Analytical Exact and Approximate Solutions for Certain Diffusion Reactions

Analytical Exact and Approximate Solutions for Certain Diffusion Reactions

1. Filobello-Nino1, M. A. Sandoval-Hernandez2, O. Alvarez-Gasca1, A.D. Contreras-Hernandez1, C. Hoyos-Reyes1,

   1. Carrillo-Ramon1, J. Sanchez-Orea1, L. Cuellar-Hernandez1, J.M. Mendez-Perez1, V.M. Jimenez-Fernandez1,

      G.J. Morales-Alarcon3, S.F. Hernandez-Machuca1, F. Martinez-Barrios4, M. Garcia-Lozano2, A.J. Mota-Hernandez2,

      N. Bagatella-Flores5, H. Vazquez-Leal1,*.

      1 Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, Xalapa, 91000, Veracruz, México.

      2 Centro de Bachillerato Tecnológico industrial y de servicios No. 190, Av. 15 Col. Venustiano Carranza 2da Sección, Boca del Río, 94297, Veracruz, México.

      3 Instituto de Psicología y Educación, Universidad Veracruzana, Agustín Melgar 2, Col. 21 de Marzo, Xalapa, 91010 Veracruz, México.

      4 Centro de Bachillerato Tecnológico industrial y de servicios No. 268, Av. La Bamba, Geovillas del Puerto, Veracruz 91777, Veracruz, México.

      5 Facultad de Física, Universidad Veracruzana, Paseo No. 112. Desarrollo Habitacional Nuevo Xalapa, Xalapa, 91097, Veracruz, México.

      Abstract:- This work presents the Taylor Series Method with shooting (STSM) with the purpose to find both approximate and exact solutions for the nonlinear problem that describes the steady state solutions of a highly nonlinear model of a coupled diffusion and nth-order chemical reaction in a spherical porous catalyst. After comparing STSM approximation with the exact solutions, we will conclude that the proposed solutions are besides of extremely handy, accurate (with relative error less than one percent in all the cases), therefore it follows that the proposed method is potentially an efficient tool to be used in practical applications instead of others cumbersome and complicated methods.

      1. INTRODUCTION

         Most of the processes in nature are nonlinear, in such a way that, the mathematical models used with the purpose to get exact and approximate solutions do not always offer the required results. On the other hand, differential equations have shown to be an appropriate tool with the end to model complicated phenomena in nature.

         Nature processes give rise to scientific problems and for the same reason the proposal of new methods with the purpose to obtain both exact and approximate solutions to the differential equations that govern these problems becomes compulsory. Given that the search for such solutions many times is a complicated task, it justifies the current research in this subject. Unlike the linear differential equations whose theory and solution methods can be found in many standard texts of differential equations [1] the case of nonlinear ordinary differential equations with exact solutions is less frequent [1]. One of the main contributions of this article is to show the potentiality of the proposed method in order to find both exact and approximate solutions with relative ease for the highly nonlinear problem that describes a coupled diffusion and nth- order chemical reaction in a spherical porous catalyst [2]. Given the great diversity of scientific problems, and their corresponding nonlinear differential equations to be solved, have been proposed several methods. Some of most employed in accordance with the literature are: tanh method [3], exp-

         function [4], Adomians decomposition method [5, 6, 7, 8], parameter expansion [9], homotopy perturbation method (HPM) [10, 11, 12, 13, 14, 15, 16, 17,18], perturbation method

         [19, 20, 21,22, 23], modified Taylor series method [24], Homotopy Analysis Method [25], Variational iteration method [26, 27], among others.

         The main goal of this work is to employ a version of Taylor Series Method with shooting (STSM) with the end to provide analytical solutions for the relevant highly nonlinear differential equation that describes the steady state solutions of a highly nonlinear model of a coupled diffusion and nth-order chemical reaction in a spherical porous catalyst [2]. As a matter of fact, the importance of the diffusion and reaction problems consists in their application in chemical and process engineering problems [28]. Respect to the process engineering field, diffusion and reaction problems arise above all in the heterogeneous catalysts by using porous structures where reaction could occur, other examples of technological interest are found in [28]. Next, we will see that STSM method is relatively easy to use and it is able to provide both, exact and analytical approximate solutions even for the case of nonlinear differential equations defined in closed intervals for which the most of the investigation works are essentially numeric. Traditionally, Taylor Series Method (TSM) is a known method which is given in terms of initial conditions for a proposed problem and it is not very employed at the moment to solve differential equations. As a matter of fact, one serious inconvenient of the problem to solve, is the presence of a

         singularity for = 0; nevertheless, we will see that STSM

         method is indeed able to adequately handle it. In brief, the

         proposed method is given in the following terms. Given an ordinary differential equation, then like TSM, STSM proposes a Taylor series for the differential equation to solve, given that the goal is to solve a boundary value problem, then the successive derivatives of the differential equation to calculate the coefficients of the Series Taylor solution will be expressed in terms of an unknown initial condition. This quantity will be tried as a shooting constant which will be determinined

         requiring that the proposed series obey the other boundary condition. We will see that even, the mentioned method is able to find exact analytical solutions.

         The rest of this work is proposed in the following way. Section 2, provides the basic idea of STSM Method. Additionally Section 3 explains the antecedents for the nonlinear differential equation that describes the steady state solutions of a highly nonlinear model of a coupled diffusion and nth-order chemical reaction in a spherical porous catalyst, Section 4 presents the application of the proposed method, in the search for an approximate solution for the relevant problem

         show that STSM method is able to provide both kinds of solutions for this complicated problem, by using essentially a single polynomial handy expression. Essentially, we will see that STSM solves an elementary algebraic equation by the proposed differential equation.

         Assuming isothermal conditions, the differential equation that governs the steady regime of the nth-order reaction- diffusion process in the spherical geometric pellet is given by [27, 29]:

         2 + 2 = ,

         above mentioned. Section 5 offers a discussion about the

         2

         (5)

         obtained solutions for this work. Finally, a brief conclusion of

         the relevant aspects of this article is given in Section 6.

      2. ELEMENTS OF SHOOTING TAYLOR SERIES METHOD

Next, we will provide the basic theory of STSM.

where, denotes the reactant concentration in pore of catalyst pellet, while is the effective diffusion coefficient for reactant, represents the distance from the pellet core and

the reaction rate constant. The reaction order belongs to the

range 0 and the boundary conditions are expressed for:

( = ) = , (6)

We start assuming a nonlinear problem with the following form

() = () (), , (1)

and

0

]

= 0. (7)

with the boundary condition

(, ) = 0, . (2)

=0

Expressing the boundary value problem (5)-(7) in terms of

the dimensionless variables

= , () = ( ) , (8)

In the above equations, is the order of the differential equation (1), represents a general operator; () denotes a known analytic function while is a boundary operator, as a

we get

0

2

matter of fact, is the boundary of the domain , and / is

+ 2 = 2, (9)

the differentiation along the normal drawn outwards from .

Following the proposed method, we will get the successive

derivatives of the differential equation to solve.

()(0), ( = 0,1, ), (3)

and

2

(1) = 1, (10)

in this expression 0 denotes the expansion point.

[ ]

=0

= 0, (11)

The series solution for (1) can be expressed as

where = ( 21/ )1/2 denotes the Thiele modulus.

(0 )

1 (0 ) 2

0

= (0) +

( ) +

1!

( ) +. . . ,

0

0

2!

(4)

1. APPLICATION OF STSM METHOD

   we note that derivatives ()(0), ( = 0,1, ) are expressed in

   terms of one of the boundary conditions of (1).

   With the purpose to get the coefficients of (4) (()(0)) ( = 0,1, ), we just apply the condition (1) = 1.

   Next, we will employ STSM method in order to find analytical approximate and exact solutions for the boundary value problem (9)-(11).

   In accordance with STSM, first we propose the following Taylor approximation:

   ( )

   ( )

   ( )

   III. ANTECEDENTS FOR THE PROBLEM THAT DESCRIBES THE STEADY STATE SOLUTIONS OF A

   NONLINEAR MODEL OF A COUPLED DIFFUSION AND

   () = (0) + 0

   2!

   2 + 0

   3!

   3 + 0

   4!

   4 +

   (12)

   NTH-ORDER CHEMICAL REACTION IN A SPHERICAL POROUS CATALYST

   where we have employed the initial condition (11).

   We will rewrite (12) as follows

   The goal of the article is to provide exact and handy analytical approximate solutions for the problem of a highly nonlinear model of a coupled diffusion and nth-order chemical

   () = + 2

   2!

   + 3 3!

   + 4 4!

   (13)

   reaction in a spherical porous catalyst. As a matter of fact, [2] presented an approach for these problems for the case of exact solutions, while [29] obtained analytical approximate solutions by using Adomian decomposition method. This article will

   after we have substituted the unknown initial conditions for

   some shooting constants.

   We note the unknown initial conditions are calculated

   about = 0, but at this point (9) has a singularity. With the

   end to avoid this problem, we multiply (9) by and after we

   apply a derivative to the resulting equation to get

   applying the condition (1) = 1, we get

   2

   2

   3

   2

   2

   2 1

   = 1 . (22)

   6

   2 + 3 + 2 2 =

   +

   , (14)

   After substituting (22) into (21) we get

   after evaluating (14) in = 0, we obtain:

   = 2. (15)

   () = 1

   2 6

   (1 2), (23)

   3

   With the purpose to follow evaluating other shooting

   constants we differentiate (14):

   that is the exact solution for this problem [2].

   Case 2. = 1, = 2

   In accordance with [2] this case possesses an exact

   3

   4

   3

   4

   + =

   4

   221

   2

   solution.

   Nevertheless, STSM will obtain a precise analytical approximate solution for this case:

   2

   +2 (1 + 1 + ( 1)2 ( ) ) ,

   After substituting = 1 and = 2 into (20) we get:

   2

   (16)

   () = + 2 2 + 2 4. (24)

   3 15

   evaluating (16) in = 0, we obtain:

   = 0. (17)

   Continuing in this form, after differentiating (16) we get:

   In accordance with the proposed method, in order to

   determinate we substitute (24) into the condition (1) = 1 to get an algebraic equation, whose solution is given by

   = 0.5555. (25)

   4 5 4

   5

   + 5 =

   2

   2

   Therefore, substituting (25) into (24) we get

   () = 0.5555 + 0.37037037 2 + 0.0740740 4. (26)

   221

   2

   + 22( 1)2 ( )

   2

   1

   2

   We note the handiness of (26).

   Next, we compare the precision of (26) with the exact

   3

   2

   + (1

   + ( 1)2 (

   ) )

   solution [2] for some values of , in order to know the

   3

   2

   2

   reliability of (26).

   +2

   x	Exact	STSM (26)	Relative error using (26)
   0	0.5510	0.5555	0.81%
   0.2	0.5660	0.5704	0.79%
   0.4	0.6121	0.6167	0.74%
   0.6	0.6933	0.6984	0.74%
   0.8	0.8187	0.8228	0.5054%
   1.0	0.4796933928	0.4700209520	0%

   +( 1)2 ( )

   .

   3

   Table 1: Comparison between (26) and exact solution for (9)-(11) using

   = 1, = 2.

   ( 2)3 ( )

   +( 1)

   2

   +22 ( )

   ( (

   2 ) )

   (18)

   Next, we evaluate (19) in = 0, to get

   2 1

   = 5

   3

   . (19)

   Therefore, by substituting (15)-(19) into (13) we get

   We note that the relative error committed by using (26) is

   2

   4 21

   scarcely less than one percent.

   () = +

   6

   2 +

   120

   4. (20)

   Case 3. = 5, = 1.

   We note the ease to obtain the approximate solution (20),

   which depends of arbitrary values of and , besides it is clear that following this procedure we can easily add more terms to

   (20). We note that the procedure is based in elementary differentiations Nevertheless, we will show the effectiveness of

   5

   (20) in order to model the proposed nonlinear problem.

   This case possesses an exact solution [2].

   We will see that STSM method will provide a handy precise analytical approximate solution for this case:

   9

   Substituting = 5 and = 1 into (20) yields

   () = +

   6

   2 +

   24

   4. (27)

   Case 1. = 0

2. CASE STUDIES

   Again, to calculate we use the condition (1) = 1 in

   order to obtain an equation

   5 9

   Next, we will obtain an exact solution for this case. We note that (20) can be simplified as:

   2

   + + = 1, (28)

   6 24

   the solution of (28) is given by

   () = +

   6

   2, (21)

   = 0.8914. (29)

   After substituting (29) into (27) we get a handy solution

   () = 0.8914 + 0.0938 2 + 0.0148 4. (30)

   Next, we will show again the precision of the proposed

   approximation (30) comparing it with the exact solution [2].

   Next, we will get a handy precise analytical approximate solution for this case.

   Substituting = 2 and = 1 into (20) yields in the

   following approximation.

   2 3

   () = + 2 +

   4. (35)

   6 60

   Table 2: Comparison between (26) and exact solution for (9)-(11)

   x	Exact	STSM (30)	Relative error using (30)
   0	0.88950	0.89140	0.20%
   0.2	0.89932	0.89517	0.21%
   0.4	0.90477	0.90678	0.22%
   0.6	0.92497	0.92708	0.22%
   0.8	0.95565	0.95749	0.19%
   1.0	1	1	0%

   using = 5, = 1.

   After applying the boundary condition (1) = 1, we get

   the algebraic equation

   2 3

   From Table 2 we see that the relative error committed by using (26) is scarcely of two tenths of one percent.

   Case 4. = 3/2, = 1.

   In accordance with [2], this case does not correspond to an

   exact solution. We will see that the proposed method provides a handy approximation with good precision.

   Substituting = 3/2 and = 1 into (20) yields in the

   following approximation.

   + + = 1. (36)

   6 60

   The solution of (36) is given by

   = 0.8646. (37)

   The obtaining of (37) results again in the solution of the

   proposed problem for the values = 2, = 1. After

   substituting (37) into (35) we get a handy accurate solution

   () = 0.8646 + 0.12458 2 + 0.010774. (38)

   We will show the precision of (38) comparing it with the

   numerical solution.

   Table 4: Comparison between (38) and numerical solution for (9)-(11)

   x	Exact	STSM (38)	Relative error using (38)
   0	0.8640	0.8646	0.069%
   0.2	0.8689	0.8696	0.080%
   0.4	0.8841	0.8848	0.080%
   0.6	0.9101	0.9108	0.082%
   0.8	0.9482	0.9487	0.057%
   1.0	1	1	0%

   using = 2, = 1.

   3/2

   () = +

   6

   2

   2 +

   80

   4. (31)

   After applying the boundary condition (1) = 1, we get

   from (31) the algebraic equation

   3/2

   +

   6

   the solution of (32) is given by

   2

   +

   80

   = 1, (32)

   From Table 4 we see that the relative error committed by using (38) is about between five hundredths and eight hundredths of one percent.

   = 0.8582. (33)

   After substituting (33) into (31) we get a handy accurate

   solution

   () = 0.8582 + 0.1325 2 + 0.00924. (34)

   We will show the precision of (34) comparing it with the

   numerical solution.

   Table 3: Comparison between (34) and numerical solution for (9)-(11)

   x	Exact	STSM (34)	Relative error using (34)
   0	0.8579	0.8582	0.034%
   0.2	0.8632	0.8635	0.036%
   0.4	0.8793	0.8796	0.038%
   0.6	0.9067	0.9070	0.033%
   0.8	0.9465	0.9467	0.021%
   1.0	1	1	0%

   using = 3/2, = 1.

   From Table 3 we see that the relative error committed by using (34) is about three hundredths of one percent.

   Case 5. = 2, = 1.

   In accordance with [2], this case does not possess an exact

   solution.

3. DISCUSSION

   In this work STSM was employed with the purpose to find both, exact and analytical approximate solutions for the rather complicated nonlinear ordinary differential equation which describes the problem of a nonlinear model of a coupled diffusion and nth-order chemical reaction in a spherical porous

   catalyst. We note that (9) has a singularity in = 0, we noted

   that STSM is much appropriate to handle this difficulty. The

   rearrangement of the equation and the systematic increasing of the order of the differential equation to solve, demonstrated its efficiency with the purpose to handle the aforementioned singularity. As a matter of fact, as result of this procedure based in just differentiations, we proposed, with little effort, to provide the general handy solution (20), which depends in

   principle of arbitrary values of the reaction order , and the Thiele modulus . In this step, the proposed procedure is very simple, for given values of and we apply the right boundary condition (1) = 1 in order to get an algebraic

   equation from the proposed solution, to determine the unknown

   initial condition whose solution provides the sought

   analytical approximate or the exact solution.

   As a matter of fact, we provide five case studies in order to show the potentiality of the proposed method. The first case

   study proposed the reaction order = 0. STSM got the correct

   exact solution for this problem [2]. For the case studies 2 and 3,

   we obtained handy precise analytical approximate solutions. In accordance with [2] these problems, which derives from the

   values = 1, = 2 and = 5, = 1 correspond to exact

   solutions. It is notable from Tables 1 and 2 that the relative

   error committed by using approximations (26) and (30) for these cases are scarcely less than one percent and two tenths of one percent respectively. Finally, Cases 4 and 5 correspond to non-exact solutions [2]. Nevertheless from Tables 3 and 4 we deduce that the relative error committed by using our approximations (34) and (38) for this case are about three hundredths and eight hundredths of one percent respectively. Therefore, our expressions for the proposed problem are not only precise but we emphasize that they are short polynomial expressions of only three terms. Besides, it is very important to emphasize that, it is possible to improve the accuracy of our STSM approximations by keeping more terms in expansion (20).

4. CONCLUSIONS

This work presented STSM with the end to provide both, exact and analytical approximate solutions for the nonlinear problem that describes the complicated nonlinear ordinary differential equation which describes the problem of a nonlinear model of a coupled diffusion and nth-order chemical reaction in a spherical porous catalyst. Despite of the fact that

(9) has a singularity in = 0, STSM was able to handle it

adequately with the end to get handy analytical solutions for

this important problem. 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 [2] shows the convenience of employing STSM as a practical tool with the purpose to obtain accurate solutions for boundary value problems instead of using other more sophisticated and cumbersome procedures. As a matter of fact [29] employed Adomian decomposition method with the end to get analytical approximate solutions for the problem (9)-(11). Nevertheless, Adomian decomposition method requires to calculate the so denominated Adomian coefficients in rder to linearize the nonlinear term and uses the concept of operator and inverse operator. As a consequence, from article [29], we noted that the employed procedure by using Adomian method is clearly more complex than the one proposed for this work. Thus, we emphasize that the use of the Taylor series is many times an adequate method to obtain handy analytical approximate solutions, and should be employed more frequently.

ACKNOWLEDGMENT

Authors would like to thank Roberto Ruiz Gomez for his contribution to this project.

REFERENCES

[1] Dennis G Zill. A first course in differential equations with modeling applications . pacific grove, ca: Brooks, 1996.

[2] Eugen Magyari. Exact analytical solutions of diffusion reaction in spherical porous catalyst. Chemical Engineering Journal, 158(2):266 270, 2010, doi: https://doi.org/10.1016/j.cej.2010.01.034.

[3] David J Evans and KR Raslan. The tanh function method for solving some important non-linear partial differential equations. International Journal of Computer Mathematics, 82(7):897905, 2005, doi: 10.1080/00207160412331336026.

[4] Fei Xu. A generalized soliton solution of the Konopelchenko- Dubrovsky equation using Hes exp-function method. Zeitschrift für

Naturforschung A, 62(12):685688, 2007, doi: doi:10.1515/zna-2007-

1202.

[5] G Adomian. A review of the decomposition method in applied mathematics. Journal of Mathematical Analysis and Applications, 135(2):501544, 1988, doi: https://doi.org/10.1016/0022-

247X(88)90170-9.

[6] E. Babolian and J. Biazar. On the order of convergence of Adomian method. Applied Mathematics and Computation, 130(2):383387, 2002, doi: https://doi.org/10.1016/S0096-3003(01)00103-5.

[7] Ali Kooch and M Abadyan. 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, 2012, doi: 10.3923/tasr.2012.57.67.

[8] S Hossien Chowdhury. A comparison between the modified homotopy perturbation method and Adomian decomposition method for solving nonlinear heat transfer equations. Journal of Applied Sciences, 11(7):14161420, 2011,

doi:https://dx.doi.org/10.3923/jas.2011.1416.1420.

[9] Li-Na Zhang and Lan Xu. Determination of the limit cycle by Hes parameter-expansion for oscillators in a u3 / (1 + u2) potential. Zeitschrift für Naturforschung A, 62(7-8):396398, 2007, doi: doi:10.1515/zna-2007-7-807.

[10] Uriel Filobello-Nino, Hector Vazquez-Leal, Agustin L Herrera-May, Roberto C Ambrosio-Lazaro, Victor M Jimenez-Fernandez, Mario A Sandoval-Hernandez, Oscar Alvarez-Gasca, and Beatriz E Palma- Grayeb. The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform. Thermal Science, 24(2 Part B): 11051115, 2020, doi: https://doi.org/10.2298/TSCI180108204F.

[11] Ji-Huan He. Homotopy perturbation method for solving boundary value problems. Physics letters A, 350(1-2):8788, 2006.

[12] Ji-Huan He. Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis, 31(2):205 209, 2008, doi: tmna/1463150264.

[13] Ji-Huan He. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics, 35(1):3743, 2000, doi: https://doi.org/10.1016/S0020-7462(98)00085-7.

[14] Hossein Aminikhah. Analytical approximation to the solution of nonlinear Blasius viscous flow equation by ltnhpm. International Scholarly Research Notices, 2012, 2012, doi: 10.5402/2012/957473.

[15] Reza Noorzad, A Tahmasebi Poor, and Mehdi Omidvar. Variational iteration method and homotopy-perturbation method for solving Burgers equation in fluid dynamics. Journal of Applied Sciences, 8(2):369373, 2008, doi: 10.3923/jas.2008.369.373.

[16] Vazquez-Leal, H., Filobello-Nino, U., Sarmiento-Reyes, A., Sandoval- Hernández, M., Pérez-Sesma, J. A. A., Pérez-Sesma, A., … & González- Martínez, F. J. (2017). Application of HPM to Solve Unsteady Squeezing Flow of a Second-Grade Fluid between Circular Plates. American Academic Scientific Research Journal for Engineering, Technology, and Sciences, 27(1), 161-178.

[17] Boubaker, K., Vazquez-Leal, H., Colantoni, A., Longo, L., Allegrini, E., Sandoval-Hernandez, M., … & Castro-González, F. (2016). Comparative of HPM and BPES solutions to Gelfand's differential equation governing chaotic dynamics in combustible gas thermal ignition. Nonlinear Science Letters A Mathematics, Physics and Mechanics, 41.

[18] Filobello-Nino, U., Vazquez-Leal, H., Huerta-Chua, J., Jimenez- Fernandez, V. M., Sandoval-Hernandez, M. A., Delgado-Alvarado, E.,

& Tlapa-Carrera, V. M. (2021). A Novel Version of HPM Coupled with the PSEM Method for Solving the Blasius Problem. Discrete Dynamics in Nature and Society, Volume 2021, Article ID 5909174, https://doi.org/10.1155/2021/5909174

[19] M. A. Sandoval-Hernandez, O. Alvarez-Gasca, A. D. Contreras- Hernandez, J. E. Pretelin-Canela, B. E. Palma-Grayeb, V. M. Jimenez- Fernandez , U. Filobello-Nino, D. Pereyra-Diaz, S. F. Hernandez- Machuca, C. E. Sampieri-Gonzalez, F. J. Gonzalez-Martinez , R. Castaneda-Sheissa, S. Hernandez-Mendez, J. Matias-Perez, L. Cuellar- Hernandez, C. Hoyos-Reyes, J. Cervantes-Perez , L. J. Varela-Lara, J. L. Vazquez-Aguirre, L. Gil-Adalid, J. L. Rocha-Fernandez, N. Bagatella- Flores, H. Vazquez-Leal, 2019, Exploring the Classic Perturbation Method for Obtaining Single and Multiple Solutions of Nonlinear Algebraic Problems with Application to Microelectronic Circuits,

INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH &

TECHNOLOGY (IJERT) Volume 08, Issue 09 (September 2019).

[20] Filobello-Nino, U., Vázquez-Leal, H., Pérez-Sesma, J. A. A., Pérez- Sesma, A., Sandoval-Hernández, M., Sarmiento-Reyes, A., … & Gonzalez-Martinez, F. J. (2017). Classical perturbation method for the solution of a model of diffusion and reaction. American Academic Scientific Research Journal for Engineering, Technology, and Sciences, 27(1), 151-160.

[21] M. Sandoval-Hernandez, H. Vazquez-Leal, U. Filobello-Nino , Elisa De-Leo-Baquero, Alexis C. Bielma-Perez, J.C. Vichi-Mendoza , O. Alvarez-Gasca, A.D. Contreras-Hernandez, N. Bagatella-Flores , B.E. Palma-Grayeb, J. Sanchez-Orea, L. Cuellar-Hernandez, 2021, The Quadratic Equation and its Numerical Roots, INTERNATIONAL JOURNAL OF ENGINEERING RESEARCH & TECHNOLOGY

(IJERT) Volume 10, Issue 06 (June 2021)

[22] U Filobello-Nino, H Vazquez-Leal, Y Khan, A Yildirim, VM Jimenez- Fernandez, AL Herrera-May, R Castaneda-Sheissa, and J Cervantes- Perez. Using perturbation methods and LaplacePadé approximation to solve nonlinear problems. Miskolc Mathematical Notes, 14(1):89101, 2013, doi: https://doi.org/10.18514/MMN.2013.517.

[23] Uriel Filobello-Nino, Hector Vazquez-Leal, Brahim Benhammouda, Luis Hernandez-Martinez, Yasir Khan, Victor Manuel Jimenez- Fernandez, Agustin Leobardo Herrera-May, Roberto Castaneda-Sheissa, Domitilo Pereyra-Diaz, Juan Cervantes-Perez, et al. A handy approximation for a mediated bioelectrocatalysis process, related to Michaelis-Menten equation. SpringerPlus, 3(1):16, 2014, doi: https://doi.org/10.1186/2193-1801-3-162.

[24] Hector Vazquez-Leal, Mario Sandoval-Hernandez, Roberto Castaneda- Sheissa, Uriel Filobello-Nino, and Arturo Sarmiento-Reyes. Modified taylor solution of equation of oxygen diffusion in a spherical cell with michaelis-menten uptake kinetics. International Journal of Applied Mathematics Research, 4(2):253, 2015, doi: http://dx.doi.org/10.14419/ijamr.v4i2.4273.

[25] Twinkle Patel, MN Mehta, and VH Pradhan. 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(1):6066, 2012, doi: 10.3923/ajaps.2012.60.66.

[26] Laila M B Assas. Approximate solutions for the generalize KdV Burgers equation by hes variational iteration method. Physica Scripta, 76(2):161164, jul 2007, doi: 10.1088/0031-8949/76/2/008.

[27] M Kazemnia, SA Zahedi, M Vaezi, and N Tolou. Assessment of modified variational iteration method in bvps of highorder differential equations. Journal of Applied Sciences, 8(22):41924197, 2008, doi: 10.3923/jas.2008.4192.4197.

[28] Mirosaw K Szukiewicz. Study of reactiondiffusion problem: modeling, exact analytical solution, and experimental verification. SN Applied Sciences, 2(7):114, 2020, doi: https://doi.org/10.1007/s42452-

020-3045-0.

[29] L. Shi-Bin, S. Yan-Ping, and K. Scott. Analytic solution of diffusion- reaction in spherical porous catalyst. Chemical Engineering & Technology, 26(1):8795, doi: https://doi.org/10.1002/ceat.200390013.