 Open Access
 Authors : U. FilobelloNino, O. AlvarezGasca, A. D. ContrerasHernandez, B. E. PalmaGrayeb , L. CuellarHernandez, C. HoyosReyes, J. M. MendezPerez, V. M. JimenezFernandez , M. A. SandovalHernandez, R. A. CallejasMolina, S. F. HernandezMachuca, J. E. PretelinCanela , J. CervantesPerez, J. L. VazquezAguirre, L. J. VarelaLara, N. BagatellaFlores, H. VazquezLeal
 Paper ID : IJERTV10IS020043
 Volume & Issue : Volume 10, Issue 02 (February 2021)
 Published (First Online): 18022021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
A Handy Analytical Approximate Solution for A Heat Transfer Problem by using A Version of Taylor Method with Boundary Conditions

FilobelloNino1, O. AlvarezGasca1, A. D. ContrerasHernandez1, B. E. PalmaGrayeb1, L. CuellarHernandez1,

HoyosReyes1, J. M. MendezPerez1, V. M. JimenezFernandez1, M. A. SandovalHernandez2,

A. CallejasMolina3, S. F. HernandezMachuca1, J. E. PretelinCanela1, J. CervantesPerez1,

L. VazquezAguirre1, L. J. VarelaLara1, N. BagatellaFlores4, H. VazquezLeal1.
1 Facultad de InstrumentaciÃ³n ElectrÃ³nica, Universidad Veracruzana, Circuito Gonzalo Aguirre BeltrÃ¡n S/N, Xalapa, Veracruz, 91000, MÃ©xico.
2 Universidad de Xalapa, Km. 2 Carretera XalapaVeracruz, Xalapa, Veracruz, 91190, MÃ©xico.
3 Doctorado en Ciencias de la IngenierÃa, TecnolÃ³gico Nacional de MÃ©xico en Celaya, Antonio GarcÃa Cubas #600, Celaya, Guanajuato, 38010, MÃ©xico.
4 Facultad de FÃsica, Universidad Veracruzana, Circuito Gonzalo Aguirre BeltrÃ¡n S/N, Xalapa, Veracruz, 91000, MÃ©xico.
Abstract This work presents the modified Taylor series method (MTSM) with the purpose to find an approximate solution for the nonlinear problem that describes the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature, After comparing MTSM approximation with the exact solutions, we will conclude that the proposed solution is besides of handy, accurate and therefore it follows that the proposed method is potentially an efficient tool for practical applications.
Keywords Nonlinear Differential Equations; Boundary Value Problems; Taylor series; Heat problems.

INTRODUCTION
The objective of this article is to find a handy analytical approximate solution for the boundary value problem of the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature [1] (see Figure 1). Given the importance of the heat transfer phenomena, both in pure problems as well as in the design and operation of equipment in applications, it is primordial to research for analytical approximate solutions for the equations describing these phenomena [1,2,9,10].
The Taylor series method (TSM) is employed with the purpose to depict a function in a certain open interval [3]. This method is based on the successive obtaining of the derivatives of the interest function, in such a way that to obtain a better approximation more terms of the series are required. As a matter of fact, an inconvenient is that the abovementioned method provides just a local convergence [4] which it means that for obtaining a good approximation for a function by using
method (MTSM) in order to find an analytical approximate solution for the nonlinear ordinary differential equation that describes the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature, which is defined on a finite interval with Dirichlet boundary conditions.
The importance of nonlinear problems is that many nature phenomena are nonlinear. For this reason, several methods have been proposed in order to find approximate solutions to nonlinear differential equations: variational approaches [6,7,29,34], tanh method [13], expfunction [14,15], Adomians decomposition method [1621], parameter expansion [22], homotopy perturbation method [2334], homotopy analysis method [35], and perturbation method [10,36,37] among many others.
The rest of the paper is as follows. In Section 2, we introduce the idea of modified Taylor series method (MTSM). For Section 3 we briefly introduce the heat problem to solve. Additionally, Section 4 presents the application of MTSM in the search for an approximate solution for nonlinear ordinary differential equation that describes the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature. Besides a discussion on the results is presented in Section 5. Finally, a brief conclusion is given in Section 6.

MTSM METHOD
To figure out how MTSM method works, consider a general nonlinear equation in the form [5]
TSM, it is required to employ many terms of the Taylor series. For this same reason, the use of TSM for obtaining solutions for nonlinear differential equations has been rather limited. Taylor series method has been employed above all to find solutions to differential equations with initial conditions
u(n) N(u) f (r),
with the following boundary condition
B(u, u / n) 0,
x , (1)
x , (2)
because TSM is directly related with them. Following the idea of [5] we will employ a version of TSM useful to solve nonlinear problems with boundary conditions, by calculating some shooting constants (in our case some initial conditions). Therefore, this work presents the modified Taylor series
where n is the order of the differential equation, N a general differential operator, B is a boundary operator, f (x) a known analytical function, is the domain boundary for
and denotes differentiation along the normal drawn outwards from .
First, we increase the order of the differential equation

MATHEMATICAL DESCRIPTION OF THE PHENOMENON
The purpose of this article is the search for an analytical
u(nk )
k
d
d
dxk N(u) f (x) ,
(3)
approximate solution for the problem, which describes the steadystate onedimension conduction of heat in a slab with thermal conductivity linearly dependent on the temperature
where k is a constant related to the number of the desired shooting constants (SC)
It is possible to express the Taylor series solution for (3) in the following form
(Figure 1).
u (x) u(x ) u(x0 ) x x u (x0 ) x x
2 …
T 0 1! 0 2! 0
, (4)
0
0
where, x0 is the expansion point and derivatives evaluated in the same point u(i) (x ) (i = 0, 1,2,3.. ) are expressed in terms
Figure 1. The 1D conduction of heat through an insulated slab
of the parameters and boundary conditions of (3)
As it is required to solve boundary condition problems,
Let T (x, y, z, t) be the temperature of the abovementioned
boundary conditions not located at the chosen expansion point
slab at a point (x, y, z)
at time t , where K, , and are the
will be replaced by shooting constants giving as result traditional DC conditions. Next, with the purpose to get the
thermal conductivity, specific heat, and density respectively.
coefficients of (4)
u(i) (x ) (i = 0, 1,), MTSM requires
It is possible to verify, following [7] that T satisfies.
0
0
(I) calculating the successive derivatives of (3) and (II) evaluating each derivative using the Dirichlet conditions.
T T T T
Finally, in order to satisfy the boundary conditions which were
x K x y K y z K z t
(6)
replaced by the shooting constants, it is required to evaluate
(4) in such points and the resulting system of equations is solved with the end to get the value of the SC constants. It is important to note that the order of the Taylor expansion (4) is
For steady conditions T / t 0, and
chosen in such a way that it includes all the shooting constants
K T K T K T 0
(7)
in the polynomal; as long as we satisfy such condition, the order of the Taylor expansion may be increased to improve accuracy.
The constants due to the extra kderivatives (see (3)) are used in order to minimize the mean square residual error defined (MSR) as
x x y y z z
For the case of steady conditions for the onedimensional conduction of heat in a slab of thickness L, we will assume
that the uniform temperatures of the faces obey the relation
T T (See Figure 1) in such a way that (7) is reduced to
f
f
x
u(n) N (u
) f (x) dx,
2 1
(5)
d K dT 0,
(8)
2
2
T T
xi
dx
with boundary conditions
dx
where uT is the approximated TSM solution (4) and
T (0) T , T(L) T .
xi , x f is the finite interval delimited by the boundary conditions.
For the case where only is required adjust one parameter, a good alternative for (5) condition is by substituting the MTSM
1 2 (9)
Assuming for simplicity that thermal conductivity varies linearly with the temperature [9] then
approximation into one of the boundary conditions (see Section 4).
K K2 (1 (T T2 )
Using the dimensionless quantities [8]
y T T2 , z x , (T T ) K1 K2 ,
(10)
T T L
1 2 K
(11)
1 2 2
into (8), and after some algebraic steps it is obtained V. DISCUSSION.
d 2 y d 2 y
dy 2
In this work MTSM was employed in order to find an
dz2

y
dz2
dz
dz
0,
(12)
approximate solution, for the nonlinear ordinary differential equation with Dirichlet boundary conditions, which describes
where (9) adopts the form:
y(0) 1,


CASE STUDY.




y(1) 0.
(13)
the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature. Since MTSM is expressed in terms of initial conditions for a differential equation, the procedure consisted of expressing the approximate solution in terms of y(0)
. We determined its value requiring that approximate solution
This section presents the application of MTSM in the search for an approximate solution for nonlinear ordinary differential equation that describes the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature (see (12) and (13)). This work will follow the general idea of MTSM but will employ just one shooting parameter correspondent to initial derivative. From this view to calculate it will be sufficient ensure that the MTSM solution satisfies the right boundary condition y(1) 0 , besides to exemplify we will work the case of
1.
We will get an approximate solution for the system (12) and
(13) with good precision performing very few effort. From (12) we obtain the following successive derivatives
satisfies the boundary condition yT (1) 0. This condition
defines an algebraic equation for , whose solution provides the sought analytical approximation solution.
Figure 2 shows the comparison between numerical solutions and approximate solution (19) for 1. It can be noticed that curves are in good agreement, whereby it is clear that MTSM
method is potentially useful in the search for approximate solutions of nonlinear problems definite with boundary conditions.
y
y2
,
1 y
y 3yy ,
1 y
2
y(4) 4 yy 3y
y(4) 4 yy 3y
,
1 y
(14)
z
z
z
z
After defining the hitherto unknown quantity y(0) , and assuming that a fourthdegree polynomial is enough we obtain from (4)
2
2
2
yT (z) 1 z 4 z
3 3
8
15 4 4
192
… , (15)
after we defined the expansion point x0 0.
Instead of (5), the only unknown parameter of (15) is easier evaluated applying the boundary condition
yT (1) 0,
(16)
Figure 2. Comparison between numerical solution and approximate
so that we get the following fourth degree algebraic equation
15 4 24 3 48 2 192 192 0.
(17)
The real solution of (17) is given by
0.768425. (18)
By substituting (17) into (15) we get the following approximate solution for (15)
T
T
y (z) 1 0.768425z 0.147619245z2
.
0.056717159z3 0.027239301z4
solution (19) for =1
As a matter of fact, Table 1 shows a more detailed comparison between numerical solution for (12) and approximations: (19) of this work, (47) of MHPMLT [9], and
(19) of PM [10] for 1. It is clear that (19), is competitive
with the second best accuracy; it's Average Absolute Relative Error (A.A.R.E) is 1.66348×102 , behind of PM fifth order
approximation PM method with accuracy1.57033×102 , but better than MHPMLT second order approximation (
2.04270×102 ) despite of the fact that MHPMLT method is
Next we will discuss the accuracy of (19).
(19)
considered more difficult to use, it requires to command MHPMLT algorithm that implies employing Laplace transform. On the other hand, PM method required solving five
recurrence differential equations. From Table 1, we note that the difference of accuracy between MTSM and Perturbation Method correspond scarcely a difference of A.A.R.E 9.315×104 . This small difference shows the convenience of employing MTSM given the difference of effort employed for both methods. While PM solved five recurrence differential equations, the proposed method required to solve essentially three derivatives and an algebraic equation. From above it is very important to emphasize that, it is possible to improve the accuracy of our MTSM approximation keeping more terms in Taylor series expansion (19).
x 
Exact 
MTSM (19) 
MHPMLT (47) [9] 
P.M (19) [10] 
0.1 
0.9270953420 
0.9216218664 
0.899999999 
0.9345155625 
0.2 
0.8489990321 
0.8399129100 
0.843983936 
0.8514560000 
0.3 
0.7654913465 
0.7544347663 
0.762539659 
0.7625813125 
0.4 
0.6763157367 
0.6646836965 
0.676971888 
0.6719040000 
0.5 
0.5811698795 
0.5700905875 
0.585906125 
0.5791015625 
0.6 
0.4796933928 
0.4700209520 
0.487967872 
0.4819840000 
0.7 
0.3714504952 
0.3637749282 
0.381782631 
0.3780163125 
0.8 
0.2559046829 
0.2505872801 
0.265975904 
0.2648960000 
0.9 
0.1323801709 
0.1296273972 
0.139173193 
0.1401855625 

Hector VazquezLeal, Mario SandovalHernandez, Roberto Castaneda Sheissa, Uriel FilobelloNino, Arturo SarmientoReyes, 2015. Modified Taylor solution of equation of oxygen diffusion in a spherical cell with MichaelisMenten uptake kinetics. International Journal of Applied Mathematical Research, 4 (2) (2015) 253258 www. Sciencepubco.com/index.php/IJAMR Science Publishing Corporation doi: 10.14419/ijamr.v4i2.4273.

Assas, L.M.B., 2007. Approximat solutions for the generalized KdV Burgers equation by Hes variational iteration method. Phys. Scr., 76: 161164. DOI: 10.1088/00318949/76/2/008

Murray R. S., Teoria y Problemas de AnÃ¡lisis de Fourier, Serie de compendios Schaum, McGraw Hill, Mexico, 1978

Marinca, V., Herisanu, N., NonLinear Dynamical Systems in Engineering, 1st ed., SpringerVerlag, Berlin Heidelberg, Germany,
2011

Uriel FilobelloNiÃ±o, HÃ©ctor VazquezLeal, AgustÃn L. HerreraMay, Roberto C. AmbrosioLazaro, Victor M. JimenezFernandez, Mario A. SandovalHernÃ¡ndez, Oscar AlvarezGasca, and Beatriz E. Palma Grayeb, The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform, Thermal Science, Vol. 24, No. 2, pp. 11051115, 2020 DOI: 10.2298/TSCI180108204F

U. FilobelloNino, H. VazquezLeal, A. SarmientoReyes, A. Perez Sesma, L. HernandezMartinez, A. HerreraMay, V. M. Jimenez Fernandez, A. MarinHernandez, D. PereyraDiaz, A. DiazSanchez., The study of heat transfer phenomena using PM for approximate solution with Dirichlet and mixed boundary conditions. Applied and Computational Mathematics2013; 2(6): 143148 doi: 10.11648/j.acm.20130206.16.

He, J.H., 2007. Variational approach for nonlinear oscillators. Chaos,
Solitons and Fractals, 34: 14301439. DOI: 10.1016/j.chaos.2006.10.026

Kazemnia, M., S.A. Zahedi, M. Vaezi and N. Tolou, 2008. Assessment
A.A.R.E 1.66348×102
2.04270×102
1.57033×102
of modified variational iteration method in BVPs highorder differential equations. Journal of Applied Sciences, 8: 4192
Table 1: Comparison between (19), exact solution and other reported approximate solutions, using 1
VI. CONCLUSIONS.
This work presented MTSM method in order to calculate an analytical approximate solution for the problem that describes the steady state one dimensional conduction of heat in a slab with thermal conductivity linearly dependent on the temperature. The method basically works calculating derivatives of several orders and expresses the solution of a differential equation in terms of the solution of one or more algebraic equations. The comparison with other methods of the literature show the convenience of employ MTSM as a practical tool with the purpose to obtain accuracy solutions for boundary value problems instead of using other more sophisticated and cumbersome procedures.
ACKNOWLEDGMENT
Authors would like to thank Roberto Ruiz Gomez for his contribution to this project.
REFERENCES

FernÃ¡ndezRojas, F., et al., Conductividad tÃ©rmica en metales, semiconductores, dielÃ©ctricos y materiales amorfos, Revista de la facultad de ingenierÃa, U .C. V., 23 (2008), 3, pp. 515.

Aminikhan, H. & Hemmatnezhad, M. (2012). A novel Effective Approach for Solving Nonlinear Heat Transfer Equations. Heat Transfer Asian Research, 41(6), 459466.

Dennis G. Zill (1997). Ecuaciones Diferenciales con Aplicaciones de Modelado (sexta ediciÃ³n) ISBN 0534955746. International Thomson Editores.

Richard L. Burden and J Douglas Faires () AnÃ¡lisis NumÃ©rico (septima ediciÃ³n). Thomson Learning.
4197.DOI:10.3923/jas.2008.4192.4197

Evans, D.J. and K.R. Raslan, 2005. The Tanh function method for solving some important nonlinear partial differential. Int. J. Computat.
Math., 82: 897905. DOI: 10.1080/00207160412331336026

Xu, F., 2007. A generalized soliton solution of the Konopelchenko Dubrovsky equation using expfunction method. Zeitschrift Naturforschung – Section A Journal of Physical Sciences, 62(12): 685 688.

Mahmoudi, J., N. Tolou, I. Khatami, A. Barari and D.D. Ganji, 2008. Explicit solution of nonlinear ZKBBM wave equation using exp function method. Journal of Applied Sciences, 8: 358 363.DOI:10.3923/jas.2008.358.363

Adomian, G., 1988. A review of decomposition method in applied mathematics. Mathematical Analysis and Applications. 135: 501 544.

Babolian, E. and J. Biazar, 2002. On the order of convergence of Adomian method. Applied Mathematics and Computation, 130(2): 383 387. DOI: 10.1016/S00963003(01)001035

Kooch, A. and M. Abadyan, 2012. Efficiency of modified Adomian decomposition for simulating the instability of nanoelectromechanical switches: comparison with the conventional decomposition method. Trends in Applied Sciences Research, 7: 57 67.DOI:10.3923/tasr.2012.57.67

Kooch, A. and M. Abadyan, 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: 34213428.
DOI:10.3923/jas.2011.3421.3428

Vanani, S. K., S. Heidari and M. Avaji, 2011. A lowcost numerical algorithm for the solution of nonlinear delay boundary integral equations. Journal of Applied Sciences, 11: 35043509.
DOI:10.3923/jas.2011.3504.3509

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:10.3923/jas.2011.1416.1420

Zhang, L.N. and L. Xu, 2007. Determination of the limit cycle by Hes parameter expansion for oscillators in a potential. Zeitschrift fÃ¼r Naturforschung – Section A Journal of Physical Sciences, 62(78): 396 398.

He, J.H., 2006. Homotopy perturbation method for solving boundary value problems. Physics Letters A, 350(12): 8788.

He, J.H., 2008. Recent Development of the Homotopy Perturbation Method. Topological Methods in Nonlinear Analysis, 31.2: 205209.

Belendez, A., C. Pascual, M.L. Alvarez, D.I. MÃ©ndez, M.S. Yebra and

HernÃ¡ndez, 2009. High order analytical approximate solutions to the nonlinear pendulum by Hes homotopy method. Physica Scripta, 79(1): 124. DOI: 10.1088/00318949/79/01/015009


He, J.H., 2000. A coupling method of a homotopy and a perturbation technique for nonlinear problems. International Journal of Nonlinear Mechanics, 35(1): 3743.

ElShaed, M., 2005.Application of Hes homotopy perturbation method to Volterras integro differential equation. International Journal of Nonlinear Sciences and Numerical Simulation, 6: 163168.

He, J.H., 2006. Some Asymptotic Methods for Strongly Nonlinear Equations. International Journal of Modern Physics B, 20(10): 1141 1199. DOI: 10.1142/S0217979206033796

Ganji, D.D, H. Babazadeh, F Noori, M.M. Pirouz, M Janipour. An Application of Homotopy Perturbation Method for Non linear Blasius Equation to Boundary Layer Flow Over a Flat Plate, ACADEMIC World Academic Union, ISNN 17493889(print), 17493897 (online). International Journal of Nonlinear Science Vol.7 (2009) No.4, pp. 309 404.

H. VazquezLeal, L. HernandezMartinez, Y. Khan, V.M. Jimenez Fernandez, U. FilobelloNino, A. DiazSanchez, A.L. HerreraMay, R. CastanedaSheissa, A. MarinHernandez, F. RabagoBernal, J. Huerta Chua, S.F. HernandezMachuca, HPM method applied to solve the model of calcium stimulated, calcium release mechanism, American Journal of Applied Mathematics, Vol. 2, No. 1, pp.2935, 2014. DOI: 10.11648/j.ajam.20140201.15

Fereidon, A., Y. Rostamiyan, M. Akbarzade and D.D. Ganji, 2010. Application of Hes homotopy perturbation method to nonlinear shock damper dynamics. Archive of Applied Mechanics, 80(6): 641649. DOI: 10.1007/s004190090334x.

Hector VazquezLeal, Arturo SarmientoReyes, Yasir Khan, Uriel FilobelloNino, and Alejandro DiazSanchez, Rational Biparameter Homotopy Perturbation Method and LaplacePadÃ© Coupled Version, Journal of Applied Mathematics, vol. 2012, Article ID 923975, 21 pages, 2012. doi:10.1155/2012/9239.

Aminikhh Hossein, 2011. Analytical Approximation to the Solution of Nonlinear Blasius Viscous Flow Equation by LTNHPM. International Scholarly Research Network ISRN Mathematical Analysis, Volume 2012, Article ID 957473, 10 pages doi: 10.5402/2012/957473

Noorzad, R., A. Tahmasebi Poor and M. Omidvar, 2008. Variational iteration method and homotopyperturbation method for solving Burgers equation in fluid dynamics. Journal of Applied Sciences, 8: 369373.
DOI:10.3923/jas.2008.369.373

Patel, T., M.N. Mehta and V.H. Pradhan, 2012. The numerical solution of Burgers equation arising into the irradiation of tumour tissue in biological diffusing system by homotopy analysis method. Asian Journal of Applied Sciences, 5: 6066.DOI:10.3923/ajaps.2012.60.66

FilobelloNiÃ±o U, H. VazquezLeal, Y. Khan, A. Yildirim, V.M. Jimenez Fernandez, A.L. Herrera May, R. CastaÃ±edaSheissa, and J.CervantesPerez. 2013, Using perturbation methods and LaplacePadÃ© approximation to solve nonlinear problems. Miskolc Mathematical Notes, 14 (1), 89101.

FilobelloNino U., VazquezLeal H., Benhammouda B., Hernandez Martinez L., Khan Y., JimenezFernandez V.M., HerreraMay A.L., CastanedaSheissa R., PereyraDiaz D., CervantesPerez J., Perez Sesma A., HernandezMachuca S.F. and CuellarHernandez L., A handy approximation for a mediated bioelectrocatalysis process, related to MichaelisMentem equation. Springer Plus, 2014 3: 162, doi:10.1186/219318013162