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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
References:
[1]

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.

[2]

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.

[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.

[4]

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.

[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.

[9]

G. Adomian, The kadomtsev-petviashvili equation, Applied Mathematics and Computation, 76 (1996), 95-97. doi: 10.1016/0096-3003(95)00186-7.

[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.

[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.

[12]

Y. Cherruault, Convergence of adomian's method, Kybernetes, 18 (1989), 31-38. doi: 10.1108/eb005812.

[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.

[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.

[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.

[16]

M. A. GondalA. S. ArifeM. 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.

[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.

[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.

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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.

[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.

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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.

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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.

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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.

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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. 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.

[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. 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.

[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.

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

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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.

[43]

M. S. MohamedF. Al-Malki and M. Al-humyani, Homotopy analysis transform method for time-space fractional gas dynamics equation, Gen. Math. Notes, 24 (2014), 1-16.

[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.

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I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

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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.

[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.

[48]

K. M. SaadD. 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.

[49]

K. M. Saad, A. Atangana and D. Baleanu, New Fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[50]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A: Statistical Mechanics and its Applications, 476 (2017), 1-14. doi: 10.1016/j.physa.2017.02.016.

[51]

K. M. Saad, E. H. AL-Shareef, M. S. Mohamed and X. J. Yang, Optimal q-homotopy analysis method for time-space fractional gas dynamics equation, The European Physical Journal Plus, 132 (2017), 23. doi: 10.1140/epjp/i2017-11303-6.

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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.

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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, The European Physical Journal Plus, 128 (2013), 133. doi: 10.1140/epjp/i2013-13133-x.

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J. Singh, D. Kumar and Sushila, Homotopy perturbation algorithm using laplace transform for gas dynamics equation, Journal of the Applied Mathematics, Statistics and Informatics, 8 (2012), 55-61. doi: 10.2478/v10294-012-0006-2.

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J. SinghD. Kumar and S. Rathore, Application of homotopy perturbation transform method for solving linear and nonlinear klein-gordon equations, Journal of Information and Computing Science, 7 (2012), 131-139.

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M. SinghM. NaseemA. Kumar and S. Kumar, Homotopy analysis transform algorithm to solve time-fractional foam drainage equation, Nonlinear Engineering, 5 (2016), 161-166. doi: 10.1515/nleng-2016-0014.

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L. A. SoltaniaE. Shivanianb and R. Ezzatia, Convection-radiation heat transfer in solar heat exchangers filled with a porous medium: Exact and shooting homotopy analysis solution, Applied Thermal Engineering, 103 (2016), 537-542.

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M. Zurigat, Solving fractional oscillators using laplace homotopy analysis method, Annals of the University of Craiova, Mathematics and Computer Science Series, 38 (2011), 1-11.

show all references

References:
[1]

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.

[2]

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.

[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.

[4]

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.

[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.

[9]

G. Adomian, The kadomtsev-petviashvili equation, Applied Mathematics and Computation, 76 (1996), 95-97. doi: 10.1016/0096-3003(95)00186-7.

[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.

[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.

[12]

Y. Cherruault, Convergence of adomian's method, Kybernetes, 18 (1989), 31-38. doi: 10.1108/eb005812.

[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.

[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.

[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.

[16]

M. A. GondalA. S. ArifeM. 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.

[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.

[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.

[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.

[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.

[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. 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.

[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. 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.

[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.

[41]

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

[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.

[43]

M. S. MohamedF. Al-Malki and M. Al-humyani, Homotopy analysis transform method for time-space fractional gas dynamics equation, Gen. Math. Notes, 24 (2014), 1-16.

[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.

[45]

I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.

[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.

[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.

[48]

K. M. SaadD. 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.

[49]

K. M. Saad, A. Atangana and D. Baleanu, New Fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[50]

K. M. Saad and J. F. Gómez-Aguilar, Analysis of reaction-diffusion system via a new fractional derivative with non-singular kernel, Physica A: Statistical Mechanics and its Applications, 476 (2017), 1-14. doi: 10.1016/j.physa.2017.02.016.

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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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