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Exact solutions of fractional nonlinear equations by generalized bell polynomials and bilinear method

For numerous fluids between elastic and viscous materials, the fractional derivative models have an advantage over the integer order models. On the basis of conformable fractional derivative and the respective useful properties, the bilinear form of time fractional Burgers equation and Boussinesq-Burgers equations are obtained using the generalized Bell polynomials and bilinear method. The kink soliton solution, anti-kink soliton solution and the single-soliton solution for different fractional order are derived respectively. The time fractional order system possesses property of time memory. And higher oscillation frequency appears as the time fractional order increasing. The fractional derivative increases the possibility of improving the control performance in complex systems with fluids between different elastic and viscous materials.
PAPER REVISED: 2020-06-20
PAPER ACCEPTED: 2020-06-20
  1. Podlubny, I., Fractional differential equations, Academic Press, New York, USA, 1999.
  2. Rudolf, H., Applications of fractional calculus in physics, World Scientific, 2000.
  3. Atangana A, Koca I. Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order. Chaos Soliton Fract, 89(2016), pp. 447-454.
  4. Chung, W. S., Fractional Newton mechanics with conformable fractional derivative, J Comput Appl Math, 290(2015), pp. 150-158.
  5. Eslami, M., et al., The first integral method applied to the Bogoyavlenskii equations by means of conformable fractional derivative, Opt Quant Electron, 49(2017), 12, 391.
  6. Akbulut, A., Kaplan, M., Auxiliary equation method for time-fractional differential equations with conformable derivative, Computers & Mathematics with Applications, 75(2018), 3, pp. 876-882.
  7. Kaya G, et al., Dynamical analysis of a discrete conformable fractional order bacteria population model in a microcosm. Physica A, 547(2020) , 123864.
  8. Heinz, S., Comments on a priori and a posteriori evaluations of sub-grid scale models for the Burgers' equation, Comput Fluids, 138(2016), pp. 35-37.
  9. Dong, M.J., et al., Nonlocal symmetries, conservation laws and interaction solutions for the classical Boussinesq-Burgers equation, Nonlinear Dynamics, 95(2019), pp. 273-291.
  10. Jiang, Y.L., Cheng, C., Lie group analysis and dynamical behavior for classical Boussinesq-Burgers system, Nonlinear Analysis: Real World Applications, 47(2019), pp. 385-397.
  11. Guo, B. Y., et al., Lump solutions and interaction solutions for the dimensionally reduced nonlinear evolution equation, Complexity, 2019, 5765061.
  12. L. Zhang, P. Yuan, J. Fu, C M Khalique. Bifurcations and exact traveling wave solutions of the Zakharov-Rubenchik equation. Disc. Cont. Dyna. Syst. - S, doi: 10.3934/dcdss.2020214
  13. Lambert, F., Loris, I., Springael, J., Willer, R., On a direct bilinearization method: Kaup's higher-order water wave equation as a modified nonlocal Boussinesq equation, J Phys A : Math Gen, 27(1994), 15, 5325.
  14. Momani, S., Non-perturbative analytical solutions of the space-and time-fractional Burgers equations, Chaos Soliton Fract, 28(2006), 4, pp. 930-937.
  15. Inc, M., The approximate and exact solutions of the space-and time-fractional Burgers equations with initial conditions by variational iteration method, J Math Anal Appl, 345(2008), 1, pp. 476-484.