Multiple solutions of the nonlinear reaction-di usion model with fractional reaction

Subject Areas : Applied Mathematics

H. Vosoughi ¹ (Department of Mathematics, Faculty of Science, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran)
E. Shivanian ² (Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran)
M. Anbarloei ³ (Department of Mathematics, Faculty of Science, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran)

Received: 2015-09-21 Accepted : 2015-09-21 Published : 2015-12-01

Keywords: Predictor homotopy analysis method, Prescribed parameter, Reaction-diusion model, multiple solutions,

Abstract :

The purpose of this letter is to revisit the nonlinear reaction-diusion modelin porous catalysts when reaction term is fractional function of the concen-tration distribution of the reactant. This model, which originates also in uidand solute transport in soft tissues and microvessels, has been recently givenanalytical solution in terms of Taylors series for dierent family of reactionterms. We apply the method so-called predictor homotopy analysis method(PHAM) which has been recently proposed to predict multiplicity of solutionsof nonlinear BVPs. Consequently, it is indicated that the problem for somevalues of the parameter admits multiple solutions. Also, error analysis of thesesolutions are given graphically.

References:

[1] A.J. Ellery, M.J. Simpson, An analytical method to solve a general class
of nonlinear reactive transport models, Chem. Eng. J. 169 (2011) 313-318.
[2] S. Abbasbandy, Approximate solution for the nonlinear model of diusion
and reaction in porous catalysts by means of the homotopy analysis
method, Chem. Eng. J. 136 (2008) 144-150.
[3] S. Abbasbandy, E. Magyari, E. Shivanian, The homotopy analysis method
for multiple solutions of nonlinear boundary value problems, Commun.
Nonlinear Sci. Numer. Simulat. 14 (2009) 3530-3536.
[4] Y.P. Sun and S.B. Liu and K. Scott, Approximate solution for the
nonlinear model of diusion and reaction in porous catalysts by the
decomposition method, Chem. Eng. J. 101 (2004) 1-10.
[5] S. Abbasbandy and E. Shivanian, Exact analytical solution of a nonlinear
equation arising in heat transfer, Phys. Lett. A, 374 (2010) 567-574.
[6] J.E. Bailey, D.E. Ollis, Biochemical Engineering Fundamentals, second
edition, McGrawHill, 1986.

7] T. P. Clement, Y. Sun, B. S. Hooker, J. N. Peterson, Modeling
multispecies reactive transport in ground water, Groundwater Monitoring
and Remediation 18 (1998) 7992.
[8] C. Zheng, G.D. Bennett, Applied Contaminant Transport Modelling,
second edition, Wiley Interscience, New York, 2002.
[9] A. Aris, The mathematical theory of diusion and reaction in permeable
catalysts, Volume 1 The Theory of Steady State, Oxford, 1975.
[10] E.J. Henley, E.M. Rosen, Material and Energy Balance Computations,
John Wiley and Sons, New York, 1969.
[11] S. Abbasbandy, E. Shivanian, Predictor homotopy analysis method and its
application to some nonlinear problems, Commun. Nonlinear Sci. Numer.
Simulat. 16 (2011) 2456-2468.
[12] S. Abbasbandy, E. Shivanian, Prediction of multiplicity of solutions
of nonlinear boundary value problems: Novel application of homotopy
analysis method, Commun. Nonlinear Sci. Numer. Simulat. 15 (2010)
3830-3846.
[13] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis
Method, Chapman Hall CRC/Press, Boca Raton, 2003.
[14] S. J. Liao, Homotopy Analysis Method in Nonlinear Dierential
Equations, Springer, 2012.
[15] T. Hayat, T. Javed, M. Sajid, Analytic solution for rotating ow and heat
transfer analysis of a third-grade uid, Acta. Mech. 191 (2007) 219-29.
[16] T. Hayat, M. Khan, M. Sajid, S. Asghar, Rotating ow of a third-grade
uid in a porous space with hall current, Nonlinear Dyn. 49 (2007) 83-91.
[17] T. Hayat, Z. Abbas, M. Sajid, S. Asghar, The in uence of thermal
radiation on MHD ow of a second grade uid, Int. J. Heat. Mass. Transf.
50 (2007) 931-41.
[18] T. Hayat, N. Ahmed, M. Sajid, S. Asghar, On the MHD ow of a second
grade uid in a porous channel, Comput. Math. Appl. 54 (2007) 14-40.
[19] T. Hayat, M. Khan, M. Ayub, The eect of the slip condition on ows of
an Oldroyd 6 constant uid, J. Comput. Appl. 202 (2007) 402-13.

[20] M. Sajid, A. Siddiqui, T. Hayat, Wire coating analysis using MHD
Oldroyd 8-constant uid, Int. J. Eng. Sci. 45 (2007) 381-92.
[21] M. Sajid, T. Hayat, S. Asghar, Non-similar analytic solution for MHD
ow and heat transfer in a third-order uid over a stretching sheet. Int.
J. Heat Mass. Transf. 50 (2007) 1723-36.
[22] L. Song, HQ. Zhang, Application of homotopy analysis method to
fractional KdV-Burgers-Kuramoto equation, Phys. Lett. A. 367 (2007)
88-94.
[23] J. Cheng, S. J. Liao, RN. Mohapatra, K. Vajravelu, Series solutions of
nano boundary layer ows by means of the homotopy analysis method, J.
Math. Anal. Appl. 343 (2008) 233-45.
[24] S. Abbasbandy, The application of the homotopy analysis method to
nonlinear equations arising in heat transfer, Phys. Lett. A. 360 (2006)
109-13.
[25] SP. Zhu, An exact and explicit solution for the valuation of American put
options, Quant. Fin. 6 (2006) 229-42.
[26] Y. Wu, KF. Cheung, Explicit solution to the exact Riemann problem and
application in nonlinear shallow-water equations, Int. J. Numer. Meth.
Fluids. 57 (2008) 1649-68.
[27] M. Yamashita, K. Yabushita, K. Tsuboi, An analytic solution of projectile
motion with the quadratic resistance law using the homotopy analysis
method, J. Phys. A. 40 (2007) 840316.
[28] Y. Bouremel, Explicit series solution for the Glauert-jet problem by
means of the homotopy analysis method, Commun. Nonlinear. Sci. Numer.
Simulat. 12(5) (2007) 714-24.
[29] L. Tao, H. Song, Chakrabarti S. Nonlinear progressive waves in water of
nite depth-an analytic approximation, Clastal. Eng. 54 (2007) 825-34.
[30] H. Song, L. Tao, Homotopy analysis of 1D unsteady, nonlinear
groundwater ow through porous media, J. Coastal. Res. 50 (2007) 292-5.
[31] A. Molabahrami, F. Khani, The homotopy analysis method to solve the
Burgers-Huxley equation. Nonlinear Anal. B: Real World Appl. 10 (2009)
589-600.

[32] A. S. Bataineh, M. S. Noorani, I. Hashim, Solutions of time-dependent
EmdenFowler type equations by homotopy analysis method, Phys. Lett.
A. 371 (2007) 7282.
[33] Z. Wang, L. Zou, H. Zhang, Applying homotopy analysis method for
solving dierential-dierence equation, Phys. Lett. A. 369 (2007) 77-84.
[34] M. Inc, On exact solution of Laplace equation with Dirichlet and Neumann
boundary conditions by the homotopy analysis method, Phys. Lett. A. 365
(2007) 412-5.
[35] W. H. Cai, Nonlinear dynamics of thermal-hydraulic networks. Ph.D.
thesis, University of Notre Dame; 2006.
[36] T. T. Zhang, L. Jia, Z. C. Wang, X. Li, The application of homotopy
analysis method for 2-dimensional steady slip ow in microchannels, Phys.
Lett. A. 372 (2008) 32237.
[37] A. K. Alomari, M. S. Noorani, R. Nazar, Adaptation of homotopy analysis
method for the numeric-analytic solution of Chen system. Commun.
Nonlinear Sci. Numer. Simul. 4 (2009) 2336-46.
[38] M. M. Rashidi, S. Dinarvand, Purely analytic approximate solutions
for steady three-dimensional problem of condensation lm on inclined
rotating disk by homotopy analysis method, Nonlinear Anal. B: Real
World Appl. 10 (2009) 2346-2356.
[39] Z. Odibat, S. Momani, H. Xu, A reliable algorithm of homotopy analysis
method for solving nonlinear fractional dierential equations, Applied
Mathematical Modelling 2010;34:593-600.
[40] S. Xinhui, Z. Liancun, Z. Xinxin, Y. Jianhong, Homotopy analysis method
for the heat transfer in a asymmetric porous channel with an expanding
or contracting wall, Appl. Math. Modell. 35 (2011) 4321-4329.
[41] R. A. Van Gorder, K. Vajravelu, Analytic and numerical solutions to the
Lane-Emden equation, Phys. Lett. A. 372 (372) 6060-5.
[42] Q. Wang, The optimal homotopy analysis method for Kawahara equation,
Nonlinear Anal. B: Real World Appl. 12(3) (2011) 1555-1561.
[43] A. R. Ghotbi, A. Bararni, G. Domairry, A. Barari, Investigation of a
powerful analytical method into natural convection boundary layer ow,
Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 2222-2228.

[44] M. Ayub, H. Zaman, M. Ahmad, Series solution of hydromagnetic ow
and heat transfer with Hall eect in a second grade uid over a stretching
sheet, Cent. Eur. J. Phys. 8 (2010) 135-49.
[45] H. Vosughi, E. Shivanian, S. Abbasbandy, A new analytical technique to
solve Volterra's integral equations, Mathematical methods in the applied
sciences, 10(34) (2011) 1243-1253.
[46] M. Ghasemi, A. Azizi, M. Fardi, Numerical solution of seven-order
Sawada-Kotara equations by homotopy perturbation method, Math. Sc.
J. 7(1) (2011) 69-77.
[47] L. Hooshangian, D. Mirzaei, A Legendre-spectral scheme for solution of
nonlinear system of Volterra-Fredholm integral equations, Math. Sc. J.
8(1) (2012) 1-14.
[48] S. Abbasbandy, E. Shivanian, K. Vajravelu, Mathematical properties of
~-curve in the frame work of the homotopy analysis method, Commun.
Nonlinear Sci. Numer. Simulat. 16 (2011) 4268-4275.