Analytical approach to solution fractional partial differential equation by optimal q-homotopy analysis method
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 2, pp. 171-183.

Voir la notice de l'article provenant de la source Math-Net.Ru

The optimal q-homotopy analysis method is employed in order to solve partial differential equations (PDEs) featuring a time-fractional derivative. Then, in order to illustrate the simplicity and ability of the suggested approach, some specific and clear examples are given. All numerical calculations in this manuscript have been carried out with Mathematica package.
@article{SJVM_2018_21_2_a3,
     author = {R. Darzi and B. Agheli},
     title = {Analytical approach to solution fractional partial differential equation by optimal q-homotopy analysis method},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {171--183},
     publisher = {mathdoc},
     volume = {21},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2018_21_2_a3/}
}
TY  - JOUR
AU  - R. Darzi
AU  - B. Agheli
TI  - Analytical approach to solution fractional partial differential equation by optimal q-homotopy analysis method
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2018
SP  - 171
EP  - 183
VL  - 21
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2018_21_2_a3/
LA  - ru
ID  - SJVM_2018_21_2_a3
ER  - 
%0 Journal Article
%A R. Darzi
%A B. Agheli
%T Analytical approach to solution fractional partial differential equation by optimal q-homotopy analysis method
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2018
%P 171-183
%V 21
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2018_21_2_a3/
%G ru
%F SJVM_2018_21_2_a3
R. Darzi; B. Agheli. Analytical approach to solution fractional partial differential equation by optimal q-homotopy analysis method. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 2, pp. 171-183. http://geodesic.mathdoc.fr/item/SJVM_2018_21_2_a3/

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

[2] El-Tawil M. A., Huseen S. N., “The q-homotopy analysis method (q-HAM)”, Int. J. Appl. Math. and Mechanics, 8:15 (2012), 51–75

[3] Huseen S. N., Grace S. R., El-Tawil M. A., “The optimal q-homotopy analysis method (Oq-HAM)”, Int. J. Comp. Technology, 11:8 (2012), 2859–2866 | DOI

[4] Kilbas A. A., Srivastava H. M., Trujillo J. J., Theory and Applications of Fractional Differential Equations, Elsevier BV, Netherlands, 2006 | MR | Zbl

[5] Miller K. S., Ross B., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993 | MR | Zbl

[6] Momani S., Shawagfeh N., “Decomposition method for solving fractional Riccati differential equations”, Appl. Math. Comput., 182 (2006), 1083–1092 | MR | Zbl

[7] Wang Q., “Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method”, Appl. Math. Comput., 182 (2006), 1048–1055 | MR | Zbl

[8] Inc M., “The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method”, J. Math. Anal. Appl., 345 (2008), 476–484 | DOI | MR | Zbl

[9] Yang X. J., Baleanu D., Khan Y., Mohyud-Din S. T., “Local fractional variational iteration method for diffusion and wave equations on Cantor sets”, Rom. J. Phys., 59 (2014), 36–48 | MR

[10] Momani S., Odibat Z., “Homotopy perturbation method for nonlinear partial differential equations of fractional order”, Phys. Lett. A, 365 (2007), 345–350 | DOI | MR | Zbl

[11] Odibat Z., Momani S., “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order”, Chaos Soliton. Fract., 36 (2008), 167–174 | DOI | MR | Zbl

[12] Hosseinnia S., Ranjbar A., Momani S., “Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part”, Comput. Math. Appl., 56 (2008), 3138–3149 | DOI | MR | Zbl

[13] Hashim I., Abdulaziz O., Momani S., “Homotopy analysis method for fractional IVPs”, Commun. Nonlinear Sci., 14 (2009), 674–684 | DOI | MR | Zbl

[14] Zurigat M., Momani S., Alawneh A., “Analytical approximate solutions of systems of fractional algebraic-differential equations by homotopy analysis method”, Comput. Math. Appl., 59 (2010), 1227–1235 | DOI | MR | Zbl

[15] Kumar P., Agrawal O. P., “An approximate method for numerical solution of fractional differential equations”, Sign. Proc., 86 (2006), 2602–2610 | DOI | Zbl

[16] Yang X. J., Baleanu D., Zhong W. P., “Approximation solutions for diffusion equation on Cantor space-time”, Proc. Romanian Academy. Series A, 14 (2013), 127–133 | MR

[17] Yuste S. B., “Weighted average finite difference methods for fractional diffusion equations”, J. Comput. Phys., 216 (2006), 264–274 | DOI | MR | Zbl

[18] Yabushita K., Yamashita M., Tsuboi K., “An analytic solution of projectile motion with the quadratic resistance law using the homotopy analysis method”, J. Phys. A: Math. and Theor., 40:29 (2007), 8403–8416 | DOI | MR | Zbl

[19] El-Tawil M. A., Huseen S. N., “On convergence of the q-homotopy analysis method”, Int. J. Contemporary Mathematical Sciences, 8:10 (2013), 481–497 | DOI | MR

[20] Neamaty A., Agheli B., Darzi R., “Variational iteration method and he's polynomials for time-fractional partial differential equations”, Progr. Fract. Differ. Appl., 1:1 (2015), 47–55 | MR