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June  2019, 12(3): 665-684. doi: 10.3934/dcdss.2019042

Comparative study of a cubic autocatalytic reaction via different analysis methods

a. 

Department of Mathematics, College of Arts and Sciences, Najran University, 61441, Najran, Saudi Arabia

b. 

Department of Mathematics, Faculty of Applied Science, Taiz University, Taiz, Yemen

* Corresponding author: khaledma_sd@hotmail.com

Received  June 2017 Revised  September 2017 Published  September 2018

In this paper we discuss an approximate solutions of the space-time fractional cubic autocatalytic chemical system (STFCACS) equations. The main objective is to find and compare approximate solutions of these equations found using Optimal q-Homotopy Analysis Method (Oq-HAM), Homotopy Analysis Transform Method (HATM), Varitional Iteration Method (VIM) and Adomian Decomposition Method (ADM).

Citation: Khaled Mohammed Saad, Eman Hussain Faissal AL-Sharif. Comparative study of a cubic autocatalytic reaction via different analysis methods. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 665-684. doi: 10.3934/dcdss.2019042
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References:
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K. Abbaoui and Y. Cherruault, Convergence of adomian's method applied to differential equations, Comput. Math. Appl., 28 (1994), 103-109.  doi: 10.1016/0898-1221(94)00144-8.  Google Scholar

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S. AbbasbandyE. Shivaniana and K. Vajravelu, Mathematical properties of h-curve in the frame work of the homotopy analysis method, Communications in Nonlinear Science and Numerical Simulation, 16 (2011), 4268-4275.  doi: 10.1016/j.cnsns.2011.03.031.  Google Scholar

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S. Abbasbandy, Numerical method for non-linear wave and diffusion equations by the variational iteration method, International Journal for Numerical Methods in Engineering, 73 (2008), 1836-1843.  doi: 10.1002/nme.2150.  Google Scholar

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S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int Commun Heat Mass Transf, 34 (2007), 380-387.   Google Scholar

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S. Abbasbandy, The application of the homotopy analysis method to nonlinear equations arising in heat transfer, Phys Lett A, 360 (2006), 109-113.  doi: 10.1016/j.physleta.2006.07.065.  Google Scholar

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S. M. Abo-Dahab, M. 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, Abstract and Applied Analysis, 2013 (2013), Art. ID 614874, 14 pp. doi: doi.org/10.1155/2013/614874.  Google Scholar

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G. Adomian, Solving the mathematical models of neurosciences and medicine, Mathematics and Computers in Simulation, 40 (1995), 107-114.  doi: 10.1016/0378-4754(95)00021-8.  Google Scholar

[9]

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[10]

A. S. ArifeS. K. Vanani and F. Soleymani, The laplace homotopy analysis method for solving a general fractional diffusion equation arising in nano- hydrodynamics, J Comput Theor Nanosci, 10 (2013), 33-36.  doi: 10.1166/jctn.2013.2653.  Google Scholar

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[14]

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[15]

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[17]

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[18]

V. G. Gupta and P. Kumar, Approximate solutions of fractional linear and nonlinear differential equations using laplace homotopy analysis method, Int. J. Nonlinear Sci., 19 (2015), 113-120.   Google Scholar

[19]

T. HayatM. Khan and S. Asghar, Homotopy analysis of mhd flows of an oldroyd 8-constant fluid, Appl. Math. Comput., 155 (2004), 417-425.  doi: 10.1016/S0096-3003(03)00787-2.  Google Scholar

[20]

T. HayatS. B. KhanM. Sajid and S. Asghar, Rotating flow of a third grade fluid in a porous space with hall current, Nonlinear Dyn, 49 (2007), 83-91.  doi: 10.1007/s11071-006-9105-1.  Google Scholar

[21]

J. H. He, Variational iteration method- a kind of non-linear analytical technique: Some examples, International Journal of Non-Linear Mechanics, 34 (1999), 699-708.  doi: 10.1016/S0020-7462(98)00048-1.  Google Scholar

[22]

J. H. He, Variational iteration method for autonomous ordinary differential systems, Appl. Math. Comput., 114 (2000), 115-123.  doi: 10.1016/S0096-3003(99)00104-6.  Google Scholar

[23]

J. H. He, Variational principle for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons & Fractals, 19 (2004), 847-851.  doi: 10.1016/S0960-0779(03)00265-0.  Google Scholar

[24]

J. H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B, 20 (2006), 1141-1199.  doi: 10.1142/S0217979206033796.  Google Scholar

[25]

O. S. Iyiola, On the solutions of non-linear time-fractional gas dynamic equations: An analytical approach, International Journal of Pure and Applied Mathematics, 98 (2015), 491-502.  doi: 10.12732/ijpam.v98i4.8.  Google Scholar

[26]

H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using adomian decomposition, Journal of Computational and Applied Mathematics, 196 (2006), 644-651.  doi: 10.1016/j.cam.2005.10.017.  Google Scholar

[27]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Vol. 204, Elsevier, Amsterdam, 2006.  Google Scholar

[28]

A. C. King, J. Billingham and S. R. Otto. Differential Equations: Linear, Nonlinear, Ordinary, Partial, Cambridge University Press, 2003. doi: 10.1017/CBO9780511755293.  Google Scholar

[29]

S. KumarJ. SinghD. Kumar and S. Kapoor, New homotopy analysis transform algorithm to solve volterra integral equation, Ain Shams Engineering Journal, 5 (2014), 243-246.  doi: 10.1016/j.asej.2013.07.004.  Google Scholar

[30]

S. Kumar and D. Kumar, Fractional modelling for bbm-burger equation by using new homotopy analysis transform method, Journal of the Association of Arab Universities for Basic and Applied Sciences, 16 (2014), 16-20.  doi: 10.1016/j.jaubas.2013.10.002.  Google Scholar

[31]

D. Kumar,J. Singh, S. Kumar and Sushila, Numerical computation of klein-gordon equations arising in quantum field theory by using homotopy analysis transform method, Alexandria Engineering Journal, 53 (2014), 469-474. doi: 10.1016/j.aej.2014.02.001.  Google Scholar

[32]

D. Kumar, J. Singh and Sushila, Application of homotopy analysis transform method to fractional biological population model, Romanian Reports in Physics, 65 (2013), 63-75. Google Scholar

[33]

S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992. Google Scholar

[34]

S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method, Boca Raton: Chapman and Hall/CRC Press, 2003.  Google Scholar

[35]

S. J. Liao, A kind of approximate solution technique which does not depend upon small parameters- Ⅱ: an application in fluid mechanics, Int J Non- Linear Mech, 32 (1997), 815-822.  doi: 10.1016/S0020-7462(96)00101-1.  Google Scholar

[36]

S.-J. Liao, An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun Nonlinear Sci Numer Simulat, 15 (2010), 2003-2016.  doi: 10.1016/j.cnsns.2009.09.002.  Google Scholar

[37]

S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl Math Comput, 147 (2004), 499-513.  doi: 10.1016/S0096-3003(02)00790-7.  Google Scholar

[38]

H. M. Liu, Generalized variational principles for ion acoustic plasma waves by He's semi-inverse method, Chaos, Solitons Fractals, 23 (2005), 573-576.   Google Scholar

[39]

M. MadaniM. FathizadehY. Khan and A. Yildirim, On the coupling of the homotopy perturbation method and laplace transformation, Math. and Comput. Model., 53 (2011), 1937-1945.  doi: 10.1016/j.mcm.2011.01.023.  Google Scholar

[40]

T. Mavoungou and Y. Cherruault, Convergence of adomian's method and applications to non-linear partial differential equation, Kybernetes, 21 (1992), 13-25.  doi: 10.1108/eb005942.  Google Scholar

[41]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons, New York, 1993.  Google Scholar

[42]

M. S. MohamedK. A. GepreelM. R. Alharthi and R. A. Alotabi, Homotopy analysis transform method for integro-differential equations, General Mathematics Notes, 32 (2016), 32-48.   Google Scholar

[43]

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Figure 1.  The absolute error between the 3-terms of Oq-HAM solutions and numerical method using Mathematica for (4)-(5) with $\alpha = 1, \beta = 1, a = 0.001, b = 0.001, h = -3.055, n = 5$
Figure 2.  The absolute error between the 3-terms of HATM solutions and numerical method using Mathematica for (4)-(5) with $ \alpha = 1, \beta = 1, a = 0.001, b = 0.001, h = -0.64$
Figure 3.  The absolute error between the second approximation by VIM and the numerical method using Mathematica for (4)-(5) with $ \alpha = 1, \beta = 1, a = 0.001, b = 0.001 .$
Figure 4.  The absolute error between the 3-terms of ADM and the numerical method using Mathematica for (4)-(5) with $ \alpha = 1, \beta = 1, a = 0.001, b = 0.001 $
Figure 7.  The absolute error between the 3-terms of Oq-HAM solutions and numerical method using Mathematica for (4)-(5) with $\alpha = 0.9, \beta = 0.99, a = 0.001, b = 0.001, h = -1.9, n = 5$
Figure 8.  The absolute error between the 3-terms of HATM solutions and numerical method using Mathematica for (4)-(5) with $\alpha = 0.9,\beta = 0.99, a = 0.001, b = 0.001, h = -0.64$
Figure 9.  The absolute error between the second approximation by VIM and the numerical method using Mathematica for (4)-(5) with $ \alpha = 0.9, \beta = 0.99, a = 0.001, b = 0.001 $
Figure 10.  The absolute error between the 3-terms of ADM and the numerical method using Mathematica for (4)-(5) with $ \alpha = 0.9, \beta = 0.99, a = 0.001, b = 0.001$
Figure 5.  The comparison of Oq-HAM, HATM, VIM and ADM for (4)-(5) with numerical method in Mathematica for $x = 0.1, 5, 20, 40,100$ respectively and $\alpha = 1, \beta = 1, a = 0.001, b = 0.001, n = 5, h_{Oq-HAM} = -3.055, h_{HATM} = 0.64.$ Dash - dotted line (Oq-HAM), dotted line (HATM), dash line (VIM), and solid line (ADM)
Figure 6.  The comparison of Oq-HAM, HATM, VIM and ADM for (4)-(5) with numerical method in Mathematica for $x = 0.1, 5, 20, 40,100$ respectively and $\alpha = 1, \beta = 1, a = 0.001, b = 0.001, n = 5, h_{Oq-HAM} = -3.055, h_{HATM} = 0.64.$ Dash - dotted line (Oq-HAM), dotted line (HATM), dash line (VIM), and solid line (ADM)
Figure 11.  The plot of Oq-HAM, HATM, VIM and ADM for (4)-(5) with $\alpha = 0.4,\beta = 0.7, a = 0.4, b = 0.2, n = 5, h_{Oq-HAM} = -3.00, h_{HATM} = -0.64 .$ Dash - dotted line (Oq-HAM), dotted line (HATM), dash line (VIM), and solid line (ADM)
Figure 12.  The plot of Oq-HAM, HATM, VIM and ADM for (4)-(5) with $\alpha = 0.7,\beta = 0.9, a = 0.4, b = 0.2, n = 5, h_{Oq-HAM} = -3.00, h_{HATM} = -0.64 .$ Dash - dotted line (Oq-HAM), dotted line (HATM), dash line (VIM), and solid line (ADM)
Figure 13.  The plot of Oq-HAM, HATM, VIM and ADM for (4)-(5) with $\alpha = 0.99,\beta = 0.99, a = 0.4, b = 0.2, n = 5, h_{q-HAM} = -3.00, h_{HATM} = -0.64 .$ Dash - dotted line (Oq-HAM), dotted line (HATM), dash line (VIM), and solid line (ADM)
Figure 14.  The surface of Oq-HAM for (4)-(5) with $\alpha = 0.5, 0.8, 1.00,\beta = 0.75,0.90, 1.00$ and $ a = 0.4, b = 0.2, n = 5, h_{Oq-HAM} = -3.00$
Figure 15.  The surface of HATM for (4)-(5) with $\alpha = 0.5, 0.8, 1.00,\beta = 0.75,0.90, 1.00$ and $ a = 0.4, b = 0.2, h_{HATM} = -0.64 $
Figure 16.  The surface of VIM for (4)-(5) with $\alpha = 0.5, 0.8, 1.00,\beta = 0.75,0.90, 1.00$ and $ a = 0.4, b = 0.2 $
Figure 17.  The surface of ADM for (4)-(5) with $\alpha = 0.5, 0.8, 1.00,\beta = 0.75,0.90, 1.00$ and $ a = 0.4, b = 0.2 $
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