Abstract
Motivated by the simplicity, natural and efficient nature of the new fractional derivative introduced by R Khalil et al in J. Comput. Appl. Math. 264, 65 (2014), analytical solution of space-time fractional Fornberg–Whitham equation is obtained in series form using the relatively new method called q-homotopy analysis method (q-HAM). The new fractional derivative makes it possible to introduce fractional order in space to the Fornberg–Whitham equation and be able to obtain its solution. This work displays the elegant nature of the application of q-HAM to solve strongly nonlinear fractional differential equations. The presence of the auxiliary parameter h helps in an effective way to obtain better approximation comparable to exact solutions. The fraction-factor in this method gives it an edge over other existing analytical methods for nonlinear differential equations. Comparisons are made on the existence of exact solutions to these models. The analysis shows that our analytical solutions converge very rapidly to the exact solutions.
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References
M Caputo, Geothermics 28, 113 (1999)
E Cumberbatch and A Fitt, Mathematical modeling: Case studies from industries (Cambridge University Press, UK, 2001)
G I Marchuk, Mathematical models in environmental problems (North-Holland, Elsevier Science Publishers, 1986)
E S Oran and J P Boris, Numerical simulation of reactive flow, 2nd edn (Cambridge University Press, UK, 2001)
K S Miller and B Ross, An introduction to the fractional calculus and fractional differential equations (John Wiley and Sons Inc., New York, 2003)
K A Gepreel and A A Al-Thobaiti, Indian J. Phys. 88, 293 (2014)
K M Furati, O S Iyiola and M Kirane, Appl. Math. Comput. 249, 24 (2014)
O S Iyiola, British J. Math. Comp. Sci. 4(10) (2014)
M Eslami, B F Vajargah, M Mirzazadeh and A Biswas, Indian J. Phys. 88(2), 177 (2014)
M A El-Tawil and S N Huseen, Int. J. Appl. Math. Mech. 8(15), 51 (2012)
O S Iyola and F D Zaman, AIP Adv. 4, 107121 (2014)
O S Iyiola, Adv. Math. Sci. J. 2(2), 71 (2013)
O S Iyiola, M E Soh and C D Enyi, Math. Eng. Sci. Aerospace. 4(4), 105 (2013)
O S Iyiola and O G Olayinka, Ain Shams Engng J. 5(3), 999 (2014)
R Khalil, M Al Horani, A Yousef and M Sababhehb, J. Comput. Appl. Math. 264, 65 (2014)
M Merdan, A Gokdogan, A Yıldırım and S T Mohyud-Din, Abstract Appl. Anal. Vol. (2012), Article ID 965367, 8 pages, DOI: 10.1155/2012/965367
Y Mahmoudi and M Kazemian, J. Basic. Appl. Sci. Res. 2(3), 2985 (2012)
S-J Liao, Int. J. Non-linear Mech. 30(3), 371 (1995)
Acknowledgements
The authors are grateful to the financial support extended by the King Fahd University of Petroleum and Minerals (KFUPM) and acknowledge the contributions of the anonymous referees which greatly helped in improving the final version of this paper.
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IYIOLA, O.S., OJO, G.O. On the analytical solution of Fornberg–Whitham equation with the new fractional derivative. Pramana - J Phys 85, 567–575 (2015). https://doi.org/10.1007/s12043-014-0915-2
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DOI: https://doi.org/10.1007/s12043-014-0915-2