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Couette flows of a viscous fluid with slip effects and non-integer order derivative without singular kernel
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).
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[52] |
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.
|
[53] |
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. |
[54] |
E. Shivanian, H. H. Alsulami, M. S. Alhuthali and S. Abbasbandy,
Predictor homotopy analysis method (PHAM) for nano boundary layer flows with nonlinear Navier boundary condition: Existence of four solutions, Filomat, 28 (2014), 1687-1697.
doi: 10.2298/FIL1408687S. |
[55] |
E. Shivanian and S. Abbasbandy,
Predictor homotopy analysis method: Two points second order boundary value problems, Nonlinear Analysis: Real World Applications, 15 (2014), 89-99.
doi: 10.1016/j.nonrwa.2013.06.003. |
[56] |
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. |
[57] |
J. Singh, D. 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. Google Scholar |
[58] |
M. Singh, M. Naseem, A. 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. |
[59] |
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, Applied Thermal Engineering, 103 (2016), 537-542. Google Scholar |
[60] |
Q. Sun,
Solving the klein-gordon equation by means of the homotopy analysis method, Appl. Math. and Comput., 169 (2005), 355-365.
doi: 10.1016/j.amc.2004.09.056. |
[61] |
J. Vahidi,
The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Computer Sci., 16 (2016), 88-102.
doi: 10.22436/jmcs.016.01.10. |
[62] |
H. Vosoughi, E. Shivanian and S. Abbasbandy,
Unique and multiple pham series solutions of a class of nonlinear reactive transport model, Numerical Algorithms, 61 (2012), 515-524.
doi: 10.1007/s11075-012-9548-z. |
[63] |
H. Vosughi, E. Shivanian and S. Abbasbandy,
A new analytical technique to solve volterra's integral equations, Mathematical methods in the applied sciences, 34 (2011), 1243-1253.
doi: 10.1002/mma.1436. |
[64] |
Y. Wu, C. Wang and S. J. Liao,
Solving the one-loop soliton solution of the vakhnenko equation by means of the homotopy analysis method, Chaos, Solitons and Fractals, 23 (2005), 1733-1740.
doi: 10.1016/S0960-0779(04)00437-0. |
[65] |
L. Xu, J. H. He and Y. Liu,
Electrospun nanoporous spheres with chinese drug, Int. J. Nonlinear Sci. Numer. Simul., 8 (2007), 199-202.
doi: 10.1515/IJNSNS.2007.8.2.199. |
[66] |
K. Yabushita, M. Yamashita 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.
doi: 10.1088/1751-8113/40/29/015. |
[67] |
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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. 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. 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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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. |
[52] |
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.
|
[53] |
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. |
[54] |
E. Shivanian, H. H. Alsulami, M. S. Alhuthali and S. Abbasbandy,
Predictor homotopy analysis method (PHAM) for nano boundary layer flows with nonlinear Navier boundary condition: Existence of four solutions, Filomat, 28 (2014), 1687-1697.
doi: 10.2298/FIL1408687S. |
[55] |
E. Shivanian and S. Abbasbandy,
Predictor homotopy analysis method: Two points second order boundary value problems, Nonlinear Analysis: Real World Applications, 15 (2014), 89-99.
doi: 10.1016/j.nonrwa.2013.06.003. |
[56] |
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. |
[57] |
J. Singh, D. 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. Google Scholar |
[58] |
M. Singh, M. Naseem, A. 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. |
[59] |
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, Applied Thermal Engineering, 103 (2016), 537-542. Google Scholar |
[60] |
Q. Sun,
Solving the klein-gordon equation by means of the homotopy analysis method, Appl. Math. and Comput., 169 (2005), 355-365.
doi: 10.1016/j.amc.2004.09.056. |
[61] |
J. Vahidi,
The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Computer Sci., 16 (2016), 88-102.
doi: 10.22436/jmcs.016.01.10. |
[62] |
H. Vosoughi, E. Shivanian and S. Abbasbandy,
Unique and multiple pham series solutions of a class of nonlinear reactive transport model, Numerical Algorithms, 61 (2012), 515-524.
doi: 10.1007/s11075-012-9548-z. |
[63] |
H. Vosughi, E. Shivanian and S. Abbasbandy,
A new analytical technique to solve volterra's integral equations, Mathematical methods in the applied sciences, 34 (2011), 1243-1253.
doi: 10.1002/mma.1436. |
[64] |
Y. Wu, C. Wang and S. J. Liao,
Solving the one-loop soliton solution of the vakhnenko equation by means of the homotopy analysis method, Chaos, Solitons and Fractals, 23 (2005), 1733-1740.
doi: 10.1016/S0960-0779(04)00437-0. |
[65] |
L. Xu, J. H. He and Y. Liu,
Electrospun nanoporous spheres with chinese drug, Int. J. Nonlinear Sci. Numer. Simul., 8 (2007), 199-202.
doi: 10.1515/IJNSNS.2007.8.2.199. |
[66] |
K. Yabushita, M. Yamashita 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.
doi: 10.1088/1751-8113/40/29/015. |
[67] |
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.
|

















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