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Analytical and semi-analytical wave solutions for longitudinal wave equation via modified auxiliary equation method and Adomian decomposition method

This paper studies the analytical and semi-analytical wave solutions for the longitudinal wave equation. Moreover, it examines the performance of the modified auxiliary equation method and Adomian decomposition method on this model. This model describes the dispersion in the circular rod that dispersion caused by the transverse Poisson's effect in electro-magnetoelastic (EME). Many explicit wave solutions are found by using the analytical technique. These solutions allow studying the physical properties of this model. The comparison between the analytical and semi-analytical solutions is discussed to show the value of the absolute error between them.
PAPER REVISED: 2019-07-10
PAPER ACCEPTED: 2019-07-30
  1. Yang, X.J. et al. "On exact traveling-wave solutions for local fractional Korteweg-de Vries equation." Chaos: An Interdisciplinary Journal of Nonlinear Science 26, no. 8 (2016): 084312.
  2. Eslami, M. "Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations." Applied Mathematics and Computation 285 (2016): 141-148.
  3. Tchier,F. et al. "Soliton solutions and conservation laws for lossy nonlinear transmission line equation." Superlattices and Microstructures 107 (2017): 320-336.
  4. Inc, M. et al. "Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis." Physica A: Statistical Mechanics and its Applications 493 (2018): 94-106.
  5. Yang, C. et al. "Transformation of soliton states for a (2+ 1) dimensional fourth-order nonlinear Schrödinger equation in the Heisenberg ferromagnetic spin chain." Laser Physics 29, no. 3 (2019): 035401.
  6. Liu, W. et al. "Interaction properties of solitonics in inhomogeneous optical fibers." Nonlinear Dynamics 95, no. 1 (2019): 557-563.
  7. Yang, C. et al. "One-soliton shaping and two-soliton interaction in the fifth-order variablecoefficient nonlinear Schrödinger equation." Nonlinear Dynamics 95, no. 1 (2019): 369-380.
  8. Liu,X. et al. "Generation and control of multiple solitons under the influence of parameters." Nonlinear Dynamics 95, no. 1 (2019): 143-150.
  9. Yu, W. et al. "Phase shift, amplification, oscillation and attenuation of solitons in nonlinear optics." Journal of advanced research 15 (2019): 69-76.
  10. Baleanu, D. et al. Optical solitons, nonlinear self-adjointness and conservation laws for Kundu- Eckhaus equation. Chinese journal of physics, 55(6) (2017) 2341-2355.
  11. Hosseini, K. et al. "New exact traveling wave solutions of the unstable nonlinear Schrödinger equations." Communications in Theoretical Physics 68, no. 6 (2017): 761.
  12. Hosseini, K. et al. "New optical solitons of cubic-quartic nonlinear Schrödinger equation." Optik 157 (2018): 1101-1105.
  13. Hosseini, K., P. Mayeli, E. Yazdani Bejarbaneh, and Qin Zhou. "New optical solitons of the longitudinal wave equation in a magnetoelectro-elastic circular rod." Acta Phys. Pol. A 133 (2018): 20-22.
  14. Hosseini, K., E. Yazdani Bejarbaneh, A. Bekir, and M. Kaplan. "New exact solutions of some nonlinear evolution equations of pseudoparabolic type." Optical and Quantum Electronics 49, no. 7 (2017): 241.
  15. Kaplan, M., P. Mayeli, and K. Hosseini. "Exact traveling wave solutions of the Wu-Zhang system describing (1+ 1)-dimensional dispersive long wave." Optical and Quantum Electronics 49, no. 12 (2017): 404.
  16. Khater, M. M., Seadawy, A. R., & Lu, D. (2017). Elliptic and solitary wave solutions for Bogoyavlenskii equations system, couple Boiti-Leon-Pempinelli equations system and Timefractional Cahn-Allen equation. Results in physics, 7, 2325-2333.
  17. Khater, M. M., Seadawy, A. R., & Lu, D. (2018). Dispersive optical soliton solutions for higher order nonlinear Sasa-Satsuma equation in mono mode fibers via new auxiliary equation method. Superlattices and Microstructures, 113, 346-358.
  18. Bibi, Sadaf, Syed Tauseef Mohyud-Din, Umar Khan, and Naveed Ahmed. (2017). Khater method for nonlinear Sharma Tasso-Olever (STO) equation of fractional order. Results in physics7, 4440- 4450.
  19. Seadawy, A. R., Lu, D., & Khater, M. M. (2017). Bifurcations of solitary wave solutions for the three dimensional Zakharov-Kuznetsov-Burgers equation and Boussinesq equation with dual dispersion. Optik, 143, 104-114.
  20. Khater, M. M., Seadawy, A. R., & Lu, D. (2018). New optical soliton solutions for nonlinear complex fractional Schrödinger equation via new auxiliary equation method and novel (G'/G)- expansion method. Pramana, 90(5), 59.
  21. Attia, R. A., Lu, D., & Khater, M. M. (2018). Structure of New Solitary Solutions for The Schwarzian Korteweg De Vries Equation And (2+ 1)-Ablowitz-Kaup-Newell-Segur Equation.
  22. Khater, M., Attia, R., & Lu, D. (2019). Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions. Mathematical and Computational Applications, 24(1), 1.
  23. Attia, R. A., Lu, D., & MA Khater, M. (2019). Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation. Mathematical and Computational Applications, 24(1), 10.
  24. Triki, H., Mirzazadeh, M., Bhrawy, A. H., Razborova, P., & Biswas, A. (2015). Solitons and other solutions to long-wave short-wave interaction equation. Rom. J. Phys, 60(1-2), 72-86.
  25. Arani, A. G., Roudbari, M. A., & Amir, S. (2016). Longitudinal magnetic field effect on wave propagation of fluid-conveyed SWCNT using Knudsen number and surface considerations. Applied Mathematical Modelling, 40(3), 2025-2038.
  26. Masemola, P., Kara, A. H., Bhrawy, A. H., & Biswas, A. (2016). Conservation laws for coupled wave equations. Rom. J. Phys, 61(3-4), 367-377.
  27. Zhen, Y., & Zhou, L. (2017). Wave propagation in fluid-conveying viscoelastic carbon nanotubes under longitudinal magnetic field with thermal and surface effect via nonlocal strain gradient theory. Modern Physics Letters B, 31(08), 1750069.
  28. Parker, K. J., & Alonso, M. A. (2016). Longitudinal iso-phase condition and needle pulses. Optics express, 24(25), 28669-28677.
  29. Seadawy, A. R., & Manafian, J. (2018). New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod. Results in physics, 8, 1158-1167.
  30. Trainiti, G., & Ruzzene, M. (2016). Non-reciprocal elastic wave propagation in spatiotemporal periodic structures. New Journal of Physics, 18(8), 083047.
  31. Bakodah, H. O., Al Qarni, A. A., Banaja, M. A., Zhou, Q., Moshokoa, S. P., & Biswas, A. (2017). Bright and dark Thirring optical solitons with improved adomian decomposition method. Optik, 130, 1115-1123.
  32. Turkyilmazoglu, M. (2016). Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method. Mediterranean Journal of Mathematics, 13(6), 4019-4037.
  33. Paripour, M., Hajilou, E., Hajilou, A., & Heidari, H. (2015). Application of Adomian decomposition method to solve hybrid fuzzy differential equations. Journal of Taibah University for Science, 9(1), 95-103.
  34. Kang, S. M., Nazeer, W., Tanveer, M., Mehmood, Q., & Rehman, K. (2015). Improvements in Newton-Rapshon method for nonlinear equations using modified Adomian decomposition method. International Journal of Mathematical Analysis, 9(39), 1919-1928.
  35. Javed, I., Ahmad, A., Hussain, M., & Iqbal, S. (2017). Some Solutions of Fractional Order Partial Differential Equations Using Adomian Decomposition Method. arXiv preprint arXiv:1712.09207.