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http://dx.doi.org/10.12941/jksiam.2021.25.132

CONFORMABLE FRACTIONAL SENSE OF FOAM DRAINAGE EQUATION AND CONSTRUCTION OF ITS SOLUTIONS  

DARVISHI, MOHAMMAD T. (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, RAZI UNIVERSITY)
NAJAFI, MOHAMMAD (DEPARTMENT OF MATHEMATICS, FACULTY OF SCIENCE, RAZI UNIVERSITY)
SHIN, BYEONG-CHUN (DEPARTMENT OF MATHEMATICS, CHONNAM NATIONAL UNIVERSITY)
Publication Information
Journal of the Korean Society for Industrial and Applied Mathematics / v.25, no.3, 2021 , pp. 132-148 More about this Journal
Abstract
The modified F-expansion method is used to construct analytical solutions of the foam drainage equation with time- and space-fractional derivatives. The conformable derivatives are considered as spacial and temporal ones. As a result, some analytical exact solutions including kink, bright-dark soliton, periodic and rational solutions are obtained.
Keywords
Fractional foam drainage equation; Soliton solution; The modified F-expansion scheme;
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1 G. Verbist, D. Weaire and A.M. Kraynik (1996). The foam drainage equation, J. Phys.: Condens. Matter 8, 3715-3731, http://iopscience.iop.org/0953-8984/8/21/002.   DOI
2 F. Khani, S. Hamedi-Nezhad, M.T. Darvishi and S.-W. Ryu (2009). New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method, Nonlinear Anal. RWA, 10, 1904-1911, doi:10.1016/j.nonrwa.2008.02.030.   DOI
3 M.T. Darvishi and F. Khani (2009). A series solution of the foam drainage equation, Comput. Math. Appl., 58, 360-368, doi:10.1016/j.camwa.2009.04.007.   DOI
4 S. Hilgenfeldt, S.A. Koehler and H.A. Stone (2001). Dynamics of coarsening foams: Accelerated and self-limiting drainage, Phys. Rev. Lett., 86, 4704-4707, https://doi.org/10.1103/PhysRevLett.86.4704.   DOI
5 M.A. Helal and M.S. Mehanna (2007). The tanh method and Adomian decomposition method for solving the foam drainage equation, Appl. Math. Comput., 190(1) 599-609, https://doi.org/10.1016/j.amc.2007.01.055.   DOI
6 O.S. Iyiola, M.E. Soh and C.D. Enyi (2013). Generalised homotopy analysis method (q-HAM) for solving foam drainage equation of time fractional type, Math. Eng. Sci. Aerospace, 4(4), 429-440.
7 L. Braun, M. Kuhnhammer and R.-V. Klitzing (2020). Stability of aqueous foam films and foams containing polymers: Discrepancies between different length scales, Current Opinion in Colloid & Interface Science, 50, 101379, https://doi.org/10.1016/j.cocis.2020.08.004.   DOI
8 S.T.R. Rizvi, K. Ali, S. Bashir, M. Younis, R. Ashraf and M.O. Ahmad (2017). Exact soliton of (2+1)-dimensional fractional Schrodinger equation, Superlattices and Microstructures, 107, 234-239, https://doi.org/10.1016/j.spmi.2017.04.029.   DOI
9 S.A. El-Wakil and E.M. Abulwafa (2015). Formulation and solution of space-time fractional Boussinesq equation, Nonlinear Dyn., 80, 167-175, https://doi.org/10.1007/s11071-014-1858-3.   DOI
10 M. Singh, M. Naseem, A. Kumar and S. Kumar (2016). Homotopy analysis transform algorithm to solve time-fractional foam drainage equation, Nonlinear Eng., 5, 161-166, https://doi.org/10.1515/nleng-2016-0014.   DOI
11 H. Hosseini Fadravi, H. Saberi-Nik and R. Buzhabadi (2011). Homotopy analysis method for solving foam drainage equation with space- and time-fractional derivatives, Int. J. Diff. Equ., 2011, 237045, https://doi.org/10.1155/2011/237045.   DOI
12 M. Alquran (2014). Analytical solutions of fractional foam drainage equation by residual power series method, Math. Sci., 8(4), 153-160.   DOI
13 A. Zulfiqar, J. Ahmad (2021). New optical solutions of conformable fractional perturbed Gerdjikov-Ivanov equation in mathematical nonlinear optics, Results in Physics, 21 Article:103825.
14 Y. Salehi and M.T. Darvishi (2016). An investigation of fractional Riccati differential equation, Optik, 127, 11505-11521, http://dx.doi.org/10.1016/j.ijleo.2016.08.008.   DOI
15 P.M. Kruglyakov, S.I. Karakashev, A.V. Nguyen and N.G. Vilkova (2008) Foam drainage, Current Opinion in Colloid & Interface Science, 13(3), 163-170, https://doi.org/10.1016/j.cocis.2007.11.003.   DOI
16 A. Trybala, N. Koursari, P. Johnson, O. Arjmandi-Tash and V. Starov (2019). Interaction of liquid foams with porous substrates, Current Opinion in Colloid & Interface Science, 39, 212-219, https://doi.org/10.1016/j.cocis.2019.01.011.   DOI
17 Z. Odibat and S. Momani (2008). Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 36(1), 167-174, https://doi.org/10.1016/j.chaos.2006.06.041.   DOI
18 A.H. Bhrawy, J.F. Alzaidy, M.A. Abdelkawy and A. Biswas (2016). Jacobi spectral collocation approximation for multi-dimensional time-fractional Schrodinger equations, Nonlinear Dyn., 84, 1553-1567, https://doi.org/10.1007/s11071-015-2588-x.   DOI
19 Z. Dahmani, M.M. Mesmoudi and R. Bebbouchi (2008). The foam drainage equation with time- and spacefractional derivatives solved by the Adomian method, Electron. J. Qual. Theor. Diff. Equ., 30, 1-10, http://www.math.u-szeged.hu/ejqtde.
20 Z. Dahmani and A. Anber (2010). The variational iteration method for solving the fractional foam drainage equation, Int. J. Nonlinear Sci., 10(1), 39-45, IJNS.2010.08.15/384.
21 A. Bouhassoun, M.H. Cherif and M. Zellal (2013). Variational homotopy perturbation method for the approximate solution of the foam drainage equation with time and space fractional derivatives, Malaya Journal of Matematik, 4, 163-170
22 L. Wang, S.F. Tian, Z.T. Zhao and X.Q. Song (2016). Lie symmetry analysis and conservation laws of a generalized time fractional foam drainage equation, Commun. Theor. Phys., 66, 35-40, https://doi.org/10.1088/0253-6102/66/1/035.   DOI
23 S. Kumar, A. Kumar and D. Baleanu (2016). Two analytical methods for time-fractional nonlinear coupled Boussinesq-Burgers' equations arise in propagation of shallow water waves, Nonlinear Dyn., 85, 699-715, https://doi.org/10.1007/s11071-016-2716-2.   DOI
24 A. Akgul, A. Kilicman and M. Inc (2013). Improved (G'/G)-expansion method for the space and time fractional foam drainage and KdV equations, Abstr. Appl. Anal., 2013, 414353, https://doi.org/10.1155/2013/414353.   DOI
25 M. Mirzazadeh, M. Eslami and A. Biswas (2014). Solitons and periodic solutions to a couple of fractional nonlinear evolution equations, Pramana, 82, 465-476, 10.1007/s12043-013-0679-0.   DOI
26 A. Ali, A.R. Seadwy, D. Baleanu (2020). Computational solutions of conformable space-time derivatives dynamical wave equations: Analytical mathematical techniques, Results in Physics, 19 Article:103419.
27 M.T. Darvishi, M. Najafi and A.M. Wazwaz (2021). Some optical soliton solutions of space-time conformable fractional Schrodinger-type models, Phys. Scr., 92(2) Article:065213, doi.org/10.1088/1402-4896/abf269.   DOI
28 G. Cai, Q. Wang and J. Huang (2014). A modified F-expansion method for solving breaking soliton equations, Int. J. Nonlinear Sci., 2, 122-128, IJNS.2006.10.15/042.
29 R. Khalil, M. Al-Horani, A. Yousef and M. Sababheh (2014). A new definition of fractional derivative, J. Comput. Appl. Math., 264, 65-70, https://doi.org/10.1016/j.cam.2014.01.002.   DOI
30 K.A. Gepreel and S. Omran (2012). Exact solutions for nonlinear partial fractional differential equations, Chin. Phys. B, 21, 110204.   DOI
31 D.L. Weaire and S. Hutzler, The physics of foams, Oxford University Press, Oxford, 2000.
32 W. Kallel, H. Almusawa, S.M. Mirhosseini-Alizamani, M. Eslami, H. Rezazadeh, M.S. Osman (2021). Optical soliton solutions for the coupled conformable Fokas-Lenells equation with spatio-temporal dispersion, Results in Physics, 26 Article:104388.
33 Z.B. Li and J.H. He (2010). Fractional complex transform for fractional differential equations, Math. Comput. Appl., 15, 970-973, https://doi.org/10.3390/mca15050970.   DOI
34 M.T. Darvishi, M. Najafi and A.M. Wazwaz (2021). Conformabl space-time fractional nonlinear (1+1)-dimensional Schrodinger-type models and their traveling wave solutions, Chaos Solitons Fractals, 150 Article:111187, doi.org/10.1016/j.chaos.2021.111187.   DOI
35 R.K. Prud'homme and S.A. Khan (Eds.), Foams: theory, measurements and applications, Dekker, New York, 1996.
36 H. Caps, S.J. Cox, H. Decauwer, D. Weaire and N. Vandewalle (2005). Capillary rise in foams under microgravity, Colloids and Surfaces A: Physicochemical Engineering Aspects, 261, 131-134, https://doi.org/10.1016/j.colsurfa.2004.10.128.   DOI
37 V. Carrier, S. Destouesse and A. Colin (2002). Foam drainage: A film contribution?, Physical Review E, 65, 061404, https://doi.org/101103/PhysRevE.65.061404.   DOI
38 N. Denkov, S. Tcholakova and N. Politova-Brinkova (2020). Physicochemical control of foam properties, Current Opinion in Colloid & Interface Science, 50, 101376 https://doi.org/10.1016/j.cocis.2020.08.001.   DOI
39 A.-L. Fameau and S. Fujii (2020) Stimuli-responsive liquid foams: From design to applications, Current Opinion in Colloid & Interface Science, 50, 101380, https://doi.org/10.1016/j.cocis.2020.08.005.   DOI
40 J.J. Bikerman, Foams: Theory and Industrial Applications, Springer, New York, 1973.
41 S. Arbabi, A. Nazari, M.T. Darvishi (2016). A semi-analytical solution of foam drainage equation by Haar wavelets method, Optik, 127, 5443-5447, http://dx.doi.org/10.1016/j.ijleo.2016.03.032.   DOI