Abstract
In this study, traveling wave solutions have been explored with the newly developed improved tanh method and modified \(\left( {1/G^{\prime}} \right)\)-expansion method for the Fractional foam drainage equation, which is famous for modeling physical phenomena such as heat conduction and acoustic waves. Abundant solutions are successfully achieved which have not been appeared ever in the literature. The found solutions are represented graphically to bring out the appearances of different types solitons. In addition, three important points are highlighted in the result and discussion section. First, the comparison of the applied methods, second, the association of the obtained solutions with the literature, and finally the effect of the change in the parameters with physical properties on the wave behavior are discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdeljawad T (2015) On conformable fractional calculus. J Comput Appl Math 279:57–66
Abdeljawad T, Al-Mdallal QM, Jarad F (2019) Fractional logistic models in the frame of fractional operators generated by conformable derivatives. Chaos Solit Fract 119:94–101
Ablowitz MJ, Clarkson PA (1991) Soliton, nonlinear evolution equations and inverse scattering. Cambridge University Press, New York
Akgul A, Kilicman A, Inc M (2013) Improved -expansion method for the space and time fractional foam drainage and KdV equations. Abst Appl Anal 2013:414353
Al-Mdallal QM, Syam MI (2007) Sine-Cosine method for finding the soliton solutions of the generalized fifth-order nonlinear equation. Chaos Solitons Fract 33(5):1610–1617
Al-Mdallal QM, Yusuf H, Ali A (2020) A novel algorithm for time-fractional foam drainage equation. Alex Eng J 59:1607–1612
Bonyah E, Gomez-Aguilar JF, Abu A (2018) Stability analysis and optimal control of a fractional human African trypanosomiasis model. Chaos Soliton Fract 117:150–160
Cantat I, Cohen-Addad S, Elias F, Graner F, Höhler R, Pitois O, Saint-Jalmes A et al (2013) Foams: structure and dynamics. OUP Oxford, Oxford
Cevikel AC (2018) New exact solutions of the space-time fractional KdV-Burgers and nonlinear fractional foam drainage equation. Therm Sci 22:15–24
Cox SJ, Weairy D, Hutzler S, Murphy J, Phelan R, Verbist G (2002) Applications and generalizations of the foam drainage equation. Math Phys Eng Sci 456:2441–2464
Das A, Ghosh N (2019) Bifurcation of traveling waves and exact solutions of Kadomtsev-Petviashvili modified equal width equation with fractional temporal evolution. Comput Appl Math 38(1):9
Duran S, Yokuş SA, Durur H (2021) Surface wave behavior and refraction simulation on the ocean for the fractional Ostrovsky–Benjamin–Bona–Mahony equation. Modern Phys Lett B 35(31):2150477
Duran S, Yokuş A, Durur H, Kaya D (2021) Refraction simulation of internal solitary waves for the fractional Benjamin-Ono equation in fluid dynamics. Mod Phys Lett B 35(26):2150363
Ege SM, Misirli E (2014) Solutions of the space-time fractional foam drainage equations and the fractional Klein-Gorgon equation by use of the modified Kudryashov method. Int J Res Advent Technol 2:384–388
Gomez-Aguilar JF, Ghanbari B, Bonyah E (2018) On the new fractional operator and application to nonlinear bloch system. Int Workshop Math Model Appl Anal Comput 2018:137–154
Hassani H, Machado JT, Naraghirad E, Sadeghi B (2020) Solving nonlinear systems of fractional-order partial differential equations using an optimization technique based on generalized polynomials. Comput Appl Math 39(4):1–19
Helal MA, Mehana MS (2006) A comparison between two different methods for solving Modified KdV-Burgers equation. Chaos Solitons Fract 28:320–326
Hoogland F, Lehmann P, Or D (2016) Drainage dynamics controlled by corner flow: application of the foam drainage equation. Water Res Res 52:8402–8412
Islam MN, Akbar MA (2018a) New exact wave solutions to the space-time fractional-coupled Burger’s equations and the space-time fractional foam drainage equation. Cogent Phys 5:1422957
Islam MN, Akbar MA (2018b) New exact wave solutions to the space-time fractional coupled Burgers equations and the space-time fractional foam drainage equation. Cogent Phys 5:1422957
Islam MT, Akter MA (2020) Further fresh and general traveling wave solutions to some fractional order nonlinear evolution equations in mathematical physics. Arab J Math Sci 26(1):5. https://doi.org/10.1108/AJMS-09.2020-0078
Islam MT, Akter MA (2021) Distinct solutions of nonlinear space-time fractional evolution equations appearing in mathematical physics via a new technique. Partial Diff Eq Appl Math 2021:3
Islam MT, Akbar MA, Azad MAK (2018a) The exact traveling wave solutions to the nonlinear space-time fractional modified Benjamin-Bona-Mahony equation. J Mech Cont Math Sci 13:56–71
Islam MT, Akbar MA, Azad AK (2018b) Traveling wave solutions to some nonlinear fractional partial differential equations through the rational—expansion method. J Ocean Eng Sci 3:76–81
Jianming L, Jie D, Wenjun Y (2011) Backlund transformation and new exact solutions of the Sharma-Tasso-Olver equation. Abstr Appl Anal 2011:935710
Khalil R, Horani MA, Yousef A, Sababheh MAM (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70
Kraynik AM, Reinelt DA, van Swol F (2003) Structure of random monodisperse foam. Phys Rev E 67(3):031403
Malfliet W (1992) Solitary wave solutions of nonlinear wave equations. Am J Phys 60(7):650–654
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
Mohyud-Din ST, Noor MA, Noor KI (2009) Modified variational iteration method for solving Sine-Gordon equations. World Appl Sci J 6:999–1004
Pandir Y, Yildirim A (2018) Analytical approach for the fractional differential equations by using the extended tanh method. Waves Ran Com Med 28:399–410
Plateau JAF (1873) Statique expérimentale et théorique des liquides soumis aux seules forces moléculaires. Gauthier-Villars, Paris
Seadawy AR (2017) Traveling-wave solution of a weakly nonlinear two-dimensional higher-order Kadomtsev-Petviashvili dynamical equation for dispersive shallow-water waves. Eur Phys J plus 132:29
Shi D, Zhang Y, Liu W (2020) Multiple exact solutions of the generalized time fractional foam drainage equation. Fractals 28:2050062
Weaire DL, Hutzler S (2001) The physics of foams. Oxford University Press, Oxford
Weaire D, Hutzler S, Cox S, Kern N, Alonso MD, Drenckhan W (2003) The fluid dynamics of foams. J Phys: Condens Matter 15(1):65–73
Yokus A (2011) Solutions of some nonlinear partial differential equations and comparison of their solutions, Ph. Diss., Fırat University
Yokus A (2018) Numerical solution for space and time fractional order Burger type equation. Alex Eng J 57:2085–2091
Yokuş A, Durur H, Duran S (2021) Simulation and refraction event of complex hyperbolic type solitary wave in plasma and obtical fiber for the perturbed Chen-Lee-Liu equation. Opt Quant Electron 53(7):1–17
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Agnieszka Malinowska.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yokuş, A., Durur, H., Duran, S. et al. Ample felicitous wave structures for fractional foam drainage equation modeling for fluid-flow mechanism. Comp. Appl. Math. 41, 174 (2022). https://doi.org/10.1007/s40314-022-01812-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01812-7
Keywords
- Improved tanh method
- Modified \(\left( {1/G^{\prime}} \right)\)-expansion method
- Fractional foam drainage equation
- Exact solutions