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

Subject Areas : Applied Mathematics

H. Vosoughi ¹ , E. Shivanian ² , M. Anbarloei ³

1 - Department of Mathematics, Faculty of Science, Islamshahr Branch, Islamic Azad University, Islamshahr, Tehran, Iran
2 - Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran
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:

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.