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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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 |
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 and Continuous Dynamical Systems - S,
2019, 12
(3)
: 665-684.
doi: 10.3934/dcdss.2019042
![]()
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S. Abbasbandy, E. Shivaniana and K. Vajravelu,
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S. Abbasbandy,
Homotopy analysis method for heat radiation equations, Int Commun Heat Mass Transf, 34 (2007), 380-387.
|
| [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. |
| [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. |
| [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. |
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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. |
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G. Adomian,
The kadomtsev-petviashvili equation, Applied Mathematics and Computation, 76 (1996), 95-97.
doi: 10.1016/0096-3003(95)00186-7. |
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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. |
| [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. |
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Y. Cherruault,
Convergence of adomian's method, Kybernetes, 18 (1989), 31-38.
doi: 10.1108/eb005812. |
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Decomposition methods: A new proof of convergence, Math. Comput. Modelling, 18 (1993), 103-106.
doi: 10.1016/0895-7177(93)90233-O. |
| [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.
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M. A. El-Tawil and S. N. Huseen,
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M. A. Gondal, A. S. Arife, M. Khan and I. Hussain,
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|
| [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. |
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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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [24] |
J. H. He,
Some asymptotic methods for strongly nonlinear equations, Int. J. Modern Phys. B, 20 (2006), 1141-1199.
doi: 10.1142/S0217979206033796. |
| [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. |
| [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. |
| [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. |
| [28] |
A. C. King, J. Billingham and S. R. Otto.
Differential Equations: Linear, Nonlinear, Ordinary, Partial, Cambridge University Press, 2003.
doi: 10.1017/CBO9780511755293. |
| [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. |
| [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. |
| [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. |
| [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. |
| [33] |
S. J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, PhD thesis, Shanghai Jiao Tong University, 1992. |
| [34] |
S. J. Liao, Beyond perturbation: introduction to the homotopy analysis method,
Boca Raton: Chapman and Hall/CRC Press, 2003. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [40] |
T. Mavoungou and Y. Cherruault,
Convergence of adomian's method and applications to non-linear partial differential equation, Kybernetes, 21 (1992), 13-25.
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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$
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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