## Journal of Applied Mathematics

### A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation

Mohamed R. Ali

#### Abstract

We deem the time-fractional Benjamin-Ono (BO) equation out of the Riemann–Liouville (RL) derivative by applying the Lie symmetry analysis (LSA). By first using prolongation theorem to investigate its similarity vectors and then using these generators to transform the time-fractional BO equation to a nonlinear ordinary differential equation (NLODE) of fractional order, we complete the solutions by utilizing the power series method (PSM).

#### Article information

Source
J. Appl. Math., Volume 2019 (2019), Article ID 3456848, 7 pages.

Dates
Revised: 2 March 2019
Accepted: 25 March 2019
First available in Project Euclid: 24 July 2019

https://projecteuclid.org/euclid.jam/1563933633

Digital Object Identifier
doi:10.1155/2019/3456848

Mathematical Reviews number (MathSciNet)
MR3949343

#### Citation

Ali, Mohamed R. A Truncation Method for Solving the Time-Fractional Benjamin-Ono Equation. J. Appl. Math. 2019 (2019), Article ID 3456848, 7 pages. doi:10.1155/2019/3456848. https://projecteuclid.org/euclid.jam/1563933633

#### References

• M. Bruzon et al., “The symmetry reductions of a turbulence model,” Journal of Physics A: Mathematical and General, vol. 34, no. 18, p. 3751, 2001.
• R. Sadat and M. Kassem, “Explicit Solutions for the (2+ 1)-dimensional jaulent–miodek equation using the integrating factors method in an unbounded domain,” Mathematical & Computational Applications, vol. 23, no. 1, p. 15, 2018.
• A. Paliathanasis and M. Tsamparlis, “Lie symmetries for systems of evolution equations,” Journal of Geometry and Physics, vol. 124, pp. 165–169, 2018.
• Y. Y. Zhang, X. Q. Liu, and G. W. Wang, “Symmetry reductions and exact solutions of the (2+ 1)-dimensional Jaulent–Miodek equation,” Applied Mathematics and Computation, vol. 219, no. 3, pp. 911–916, 2012.
• M. Mirzazadeh and M. Eslami, “Exact solutions of the Kudryashov-Sinelshchikov equation and nonlinear telegraph equation via the first integral method,” Nonlinear Analysis: Modelling and Control, vol. 17, no. 4, pp. 481–488, 2012.
• M. Mirzazadeh, M. Eslami, A. H. Bhrawy, B. Ahmed, and A. Biswas, “Solitons and other solutions to complex-valued Klein-Gordon equation in $\Phi$-4 field theory,” Applied Mathematics & Information Sciences, vol. 9, no. 6, pp. 2793–2801, 2015.
• A. Nazarzadeh, M. Eslami, and M. Mirzazadeh, “Exact solutions of some nonlinear partial differential equations using functional variable method,” Pramana–-Journal of Physics, vol. 81, no. 2, pp. 225–236, 2013.
• M. Eslami, A. Neyrame, and M. Ebrahimi, “Explicit solutions of nonlinear (2+1)-dimensional dispersive long wave equation,” Journal of King Saud University - Science, vol. 24, no. 1, pp. 69–71, 2012.
• M. Mirzazadeh, M. Eslami, and A. H. Arnous, “Dark optical solitons of Biswas-Milovic equation with dual-power law nonlinearity,” The European Physical Journal Plus, vol. 130, 4, no. 1, 2015.
• E. Aksoy, M. Kaplan, and A. Bekir, “Exponential rational function method for space-time fractional differential equations,” Waves in random and complex media : propagation, scattering and imaging., vol. 26, no. 2, pp. 142–151, 2016.
• A. Bekir and A. C. Cevikel, “New exact travelling wave solutions of nonlinear physical models,” Chaos, Solitons and Fractals, vol. 41, no. 4, pp. 1733–1739, 2009.
• M. Eslami, M. Mirzazadeh, B. Fathi Vajargah, and A. Biswas, “Optical solitons for the resonant nonlinear Schrödinger's equation with time-dependent coefficients by the first integral method,” Optik - International Journal for Light and Electron Optics, vol. 125, no. 13, pp. 3107–3116, 2014.
• M. F. El-Sabbagh, R. Zait, and R. M. Abdelazeem, “New exact solutions of some nonlinear partial differential equations via the improved exp-function method,” IJRRAS, vol. 18, no. 2, pp. 132–144, 2014.
• Y. Gurefe, E. Misirli, A. Sonmezoglu, and M. Ekici, “Extended trial equation method to generalized nonlinear partial differential equations,” Applied Mathematics and Computation, vol. 219, no. 10, pp. 5253–5260, 2013.
• A. H. Khater, M. H. Moussa, and S. F. Abdul-Aziz, “Invariant variational principles and conservation laws for some nonlinear partial differential equations with constant coefficients - I,” Chaos, Solitons and Fractals, vol. 14, no. 9, pp. 1389–1401, 2002.
• Q. Feng and F. Meng, “Traveling wave solutions for fractional partial differential equations arising in mathematical physics by an improved fractional Jacobi elliptic equation method,” Mathematical Methods in the Applied Sciences, vol. 40, no. 10, pp. 3676–3686, 2017.
• R. K. Gazizov, A. A. Kasatkin, and S. Y. Lukashchuk, “Continuous transformation groups of fractional differential equations,” Vestnik Usatu, vol. 9, no. 3, p. 21, 2007.
• S. Zhang and H. Q. Zhang, “Fractional sub-equation method and its applications to nonlinear fractional PDEs,” Physics Letters A, vol. 375, no. 7, pp. 1069–1073, 2011.
• B. Zheng, “A new fractional Jacobi elliptic equation method for solving fractional partial differential equations,” Advances in Difference Equations, vol. 2014, no. 1, p. 228, 2014.
• A. Neyrame, A. Roozi, S. S. Hosseini, and S. M. Shafiof, “Exact travelling wave solutions for some nonlinear partial differential equations,” Journal of King Saud University - Science, vol. 22, no. 4, pp. 275–278, 2010.
• K. B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, vol. 111, Academic Press, New York, NY, USA, London, UK, 1974.
• B. Ahmad, S. K. Ntouyas, and A. Alsaedi, “On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions,” Chaos, Solitons & Fractals, vol. 83, pp. 234–241, 2016.
• V. S. Kiryakova, Generalized Fractional Calculus and Applications, CRC press, Botan Roca, Fl, USA, 1993.
• G.-W. Wang, X.-Q. Liu, and Y.-Y. Zhang, “Lie symmetry analysis to the time fractional generalized fifth-order KdV equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 9, pp. 2321–2326, 2013.
• S. S. Ray and S. Sahoo, “Invariant analysis and conservation laws of (2+ 1) dimensional time-fractional ZKBBM equation in gravity water waves,” Computers & Mathematics with Applications, 2017.
• G.-W. Wang and T.-Z. Xu, “Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis,” Nonlinear Dynamics, vol. 76, no. 1, pp. 571–580, 2014.
• M. Inc, A. Yusuf, A. I. Aliyu, and D. Baleanu, “Lie symmetry analysis, explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations,” Physica A: Statistical Mechanics and its Applications, vol. 496, pp. 371–383, 2018. \endinput