotopy Perturbation Method; Troesch’s Problem

Resumen

Este artículo propone el método de perturbación homotópica con Transformada de Laplace para encontrar una solución aproximada al problema de Troesch. Después de comparar las figuras entre la solución aproximada y la exacta, así como el error relativo absoluto promedio (AARE) de las soluciones aproximadas en este estudio, con otras reportadas en la literatura, se puede decir que la solución propuesta es, además de práctica, precisa. Se concluye que el método LTHPM es potencialmente útil.

Palabras clave: Transformada de Laplace; Método de Perturbación Homotópica; Problema de Troesch

Introducción

Troesch’s equation is important in physics because it models the confinement of a plasma column by radiation pressure. Thus, it is relevant to search for precise solutions for this equation. Unfortunately, Troesch’s equation is difficult to solve, as it happens with several nonlinear differential equations that appear in the physical sciences.

Laplace transform (LT) has played a relevant role, not only for its theoretical interest but also for its methods that allow to solve, in a simpler way, ordinary differential equations (ODE) that model many problems in the fields of engineering and science, in comparison with other mathematical methods (Spiegel, 1988). Specifically, LT has been useful for the solution of ODE with constant coefficients and initial conditions; also, LT can be employed in the case of certain differential equations with variable coefficients and partial differential equations (Spiegel, 1988). The applications of LT for nonlinear ODES has been focused, mainly, on getting analytical approximate solutions. Thus, Aminikhah & Hemmatnezhad (2012) reported LT-HPM, which is a coupling of the homotopy perturbation method (HPM) and LT methods aiming to get precise approximate solutions for these equations. Nevertheless, just as it happens with LT, LT-HPM has been employed, above all, to solve problems with initial conditions (Aminikhah, 2012a; Aminikhah & Hemmatnezhad, 2012), because LT-HPM is directly related to these problems. This work presents LT-HPM with the goal of finding an approximate solution for the nonlinear ODE that describes Troesch’s problem, which is defined on a finite interval with Dirichlet boundary conditions.

The organization of this work is as follows. Section Standard HPM introduces the basic idea of the Homotopy Perturbation Method. The next Section presents the LT-HPM method. In Case Study section, LT-HPM is presented in order to find an approximate solution for Troesch’s equation. Additionally, a discussion on the results is presented. Finally, a conclusions section presents the results obtained from this work.

Standard HPM

The HPM method was introduced with the aim to find approximate solutions to various kinds of nonlinear problems. Indeed, HPM is a combination of the classical perturbative technique and the homotopy, whose origin is found in the topology, but it is not restricted to small parameters. Besides, HPM often requires a few iterations to find accurate approximate solutions (He, 2000; 1999).

To understand how the HPM method works, consider a general nonlinear differential equation (He, 2000) expressed as:

A(U)-f(r)=0,                rΩ (1)

with boundary conditions

B(U,U/n)=0,               rΓ (2)

Where A is a differential operator, B is a boundary operator, f (r) is a known analytical function and Γ is the domain boundary for Ω. Besides, (1) can be expressed as:

L(U)+N(U)-f(r)=0, (3)

where L is linear and N nonlinear.

In general, a homotopy is constructed as follows (He, 1999; 2000):

H(U,p)=(1-p)[L(U)-L(u0)]+p[L(U)+N(U)-f(r)]=0,     p[0,1],      rΩ (4)

or

H(U,p)=L(U)-L(u0)+p[L(u0)+N(U)-f(r)]=0,     p[0,1],      rΩ (5)

Where p is called the homotopy parameter, while u 0 is the first approximation for the solution of (3) that satisfies the boundary conditions.

The solution for (4) or (5) is expressed as:

U=v0+v1p+v2p2+... (6)

By substituting (6) into (5) and comparing identical powers of p terms, there can be found values for ν0, ν1, ν2,

After considering p→1, the following approximate solution for (1) is obtained:

U=v0+v1+v2+v3... (7)

Basic Idea of Laplace Transform Homotopy Perturbation Method (LT-HPM)

The aim of this section is to show how LT-HPM is used with the purpose of obtaining approximate solutions for ODE, as (3) (Aminikhah, 2012b; Filobello-Nino et al., 2015).

The procedure for LT-HPM follows the same steps as HPM, but only until (5); next, LT is applied on both sides of (5) to get:

IL(U)-L(u0)+p[L(u0)+N(U)-f(r)=0 (8)

From Spiegel (1988), the following is obtained:

snIU-sn-1U(0)-sn-2U'(0)-...-U(n-1)(0)=IL(u0)-pL(u0)+p-N(U)+f(r) (9)

or

I(U)=1snsn-1U(0)+sn-2U'(0)+..+U(n-1)(0)+1snIL(u0)-pL(u0)+p-N(U)+f(r). (10)

Applying inverse Laplace transform to both sides of (10), the following is obtained:

U=I-11snsn-1U(0)+sn-2U'(0)+..+U(n-1)(0)+1snIL(u0)-pL(u0)+p-N(U)+f(r) (11)

Next, it is assumed that the solutions of (3) are expressed as:

U=n=0pnvn (12)

so that substituting (12) into (11) be:

n=0pnνn=I-11snsn-1U(0)+sn-2U'(0)+..+U(n-1)(0)+1snIL(u0)-pL(u0)+p-N( n=0pnνn)+f(r). (13)

After comparing coefficients with the same power of p :

p0:ν0=I-11snsn-1U(0)+sn-2U'(0)+..+U(n-1)(0))+IL(u0)p1:ν1=I-11snI-N(ν0)-Lu0+f(r),p2:ν2=I-11snI-N(ν0,ν1),p3:ν3=I-11snI-N(ν0,ν1,ν2),pj:νj=I-11snI-N(ν0,ν1,ν2,...,νj) (14)

Assuming that the initial approximation adopts the form U(0)=u0=α0, U'(0)=α1,..,Un-1(0) =αn-1; then an approximate solution is expressed in terms of the following limit:

u=limp1U=ν0+ν1+ν2+... (15)

Case Study

Next, LT-HPM is used in order to get a handy analytical approximate solution for the nonlinear problem:

d2y(x)dx2=εsenh(εy), 0x1, y(0)=0, y(1)=1 (16)

where ε is a positive parameter (Troeschs parameter).

The terms to be identified are

L(y)=y(x) (17)

and

N(y)=-εsenh(εy), (18)

where prime denotes differentiation respect to x.

Next, on the right side of (16), in terms of its Taylor series, the following is expressed:

y=ε2y+ε4y36+ε6y5120+ε8y75040+... (19)

In order to show the usefulness of the proposed method, an accurate approximate solution will be obtained, keeping only the first two terms on the right side of (19).

Next, the following homotopy is constructed (see 4):

(1-p)(y-y0)+py-ε2y-ε4y36=0 (20)

or

y=y0+p-y0+ε2y+ε4y36 (21)

After applying LT:

Iy=Iy0+p-y0+ε2y+ε4y36 (22)

Following Spiegel (1988), the (22) is rewritten as:

s2Y(s)-sy(0)-y'(0)=Iy0+p-y0+ε2y+ε4y36 , (23)

where Y(s)=I(y(x)) has been defined.

After using y(0)=0 , (23) is expressed as:

s2Y(s)-A=Iy0+p-y0+ε2y+ε4y36, (24)

Where A=y'(0)[/p]

Solving for Y(s) and applying Laplace inverse transform I-1 , the next is obtained:

y(x)=I-1As2+1s2Iy0+p-y0+ε2y+ε4y36 (25)

Next, the following series solution for y(x) will be assumed as

y(x)=n=0pnνn (26)

It is noted that

ν0(x)=Ax (27)

is the first approximation for the solution for (16), which satisfies the conditions y(0)=0, y'(0)=A.

Thus, after substituting (26) into (25), the following is obtained:

n=0pnνn=I-1As2+1s2Iy0+p-y0+ε2ν0+pν1+p2ν2+..+ε4ν0+pν1+p2ν2+..36 (28)

After comparing coefficients of identical powers of p terms:

p0:ν0(x)=I-1As2, (29)

p1:ν1(x)=I-1ε2s2I{ν0}+ε46s2I{ν03} , (30)

p2:ν2(x)=I-1ε2s2Iν1+ε42s2Iν1ν02, (31)

After solving Laplace transforms (29), (30), and (31):

p0: ν0(x)=Ax, (32)

p1: ν1(x)=ε2Ax36+A3ε4x5120, (33)

p2: ν2(x)=ε4A120x5+11ε6A35040x7+ε8A517280x9. (34)

and so on.

By substituting (32)-(34) into (15), and considering the limit value p1, the handy approximation is obtained:

y(x)=Ax+ε2A6x3+ε4A1+A2120x5+11ε6A35040x7+ε8A517280x9 (35)

In order to calculate the value of A , it is necessary that (35) satisfies the boundary conditiony(1)=1. This condition gives rise to an algebraic equation for the unknown A. Proposing as case studies ε=0.5, ε=1 , and ε=2, the values

A=0.9590503635                 (ε=0.5), (36)

A=0.8456306348                  (ε=1), (37)

and

A=0.5323243593                  (ε=2) (38)

are obtained, respectively.

By substituting (36), (37), and (38) into (35), the following expressions are obtained:

y(x)=0.9590503635x+0.03996043182x3+0.0009589392735x5+0.00003008197995x7+1.834101508x10-7x9. (39)

y(x)=0.8456306348x+0.1409384391x3+0.01208611363x5+0.001319788295x7+0.00002502417688x9. (40)

y(x)=0.5323243593x+0.3548829062x3+0.09108915995x5+0.02107032055x7+0.0006332539792x9. (41)

Discussion

This article employed LT-HPM in the search for a handy and precise analytical approximate solution for the nonlinear ODE defined with finite boundary conditions that describe the Troesch’s problem. Since the proposed method is expressed in terms of initial conditions for a given problem (see (14), this procedure consisted of expressing the approximated solutions in terms of A=y'(0). It is noted that in order to determine the value of A, the boundary condition y(1) = 1 has to be satisfied. This condition defines an algebraic equation for the unknown A.

Figure 1 shows the comparison between numerical and approximate solutions (39)-(41) for the cases ε=0.5 ,ε=1 , and ε=2 , respectively. It is also noted that curves are in good agreement; thus, the potentiality of the proposed method in the search for solutions for nonlinear ODE with finite boundary conditions is clear.

Table 1 shows the comparison between the exact solution reported by Erdogan & Ozis (2011), and approximations (39), ADM (Deeba et al., 2000), HPM (Feng et al., 2007), HPM (Mirmoradia et al., 2009), and HAM (Hassana & El-Tawil, 2011) for the case ε=0.5 . It is clear that (39) is competitive with the second best accuracy; its average absolute relative error (AARE) is scarcely of 4.6x10-6 , only HAM (Hassana & El-Tawil, 2011) was more accurate, since its AARE is 2.51374x10-6 . Besides, HPM, ADM, and HAM methods are considered more difficult to use, because LT-HPM does not require solving recurrence differential equations. HAM usually requires more iterations and, consequently, more terms in comparison with LT-HPM; therefore, its expressions are many times long and cumbersome. Table 2 shows that, for ε=1 , approximation (40) possesses the lowest AARE 4.5x10-4 even though ε=1 is not small (see below).

Table 1 Comparison between (39), the exact solution (Erdogan & Ozis, 2011), and other reported approximate solutions using ε =0.5

x Exact (Erdogan & Ozis, 2011) This work, LT-HPM (39) ADM (Deeba et al., 2000) HPM (Feng et al., 2007) HPM (Mirmoradia et al., 2009) HAM (Hassana & El-Tawil, 2011)
0.1 0.0959443493 0.09594500637 0.0959383534 0.0959395656 0.095948026 0.0959446190
0.2 0.1921287477 0.1921300635 0.1921180592 0.1921193244 0.192135797 0.1921292845
0.3 0.2887944009 0.2887963775 0.2887803297 0.2887806940 0.288804238 0.2887952148
0.4 0.3861848464 0.3861874818 0.3861687095 0.3861675428 0.386196642 0.3861859313
0.5 0.4845471647 0.4845504381 0.4845302901 0.4845274183 0.4845599 0.4845485110
0.6 0.5841332484 0.5841370824 0.5841169798 0.5841127822 0.584145785 0.5841348222
0.7 0.6852011483 0.6852053362 0.6851868451 0.6851822495 0.685212297 0.6852028604
0.8 0.7880165227 0.7880205903 0.7880055691 0.7880018367 0.788025104 0.7880181729
0.9 0.8928542161 0.89285571853 0.8928480234 0.8928462193 0.892859085 0.8928553997
AARE 4.6x10-6 3.47802x10-5 3.57932x10-5 2.44418x10-5 2.51374x10-6

Source: Authors’ own elaboration.

Table 2 Comparison between (40), the exact solution (Erdogan & Ozis, 2011), and other reported approximate solutions, using ε = 1

x Exact (Erdogan & Ozis, 2011) This work, LT-HPM (40) ADM (Deeba et al., 2000) HPM (Feng, et al., 2007) HPM (Mirmoradia,et al., 2009) HAM (Hassana & El-Tawil, 2011)
0.1 0.0846612565 0.08470412291 0.084248760 0.0843817004 0.084934415 0.0846732692
0.2 0.1701713582 0.1702575190 0.169430700 0.1696207644 0.170697546 0.1701954538
0.3 0.2573939080 0.2575241867 0.256414500 0.2565929224 0.258133224 0.2574302342
0.4 0.3472228551 0.3473982447 0.346085720 0.3462107378 0.348116627 0.3472715981
0.5 0.4405998351 0.4408206731 0.439401985 0.4394422743 0.44157274 0.4406610140
0.6 0.5385343980 0.5387980978 0.537365700 0.5373300622 0.539498234 0.5386072529
0.7 0.6421286091 0.6424243421 0.641083800 0.6410104651 0.642987984 0.7526899495
0.8 0.7526080939 0.7529055047 0.751788000 0.7517335467 0.753267551 0.7526899495
0.9 0.8713625196 0.8715893682 0.870908700 0.8708835371 0.871733059 0.8714249118
AARE 4.5x10-4 0.002714577 0.002320107 0.002044737 0.0019244326

Source: Authors’ own elaboration.

An important feature from the proposed method is deduced from equations like (16), which are expressed in the following form L(x)+εN(x)=0, (L(x) linear, N(x) nonlinear). It is known that some methods as PM (Chow, 1995; Holmes, 1995) work better for values of ε <<1. For the problem presented here, reference is made to small values of Troesch’s parameter. As a matter of fact, ε can be conceived as a parameter of smallness, which determines how greater the contribution of L(x) is in comparison with N(x). Generally, it is easier to find accurate analytical approximate solutions for small values of ε than for large values of this parameter. Figure 1 shows that (40) and (41) provide good approximations for the nonlinear problem (16), although perturbation parameters ε=1 and ε=2 are indeed large. Therefore, the proposed method is not limited to small parameters.

Finally, it is very important to emphasize that it is possible to improve the accuracy of the obtained approximations, adding higher order approximations to the solution (35) and keeping more terms in the Taylor series expansion (19).

Conclusions

In this article, LT-HPM was used to find a polynomial analytical approximate solution of five terms for the second order nonlinear ODE that describes Troesch’s problem with Dirichlet boundary conditions defined on a finite interval. In general, the proposed method expresses the problem of obtaining an approximate solution for a nonlinear ordinary differential equation, in terms of solving an algebraic equation for some unknown initial condition. From Figure 1, and Table 1 and Table 2, it is concluded that LT-HPM is a method with potential in the search for solutions for boundary value nonlinear problems. As it was already mentioned, an additional advantage of LT-HPM is that it does not require to solve several recurrence differential equations; therefore, it can be said that LT-HPM is a useful tool for practical applications.

Acknowledgments

All the authors express appreciation to the late Professor Jose Antonio Agustin Perez-Sesma whose contribution to this work was of great significance. The authors would like to thank Rogelio Alejandro Callejas-Molina for his contribution to this project.

References

Amezcua, M., & Gálvez, A. (2002). Los modos de análisis en investigación cualitativa en salud: perspectiva crítica y reflexiones en voz alta. Revista Española de Salud Pública, 76(5), 423-436. [ Links ]

Asistencia, Asesoría y Administración de Espectáculos (AAA). (2013). Lucha Libre contra la obesidad. 1, 2, 3 saludable otra vez [anuncio de televisión]. Ciudad de México, México. Recuperado el 8 de febrero de 2016 de http://www.gob.mx/salud/prensa/inicia-la-campana-contra-obesidad-1-2-3-saludable-otra-vez-10484?idiom=esLinks ]

Barrientos-Pérez, M., & Flores-Huerta, S. (2008). ¿Es la obesidad un problema médico individual y social? Políticas públicas que se requieren para su prevención. Boletín Médico Hospital Infantil México 2008, 65(6), 638-651. [ Links ]

Bazán, I., & Miño, R. (2015). La imagen corporal en los medios de comunicación masiva. Psicodebate, 15(1), 23-42. [ Links ]

Bertran, M., & Arroyo, P. (2006). Antropología y nutrición. México: Universidad Autónoma Metropolitana y Fundación Mexicana para la Salud. [ Links ]

Castiel, L., & Álvarez-Dardet, C. (2007). La salud persecutoria. Revista de Saúde Pública, 41(3), 461-466. [ Links ]

Cohen, L., Perales, D., & Steadman, C. (2005). The O word: why the focus on obesity is harmful to community health. Californian Journal of Health Promotion, 3(3), 154-161. [ Links ]

Dávila-Torres, J., González-Izquierdo, J. J., & Barrera-Cruz, A. (2015). Panorama de la obesidad en México. Revista Médica del Instituto Mexicano del Seguro Social, 53(2), 240-249. [ Links ]

Diario Oficial de la Federación (DOF). (23 de agosto de 2010). Lineamientos Generales para el expendio o distribución de alimentos y bebidas en los establecimientos de consumo escolar de los planteles de educación básica. Recuperado el 8 de febrero de 2016 de http://dof.gob.mx/nota_detalle.php?codigo=5156173&fecha=23/08/2010Links ]

Diario Oficial de la Federación (DOF). (15 de abril de 2014a). Lineamientos de los criterios nutrimentales y de publicidad que deberán observar los anunciantes de alimentos y bebidas no alcohólicas para publicitar sus productos en televisión abierta y restringida, así como en salas de exhibición cinematográfica. Recuperado el 8 de febrero de 2016 de http://www.dof.gob.mx/nota_detalle.php?codigo=5340694&fecha=15/04/2014Links ]

Adomian, G. (1988). A review of decomposition method in applied mathematics. Mathematical Analysis and Applications, 135(2), 501-544. doi: https://doi.org/10.1016/0022-247X(88)90170-9 [ Links ]

Aminikhah, H. (2012a). The combined Laplace transform and new homotopy perturbation method for stiff systems of ODE s. Applied Mathematical Modelling, 36(8), 3638-3644. doi: https://doi.org/10.1016/j.apm.2011.10.014 [ Links ]

Aminikhah, H. (2012b). Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, 2012, 1-10. Article ID 957473. doi: https://doi.org/10.5402/2012/957473 [ Links ]

Aminikhah, H., & Hemmatnezhad, M. (2012). A novel Effective Approach for Solving Nonlinear Heat Transfer Equations. Heat Transfer- Asian Research, 41(6), 459-466. doi: https://doi.org/10.1002/htj.20411 [ Links ]

Araghi, M. F., & Rezapour, B. (2011). Application of homotopy perturbation method to solve multidimensional schrodingers equations. International Journal of Mathematical Archive (IJMA), 2, 1-6. http://www.ijma.info/index.php/ijma/article/view/665. [ Links ]

Araghi, M. F., & Sotoodeh, M. (2012). An enhanced modified homotopy perturbation method for solving nonlinear volterra and fredholm integro-differential equation. World Applied Sciences Journal, 20(12), 1646-1655. [ Links ]

Assas, L. M. B. (2007). Approximate solutions for the generalized K-dV- Burgers’ equation by He’s variational iteration method. Physica Scripta, 76(2), 161-164. doi: https://doi.org/10.1088/0031-8949/76/2/008 [ Links ]

Babolian, E., & Biazar, J. (2002). On the order of convergence of Adomian method. Applied Mathematics and Computation, 130(2), 383-387. doi: https://doi.org/10.1016/S0096-3003(01)00103-5 [ Links ]

Bayat, M., Bayat, M., & Pakar, I. (2014). Nonlinear vibration of an electrostatically actuated microbeam. Latin American Journal of Solids and Structures, 11, 534 - 544. http://www.scielo.br/scielo.php?script=sci_arttext&pid=S1679-78252014000300009Links ]

Bayat, M., Pakar, I., & Emadi, A. (2013). Vibration of electrostatically actuated microbeam by means of homotopy perturbation method. Structural Engineering and Mechanics, 48, 823-831. doi: https://doi.org/10.12989/sem.2013.48.6.823 [ Links ]

Belendez, A., Pascual, C., Alvarez, M. L., Méndez, D. I., Yebra, M. S., & Hernández, A. (2009). High order analytical approximate solutions to the nonlinear pendulum by He’s homotopy method. Physica Scripta , 79(1), 1-24. doi: https://doi.org/10.1088/0031-8949/79/01/015009 [ Links ]

Biazar, J., & Aminikhah, H. (2009). Study of convergence of homotopy perturbation method for systems of partial differential equations. Computers and Mathematics with Applications, 58(11-12), 2221-2230. doi: https://doi.org/10.1016/j.camwa.2009.03.030 [ Links ]

Biazar, J., & Eslami, M. (2012). A new homotopy perturbation method for solving systems of partial differential equations. Computers & Mathematics with Applications, 62, 225-234. doi: https://doi.org/10.1016/j.camwa.2011.04.070 [ Links ]

Biazar, J., & Ghanbari, B. (2012). The homotopy perturbation method for solving neutral functional differential equations with proportional delays. Journal of King Saud University - Science, 24, 33-37. doi: https://doi.org/10.1016/j.jksus.2010.07.026 [ Links ]

Biazar, J., & Ghazvini, H. (2009). Convergence of the homotopy perturbation method for partial differential equations. Nonlinear Analysis: Real World Applications, 10(5), 2633-2640. doi: https://doi.org/10.1016/j.nonrwa.2008.07.002 [ Links ]

Chow, T. L. (1995). Classical Mechanics. New York: John Wiley and Sons Inc. [ Links ]

Chowdhury, S. H. (2011). A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. Journal of Applied Sciences, 11, 1416-1420. doi: https://doi.org/10.3923/jas.2011.1416.1420 [ Links ]

Deeba, E., Khuri, S. A., & Xie, S. (2000). An algorithm for solving boundary value problems. Journal of Computational Physics, 159, 125-138. doi: https://doi.org/10.1006/jcph.2000.6452 [ Links ]

El-Shaed, M. (2005). Application of He’s homotopy perturbation method to Volterra’s integro differential equation. International Journal of Nonlinear Sciences and Numerical Simulation, 6, 163-168. doi: https://doi.org/10.1515/IJNSNS.2005.6.2.163 [ Links ]

Erdogan U., & Ozis, T. (2011). A smart nonstandard finite difference scheme for second order nonlinear boundary value problems. Journal of Computational Physics , 230(17), 6464-6474. doi: https://doi.org/10.1016/j.jcp.2011.04.033 [ Links ]

Evans, D. J., & Raslan, K. R. (2005). The Tanh function method for solving some important nonlinear partial differential equations. International Journal of Computer Mathematics ., 82, 897-905. doi: https://doi.org/10.1080/00207160412331336026 [ Links ]

Feng, X., Mei, L., & He, G. (2007). An efficient algorithm for solving Troesch’s problem. Applied Mathematics and Computation , 189(1), 500-507. doi: https://doi.org/10.1016/j.amc.2006.11.161 [ Links ]

Fereidon, A., Rostamiyan, Y., Akbarzade, M., & Ganji, D. D. (2010). Application of He’s homotopy perturbation method to nonlinear shock damper dynamics. Archive of Applied Mechanics, 80(6), 641-649. doi: https://doi.org/10.1007/s00419-009-0334-x [ Links ]

Filobello-Niño, U., Vazquez-Leal, H., Castañeda-Sheissa, R., Yildirim, A., Hernandez Martinez, L., Pereyra-Díaz, D., Pérez-Sesma, A., & Hoyos-Reyes, C. (2012a). An approximate solution of Blasius equation by using HPM method. Asian Journal of Mathematics and Statistics, 2012 5(2), 1-10. doi: https://doi.org/10.3923/ajms.2012.50.59 [ Links ]

Filobello-Nino, U., Vazquez-Leal, H., Khan, Y., Yildirim, A., Pereyra-Diaz, D., Perez-Sesma, A., Hernandez-Martinez, L., Sanchez-Orea, J., Castaneda-Sheissa, R., & Rabago-Bernal, F. (2012b) HPM applied to solve nonlinear circuits: a study case. Applied Mathematical Sciences, 6(85-88), 4331-4344. http://m-hikari.com/ams/ams-2012/ams-85-88-2012/sheissaAMS85-88-2012.pdfLinks ]

Filobello-Nino, U., Vazquez-Leal, H., Khan, Y., Yildirim, A., Jimenez-Fernandez, V. M., Herrera-May, A. L., Castaneda-Sheissa, R., & Cervantes-Perez, J. (2013a). Using Perturbation method and Laplace-Padé approximation to solve nonlinear problems. Miskolc Mathematical Notes, 14(1), 89-101. doi: https://doi.org/10.18514/MMN.2013.517 [ Links ]

Filobello-Niño, U., Vazquez-Leal, H., Khan, Y., Perez-Sesma, A., Diaz-Sanchez, A., Herrera-May, A., Pereyra-Diaz, D., Castañeda-Sheissa, R., Jimenez-Fernandez, V. M., & Cervantes-Perez, J. (2013b). A handy exact solution for flow due to a stretching boundary with partial slip. Revista Mexicana de Física E, 59, 51-55. [ Links ]

Filobello-Nino, U., Vazquez-Leal, H., Khan, Y., Perez-Sesma, A., Diaz-Sanchez, A., Jimenez-Fernandez, V. M., Herrera-May, A., Pereyra-Diaz, D., Mendez-Perez, J. M. & Sanchez-Orea, J. (2015) Laplace transform-homotopy perturbation method as a powerful tool to solve nonlinear problems with boundary conditions defined on finite intervals. Computational and Applied Mathematics, 34(1), 1-16. doi: https://doi.org/10.1007/s40314-013-0073-z [ Links ]

Ganji, D. D., Mirgolbabaei, H., Miansari, M., & Miansari, M. (2008). Application of homotopy perturbation method to solve linear and non-linear systems of ordinary differential equations and differential equation of order three. Journal of Applied Sciences , 8(7), 1256-1261. doi: https://doi.org/10.3923/jas.2008.1256.1261 [ Links ]

Hassana, H. N., & El-Tawil. M. A. (2011). An efficient analytic approach for solving two point nonlinear boundary value problems by homotopy analysis method. Mathematical Methods in the Applied Sciences, 34, 977-989. doi: https://doi.org/10.1002/mma.1416 [ Links ]

He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4), 257-262. doi: https://doi.org/10.1016/S0045-7825(99)00018-3 [ Links ]

He, J. H. (2000). A coupling method of a homotopy and a perturbation technique for nonlinear problems. International Journal of Nonlinear Mechanics, 35(1), 37-43. doi: https://doi.org/10.1016/S0020-7462(98)00085-7 [ Links ]

He, J. H. (2006a). Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(1-2), 87-88. doi: https://doi.org/10.1016/j.physleta.2005.10.005 [ Links ]

He, J. H. (2006b). Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of Modern Physics B, 20(10), 1141-1199. doi: https://doi.org/10.1142/S0217979206033796 [ Links ]

He, J. H. (2007). Variational approach for nonlinear oscillators. Chaos, Solitons and Fractals, 34(5),1430-1439. doi: https://doi.org/10.1016/j.chaos.2006.10.026 [ Links ]

Holmes, M. H. (1995). Introduction to Perturbation Methods, Germany: Springer-Verlag. [ Links ]

Kazemnia, M., Zahedi, S. A., Vaezi, M., & Tolou, N. (2008). Assessment of modified variational iteration method in BVPs high-order differential equations. Journal of Applied Sciences , 8(22), 4192-4197. doi: https://doi.org/10.3923/jas.2008.4192.4197 [ Links ]

Khan, M., Gondal, M. A., Hussain, I., & Vanani, S. K. (2011). A new study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain. Mathematical and Computer Modelling, 55(3-4 ), 1143-1150. doi: https://doi.org/10.1016/j.mcm.2011.09.038 [ Links ]

Khan, Y., & Wu, Q. (2011). Homotopy perturbation transform method for nonlinear equations using He’s polynomials. Computers and Mathematics with Applications , 61(8), 1963-1967. doi: https://doi.org/10.1016/j.camwa.2010.08.022 [ Links ]

Kooch, A., & Abadyan, M. (2011). Evaluating the ability of modified Adomian decomposition method to simulate the instability of freestanding carbon nanotube: comparison with conventional decomposition method. Journal of Applied Sciences , 11(19), 3421-3428. doi: https://doi.org/10.3923/jas.2011.3421.3428 [ Links ]

Kooch, A., & Abadyan, M. (2012). Efficiency of modified Adomian decomposition for simulating the instability of nano-electromechanical switches: comparison with the conventional decomposition method. Trends in Applied Sciences Research, 7(1), 57-67. doi: https://doi.org/10.3923/tasr.2012.57.67 [ Links ]

Madani, M., Fathizadeh, M., Khan, Y. & Yildirim, A. (2011). On the coupling of the homotopy perturbation method and Laplace transformation. Mathematical and Computer Modelling , 53(9-10), 1937-1945. doi: https://doi.org/10.1016/j.mcm.2011.01.023 [ Links ]

Mahmoudi, J., Tolou, N., Khatami, I., Barari, A., & Ganji, D. D. (2008). Explicit solution of nonlinear ZK-BBM wave equation using Exp-function method. Journal of Applied Sciences , 8(2), 358-363. doi: https://doi.org/10.3923/jas.2008.358.363 [ Links ]

Marinca, V., & Herisanu, N. (2011). Nonlinear Dynamical Systems in Engineering. Berlin Heidelberg: Springer-Verlag. [ Links ]

Mirmoradia, S. H., Hosseinpoura, I., Ghanbarpour, S., & Barari, A. (2009). Application of an approximate analytical method to nonlinear Troesch, s problem. Applied Mathematical Sciences , 3(32), 1579-1585. http://www.m-hikari.com/ams/ams-password-2009/ams-password29-32-2009/mirmoradiAMS29-32-2009.pdfLinks ]

Noorzad, R., Poor, A. T., & Omidvar, M. (2008). Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. Journal of Applied Sciences , 8(2), 369-373. doi: https://doi.org/10.3923/jas.2008.369.373 [ Links ]

Patel, T., Mehta, M. N., & Pradhan, V. H. (2012). The numerical solution of Burger’s equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian Journal of Applied Sciences , 5(1), 60-66. doi: https://doi.org/10.3923/ajaps.2012.60.66 [ Links ]

Rashidi, M., Pour, S. M., Hayat, T., & Obaidat, S. (2012). Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method. Computers & Fluids, 54, 1-9. doi: https://doi.org/10.1016/j.compfluid.2011.08.001 [ Links ]

Rashidi, M. M., Rastegari, M. T., Asadi, M., & Beg, O. A. (2012). A study of non-newtonian flow and heat transfer over a non-isothermal wedge using the homotopy analysis method. Chemical Engineering Communications, 199, 231-256. doi: https://doi.org/10.1080/00986445.2011.586756 [ Links ]

Sharma, P. R., & Methi, G. (2011). Applications of homotopy perturbation method to partial differential equations. Asian Journal of Mathematics & Statistics, 4(3), 140-150. doi: https://doi.org/10.3923/ajms.2011.140.150 [ Links ]

Spiegel, M. R. (1988). Teoría y Problemas de Transformadas de Laplace (primera edición. Serie de compendios Schaum). México: MacGraw Hill. [ Links ]

Vanani, S. K., Heidari, S., & Avaji, M. (2011). A low-cost numerical algorithm for the solution of nonlinear delay boundary integral equations. Journal of Applied Sciences , 11(20), 3504-3509. doi: https://doi.org/10.3923/jas.2011.3504.3509 [ Links ]

Vazquez-Leal, H., Filobello-Niño, U., Castañeda-Sheissa, R., Hernandez Martinez, L., & Sarmiento-Reyes, A. (2012). Modified HPMs inspired by homotopy continuation methods, Mathematical Problems in Engineering, 2012, 1-20. doi: https://doi.org/10.1155/2012/309123. [ Links ]

Vazquez-Leal, H., Hernandez-Martinez, L., Khan, Y., Jimenez-Fernandez, V. M., Filobello-Nino, U., Diaz-Sanchez, A., Herrera-May, A. L., Castaneda-Sheissa, R., Marin-Hernandez, A., Rabago-Bernal, F., & Huerta-Chua, J. (2014). Multistage HPM applied to path tracking damped oscillations of a model for HIV infection of CD4+ T cells. British Journal of Mathematics & Computer Science, 4(8), 1035-1047. doi: https://doi.org/10.9734/BJMCS/2014/7714 [ Links ]

Vazquez-Leal, H., Castañeda-Sheissa, R., Filobello-Niño, U., Sarmiento-Reyes, A., & Sánchez-Orea, J. (2012). High accurate simple approximation of normal distribution related integrals. Mathematical Problems in Engineering, 2012, 1-22. doi: https://doi.org/10.1155/2012/124029 [ Links ]

Vazquez-Leal, H., Khan, Y., Fernández-Anaya, G., Herrera-May, A., Sarmiento-Reyes, A., Filobello-Nino, U., Jimenez-Fernández, V. M., & Pereyra-Díaz, D. (2012). A General Solution for Troesch’s Problem. Mathematical Problems in Engineering, 2012, 1-14. doi: https://doi.org/10.1155/2012/208375 [ Links ]

Vazquez-Leal, H., Khan, Y., Filobello-Nino, U., Sarmiento-Reyes, A., Diaz-Sanchez, A., & Cisneros-Sinencio, L. F. (2013). Fixed-Term Homotopy. Journal of Applied Mathematics, 2013, 1-11. doi: https://doi.org/10.1155/2013/972704 [ Links ]

Vazquez-Leal, H., Sarmiento-Reyes, A., Khan, Y., Filobello-Nino, U., & Diaz-Sanchez, A. (2012) Rational Biparameter Homotopy Perturbation Method and Laplace-Padé Coupled Version. Journal of Applied Mathematics, 2012, 1-21, Article ID 923975. doi: https://doi.org/10.1155/2012/923975 [ Links ]

Xu, F. (2007). A generalized soliton solution of the Konopelchenko-Dubrovsky equation using exp-function method. Zeitschrift Naturforschung - Section A Journal of Physical Sciences, 62(12), 685-688. doi: https://doi.org/10.1515/zna-2007-1202 [ Links ]

Zhang, L. N., & Xu, L. (2007). Determination of the limit cycle by He’s parameter expansion for oscillators in a potential. Zeitschrift für Naturforschung - Section A Journal of Physical Sciences, 62(7-8), 396-398. doi: https://doi.org/10.1515/zna-2007-7-807 [ Links ]

Como citar: Filobello-Nino, U., Vazquez-Leal, H., Herrera-May, A. L., Jimenez-Fernandez, V. M., Cervantes-Perez, J., Pereyra-Diaz, D., Hoyos-Reyes, C., Sandoval-Hernández, M. A., Huerta-Chua, J., & Ruiz-Gómez, R. (2019). An easy and computable approximation for Troesch’s problem by using the Laplace Transform-Homotopy Perturbation Method. Acta Universitaria 29, e2065. doi. http://doi.org/10.15174.au.2019.2065

Received: September 07, 2017; Accepted: October 05, 2018; Published: September 02, 2019

*Autor de correspondencia: *E-Mail: hvazquez@uv.mx

This is an open-access article distributed under the terms of the Creative Commons Attribution License