An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system

K. M. Saad; O. S. Iyiola; P. Agarwal; K. M. Saad; O. S. Iyiola; P. Agarwal

doi:10.3934/Math.2018.1.183

AIMS Mathematics

2018, Volume 3, Issue 1: 183-194. doi: 10.3934/Math.2018.1.183

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Research article

An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system

1.
Department of Mathematics, Faculty of Arts and Sciences, Najran, Najran University, Saudi Arabia
2.
Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen
3.
Department of Mathematical Sciences, Minnesota State University Moorhead, MN USA
4.
Department of Mathematical, Anand International College of Engineering, Jaipur-303012, India
5.
Center for Basic and Applied Sciences, Jaipur-302029, India

Received: 17 September 2017 Accepted: 31 January 2018 Published: 21 March 2018

We established an effective algorithm for the homotopy analysis method (HAM) to solve a cubic isothermal auto-catalytic chemical system (CIACS). Our solution comes in a rapidly convergent series where the intervals of convergence given by h-curves and to find the optimal values of h, we used the averaged residual errors. The HAM solutions are compared with the solutions obtained by Mathematica in-built numerical solver. We also show the behavior of the HAM solution.

Keywords:

Citation: K. M. Saad, O. S. Iyiola, P. Agarwal. An effective homotopy analysis method to solve the cubic isothermal auto-catalytic chemical system[J]. AIMS Mathematics, 2018, 3(1): 183-194. doi: 10.3934/Math.2018.1.183

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Abstract

References

[1]	S. Abbasbandy and M. Jalili, Determination of optimal convergence-control parameter value in homotopy analysis method, Numer. Algorithms, 64 (2013), 593-605.
[2]	S. Abbasbandya, E. Shivaniana and K. Vajravelub, Mathematical properties of h-curve in the frame work of the homotopy analysis method, Commun. Nonlinear Sci., 16 (2011), 4268-4275.
[3]	S. M. Abo-Dahab, S. Mohamed and T. A. Nofal, A One Step Optimal Homotopy Analysis Method for propagation of harmonic waves in nonlinear generalized magneto-thermoelasticity with two relaxation times under influence of rotation, Abstr. Appl. Anal., 2013 (2013), 1-14.
[4]	A. Sami Bataineh, M. S. M. Noorani and I. Hashim, The homotopy analysis method for Cauchy reaction diffusion problems, Phys. Lett. A, 372 (2008), 613-618.
[5]	K. A. Gepreel and M. S. Mohamed, An optimal homotopy analysis method nonlinear fractional differential equation, Journal of Advanced Research in Dynamical and Control Systems, 6 (2014), 1-10.
[6]	M. Ghanbari, S. Abbasbandy and T. Allahviranloo, A new approach to determine the convergencecontrol parameter in the application of the homotopy analysis method to systems of linear equations, Appl. Comput. Math., 12 (2013), 355-364.
[7]	J. H. Merkin, D. J. Needham and S. K. Scott, Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system, IMA J. Appl. Math., 50 (1993), 43-76.
[8]	O. S. Iyiola, A numerical study of ito equation and Sawada-Kotera equation both of time-fractional type, Adv. Math., 2 (2013), 71-79.
[9]	O. S. Iyiola, A fractional diffusion equation model for cancer tumor, American Institute of Physics Advances, 4 (2014), 107121.
[10]	O. S. Iyiola, Exact and Approximate Solutions of Fractional Diffusion Equations with Fractional Reaction Terms, Progress in Fractional Di erentiation and Applications, 2 (2016), 21-30.
[11]	O. S. Iyiola and G. O. Ojo, On the analytical solution of Fornberg-Whitham equation with the new fractional derivative, Pramana, 85 (2015), 567-575.
[12]	O.S. Iyiola, O. Tasbozan, A. Kurt, et al. On the analytical solutions of the system of conformable time-fractional Robertson equations with 1-D diffusion, Chaos, Solitons and Fractals, 94 (2017), 1-7.
[13]	S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992.
[14]	S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, Boca Raton: Chapman and Hall/CRC Press, 2003.
[15]	S. J. Liao, An optimal homotopy analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci., 15 (2010), 2003-2016.
[16]	M. Russo and R. V. Gorder, Control of error in the homotopy analysis of nonlinear Klein-Gordon initial value problems, Appl. Math. Comput., 219 (2013), 6494-6509.
[17]	K. M. Saad, An approximate analytical solutions of coupled nonlinear fractional diffusion equations, Journal of Fractional Calculus and Applications, 5 (2014), 58-72.
[18]	K. M. Saad, E. H. AL-Shareef, S. Mohamed, et al. Optimal q-homotopy analysis method for timespace fractional gas dynamics equation, Eur. Phys. J. Plus, 132 (2017), 23.
[19]	K. M. Saad and A. A. AL-Shomrani, An application of homotopy analysis transform method for riccati differential equation of fractional order, Journal of Fractional Calculus and Applications, 7 (2016), 61-72.
[20]	M. Shaban, E. Shivanian and S. Abbasbandy, Analyzing magneto-hydrodynamic squeezing flow between two parallel disks with suction or injection by a new hybrid method based on the Tau method and the homotopy analysis method, Eur. Phys. J. Plus, 128 (2013), 133.
[21]	E. Shivanian and S. Abbasbandy, Predictor homotopy analysis method: Two points second order boundary value problems, Nonlinear Anal-Real, 15 (2014), 89-99.
[22]	E. Shivanian, H. H. Alsulami, M. S Alhuthali, et al. Predictor Homotopy Analysis Method (Pham) for Nano Boundary Layer Flows with Nonlinear Navier Boundary Condition: Existence of Four Solutions, Filomat, 28 (2014), 1687-1697.
[23]	L. A. Soltania, E. Shivanianb and R. Ezzatia, Convection-radiation heat transfer in solar heat exchangers filled with a porous medium: Exact and shooting homotopy analysis solution, Appl. Therm. Eng., 103 (2016), 537-542.
[24]	H. Vosoughi, E. Shivanian and S. Abbasbandy, Unique and multiple PHAM series solutions of a class of nonlinear reactive transport model, Numer. Algorithms, 61 (2012), 515-524.
[25]	H. Vosughi, E. Shivanian and S. Abbasbandy, A new analytical technique to solve Volterra's integral equations, Math. methods appl. sci., 34 (2011), 1243-1253.
[26]	M. Yamashita, K. Yabushita and K. Tsuboi, An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method, J. Phys. A, 40 (2007), 8403-8416.
[27]	X. Zhang, P. Agarwal, Z. Liu, et al. Existence and uniqueness of solutions for stochastic differential equations of fractional-order q > 1 with finite delays, Adv. Di er. Equ-NY, 123 (2017), 1-18.
[28]	M. Ruzhansky, Y. J. Cho, P. Agarwal, et al. Advances in Real and Complex Analysis with Applications, Springer Singapore, 2017.
[29]	S. Salahshour, A. Ahmadian, N. Senu, et al. On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem, Entropy, 17 (2015), 885-902.

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