Vosoughi, H., Shivanian, E., Anbarloei, M. (2013). Multiple solutions of the nonlinear reaction-diusion model with fractional reaction. Theory of Approximation and Applications, 9(2), 159-170.

H. Vosoughi; E. Shivanian; M. Anbarloei. "Multiple solutions of the nonlinear reaction-diusion model with fractional reaction". Theory of Approximation and Applications, 9, 2, 2013, 159-170.

Vosoughi, H., Shivanian, E., Anbarloei, M. (2013). 'Multiple solutions of the nonlinear reaction-diusion model with fractional reaction', Theory of Approximation and Applications, 9(2), pp. 159-170.

Vosoughi, H., Shivanian, E., Anbarloei, M. Multiple solutions of the nonlinear reaction-diusion model with fractional reaction. Theory of Approximation and Applications, 2013; 9(2): 159-170.

Multiple solutions of the nonlinear reaction-diusion model with fractional reaction

^{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

Abstract

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