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industrial processes, and are also very important in many technological processes and applications [ 9 ]. The well-known foam-drainage equation is given by [ 10 ] (1) u t − ( u 1 / 2 u x ) x + 2 u u x = 0 , $${u_t} - {({u^{1/2}}{u_x})_x} + 2u{u_x} = 0{\rm{,}}$$ which has been studied by tanh-coth method, Exp-function method, and Adomian method [ 11 ], [ 12 ], [ 13 ]. Symmetry reductions and conservation laws were analysed in [ 14 ]. This article concentrates on the symmetry and conservation law of the extended form of foam-drainage equation, namely, the generalised foam-drainage

equation governs gas dynamics. A great potential of a research work has been invested to Burgers equation. Several exact solutions have been obtained by using distinct approaches, such as tanh method, decomposition method, variation iteration method and so on [ 1 , 3 , 24 ]. (II.) The foam drainage equation has the following form [ 8 ] u t + ( u 2 − u 2 u x ) x = 0. $${u_t} + {({u^2} - {{\sqrt u } \over 2}{u_x})_x} = 0.$$ (1.3) Foaming arises in many distillation processes. The drainage of fluid foams involves the reciprocation viscous forces, gravity and surface

transform, Appl. Math. Modell., 38 (2014) 3154– 3163. Kumar S. A new analytical modelling for telegraph equation via Laplace transform Appl. Math. Modell 38 2014 3154 3163 [17] G. Verbist, D. Weuire, A.M. Kraynik, The foam drainage equation, J. Phys. Condens. Matter 8 (1996) 3715–3731. Verbist G. Weuire D. Kraynik A.M. The foam drainage equation J. Phys. Condens. Matter 8 1996 3715 3731 [18] D. Weaire, S. Hutzler, S. Cox, M.D. Alonso, D. Drenckhan, The fluid dynamics of foams, J. Phys. Condens. Matter 15(2003) S65– S73. Weaire D. Hutzler S. Cox S. Alonso M.D. Drenckhan D

Abstract This paper couples Laplace transformation with He's polynomials to solve foam drainage equation. The proposed scheme is fully capable of handling such problems. Numerical results clearly reveal the reliability of this elegant coupling. Keywords: He's polynomials, partial differential equations, foam drainage equations. 1. Introduction In the recent past, He [2-8] developed and formulated homotopy perturbation method (HPM) by merging the standard homotopy and perturbation. The HPM is fully compatible with the versatile nature of the physical problems and has been


Kumar Homotopy analysis transform algorithm to solve time-fractional foam drainage equation  161 Ch. RamReddy, O. Surender, Ch. Venkata Rao Effects of Soret, Hall and Ion-slip on mixed convection in an electrically conducting Casson fluid in a vertical channel  167 Karthikeyan Rajagopal, Anitha Karthikeyan Chaos suppression of Fractional order Willamowski– Rössler Chemical system and its synchronization using Sliding Mode Control  177 Ajit K. Singh, Vijay K. Yadav, S. Das Comparative study of synchronization methods of fractional order chaotic systems  185 Ch

., 85, (2006), 1459-1470. [4] M. Benchohra, J. Henderson, S.K. Ntouyas, A. Ouahab, Existence results for fractional order functional differential equations with infinite delay. J. Math. Anal. Appl., 338(2), (2008), 1340-1350. [5] M. Benchohra, S. Hamania, S.K. Ntouyas, Boundary value problems for differential equations with fractional order and nonlocal conditions. Non­linear Analysis, 71, (2009), 2391-2396. [6] Z. Dahmani, M.M. Mesmoudi, R. Bebbouchi, The foam drainage equation with time and space fractional derivatives solved by the ADM method. E. J. Qualitative


Sajid, Khurram Javid and Raheel Ahmed Peristaltic Flow of Rabinowitsch Fluid in a Curved Channel: Mathematical Analysis Revisited 245 Igor Pažanin and Pradeep G. Siddheshwar Analysis of the Laminar Newtonian Fluid Flow Through a Thin Fracture Modelled as a Fluid-Saturated Sparsely Packed Porous Medium 253 Zhi-Yong Zhang and Kai-Hua Ma Lie Symmetries and Conservation Laws of the Generalised Foam-Drainage Equation 261 Xiu-Bin Wang, Shou-Fu Tian, Chun-Yan Qin and Tian-Tian Zhang Lie Symmetry Analysis, Analytical Solutions, and Conservation Laws of the

.T. Darvishi, F. Khani, He's variational iteration method for some nonlinear partial differential equations, Computers and Mathematics with Applications, (to appear). [7]F. Khani, S. Hamedi-Nezhad, M.T. Darvishi, Sang-Wan Ryu, New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method, Nonlinear Analysis: Real World Applications (in press), doi: 10.1016/j .nonrwa.2008.02.030. [8]B.-C. Shin, M.T. Darvishi, A. Barati, Some exact and new solutions of the Nizhnik-Novikov-Vesselov equation using the Exp-function method, Computers and

homotopy analysis method (q-HAM) for solving foam drainage equation of time fractional type, Math. Eng. Sci. Aerospace, 4(4), 105–116. Iyiola O. Soh M.E. Enyi C.D. 2013 Generalised homotopy analysis method (q-HAM) for solving foam drainage equation of time fractional type Math. Eng. Sci. Aerospace 4 4 105 116 [12] Mohamed, M. S. and Hamed, Y. S. (2016), Solving the convection-diffusion equation by means of the optimal qhomotopy analysis method (Oq-HAM), Results Phys. 6, 20–25. 10.1016/j.rinp.2015.12.008 Mohamed M.S. Hamed Y.S. 2016 Solving the convection

flow equation, J. Adv. Res. Sei. Comput. 2(2010)35-43. 34. M. Khan, M. A. Gondal, A new analytical solution of foam drainage equation by Laplace decomposition method, J. Adv. Res. Differ. Eqs. 2(2010) 53-64. 35. M. Khan, M. A. Gondal, Application of Laplace decomposition method to solve sys- tems of nonlinear coupled partial differential equations, J. Adv. Res. Sei. Comput. 2(2010) 1-14. 36. Y. Khan, An Effective Modification of the Laplace Decomposition Method for Non- linear Equations. Int. J. Nonlinear. Sei. Numer. Simul. 10(2009) 1373-1376. 37. M