Comparative study of a cubic autocatalytic reaction via different analysis methods

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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

Khaled Mohammed Saad ^a,b,, and Eman Hussain Faissal AL-Sharif ^a,

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).

Keywords: Optimal q-Homotopy Analysis Method, homotopy analysis transform method, varitional iteration method and Adomian decomposition method, cubic an isothermal autocatalytic.

Mathematics Subject Classification: Primary: 35-XX, 35Kxx; Secondary: 35R11.

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

References:

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[2]	S. Abbasbandy, E. 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
[3]	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
[4]	S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int Commun Heat Mass Transf, 34 (2007), 380-387. Google Scholar
[5]	S. Abbasbandy, Soliton solutions for the 5th-order kdv equation with the homotopy analysis method, Nonlinear Dyn, 51 (2008), 83-87. doi: 10.1007/s11071-006-9193-y. Google Scholar
[6]	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
[7]	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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show all references

References:

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[2]	S. Abbasbandy, E. 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
[3]	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
[4]	S. Abbasbandy, Homotopy analysis method for heat radiation equations, Int Commun Heat Mass Transf, 34 (2007), 380-387. Google Scholar
[5]	S. Abbasbandy, Soliton solutions for the 5th-order kdv equation with the homotopy analysis method, Nonlinear Dyn, 51 (2008), 83-87. doi: 10.1007/s11071-006-9193-y. Google Scholar
[6]	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
[7]	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
[8]	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]	G. Adomian, The kadomtsev-petviashvili equation, Applied Mathematics and Computation, 76 (1996), 95-97. doi: 10.1016/0096-3003(95)00186-7. Google Scholar
[10]	A. S. Arife, S. 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
[11]	M. Caputo, Linear models of dissipation whose q is almost frequency independent, Geophysical Journal, 13 (1967), 529-539. doi: 10.1111/j.1365-246X.1967.tb02303.x. Google Scholar
[12]	Y. Cherruault, Convergence of adomian's method, Kybernetes, 18 (1989), 31-38. doi: 10.1108/eb005812. Google Scholar
[13]	Y. Cherruault and G. Adomians, Decomposition methods: A new proof of convergence, Math. Comput. Modelling, 18 (1993), 103-106. doi: 10.1016/0895-7177(93)90233-O. Google Scholar
[14]	V. F. M. Delgado, J. F. Gómez-Aguilar, H. Y. Martez and D. Baleanu, Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Adv. Differ. Equ. , 2016 (2016), Paper No. 164, 17 pp. doi: 10.1186/s13662-016-0891-6. Google Scholar
[15]	M. A. El-Tawil and S. N. Huseen, On convergence of the q-homotopy analysis method, Int. J. of Contemp. Math. Scies., 8 (2013), 481-497. doi: 10.12988/ijcms.2013.13048. Google Scholar
[16]	M. A. Gondal, A. S. Arife, M. Khan and I. Hussain, An efficient numerical method for solving linear and nonlinear partial differential equations by combining homotopy analysis and transform method, World Applied Sciences Journal, 14 (2011), 1786-1791. Google Scholar
[17]	C. Gong, W. Bao, G. Tang, Y. Jiang and J. Liu, A domain decomposition method for time fractional reaction-diffusion equation, The Scientific World Journal, 2014 (2014), Article ID 681707, 5 pages. doi: 10.1155/2014/681707. Google Scholar
[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. Hayat, M. 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. Hayat, S. B. Khan, M. 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
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[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. Kumar, J. Singh, D. 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. Madani, M. Fathizadeh, Y. 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. Mohamed, K. A. Gepreel, M. R. Alharthi and R. A. Alotabi, Homotopy analysis transform method for integro-differential equations, General Mathematics Notes, 32 (2016), 32-48. Google Scholar
[43]	M. S. Mohamed, F. Al-Malki and M. Al-humyani, Homotopy analysis transform method for time-space fractional gas dynamics equation, Gen. Math. Notes, 24 (2014), 1-16. Google Scholar
[44]	Z. Odibat, A new modification of the homotopy perturbation method for linear and nonlinear operators, Appl. Math. Comput., 189 (2007), 746-753. doi: 10.1016/j.amc.2006.11.188. Google Scholar
[45]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. Google Scholar
[46]	A. Répaci, Nonlinear dynamical systems: On the accuracy of adomian's decomposition method, Appl. Mth. Lett., 3 (1990), 35-39. doi: 10.1016/0893-9659(90)90042-A. Google Scholar
[47]	K. M. Saad, Comparing the Caputo, Caputo-Fabrizio and Atangana-Baleanu derivative with fractional order: Fractional Cubic Isothermal Auto-catalytic Chemical System, Eur. Phys. J. Plus 133 (2018), p49. doi: 10.1140/epjp/i2018-11947-6. Google Scholar
[48]	K. M. Saad, D. Baleanu and A. Atangana, New fractional derivatives applied to the Korteweg-de Vries and Korteweg-de Vries-Burger's equations, A. Comp. Appl. Math., (2018), 1-14. doi: 10.1007/s40314-018-0627-1. Google Scholar
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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 $\alpha = 1, \beta = 1, a = 0.001, b = 0.001, h = -3.055, n = 5$

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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$

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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 .$

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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 $

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Figure 7. The absolute error between the 3-terms of Oq-HAM solutions and numerical method using Mathematica for (4)-(5) with $\alpha = 0.9, \beta = 0.99, a = 0.001, b = 0.001, h = -1.9, n = 5$

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Figure 8. The absolute error between the 3-terms of HATM solutions and numerical method using Mathematica for (4)-(5) with $\alpha = 0.9,\beta = 0.99, a = 0.001, b = 0.001, h = -0.64$

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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 $

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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$

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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)

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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)

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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)

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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)

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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)

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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$

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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 $

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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 $

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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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