Solitary Pattern Solutions of gBBM Equation using New Iterative Method (NIM)

  • Authors

    • Hamzeh Zureigat Department of Mathematics, Faculty of Science and Technology, Jadara University, 21110 Irbid, Jord
    2023-11-05
    https://doi.org/10.14419/5j1jpw81

  • In this study, we focus on the solitary pattern solutions of the generalized Benjamin-Bona-Mahony equations (gBBM) and propose a new iterative method (NIM) for their numerical solution, given suitable initial conditions. Our proposed NIM approach generates numerical solutions in the form of a convergent power series with computationally simple components. Our results demonstrate that the NIM approach exhibits exceptional performance in terms of accuracy, efficiency, simplicity, stability, and reliability.

  • References

    1. Wang XY. Exact and explicit solitary wave solutions for the generalized Fisher equation. Phys Lett A 1988; 131 (4-5): 277-279.
    2. Jeffrey A, Mohamad MNB. Exact solutions to the KdV-Burgers equation. Wave Motion 1991;14:369–375.
    3. Wadati M. The exact solution of the modified Korteweg-de Vries equation. J Phys Soc Jpn 1972;32:1681–1687.
    4. Kivshar YS, Pelinovsky DE. Self-focusing and transverse instabilities of solitary waves. Phys Rep 2000;331:117–195.
    5. Hereman W, Takaoka M. Solitary wave solutions of nonlinear evolution and wave equations using a direct method and MACSYMA. J Phys A 1990;23:4805–4822.
    6. Khaled Batiha, Belal Batiha, A new Algorithm for solving linear ordinary differential equations, World Applied Sciences Journal, Volume 15, Issue 12, 2011, Pages 1774-1779.
    7. Belal Batiha, Numerical solution of a class of singular second-order IVPs by variational iteration method, International Journal of Mathematical Analysis, 2009, 3(37-40), pp. 1953-1968.
    8. Javidi M. A numerical solution of the generalized Burger’s-Huxley equation by pseudospectral method and Darvishi’s preconditioning. Appl Math Comput 1990;175:1619–1628.
    9. Javidi M. A numerical solution of the generalized Burger’s-Huxley equation by spectral collocation method. Appl Math Comput,
    10. doi:10.1016.amc.2005.11.051, in press
    11. Adomian G. Solving frontier problems of physics: the decomposition method. Boston: Kluwer Academic; 1994.
    12. I. Hashim, M.S.M. Noorani, B. Batiha, A note on the Adomian decomposition method for the generalized Huxley equation, Applied Mathematics and Computation, Volume 181, Issue 2, 2006, Pages 1439-1445
    13. He JH. Application of homotopy perturbation method to nonlinear wave equations. Chaos Soliton Fract 2005;26(3):695–700.
    14. He JH. Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer Simul 2005;6(2):207–8.
    15. He JH. Limit cycle and bifurcation of nonlinear problems. Chaos Soliton Fract 2005;26(3):827–33.
    16. Batiha, B. A variational iteration method for solving the nonlinear Klein-Gordon equation, Australian Journal of Basic and Applied Sciences, 2009, 3(4), pp. 3876–3890.
    17. Batiha, B., Numerical solution of a class of singular second-order IVPs by variational iteration method, International Journal of Mathematical Analysis, 2009, 3(37-40), pp. 1953–1968.
    18. Batiha, B., Application of variational iteration method to linear partial differential equations, Applied Mathematical Sciences, 2009, 3(49-52), pp. 2491–2498.
    19. Batiha, A.-M., Batiha, B., A new method for solving epidemic model, Australian Journal of Basic and Applied Sciences, 2011, 5(12), pp. 3122–3126.
    20. Batiha, K., Batiha, B. ,A new algorithm for solving linear ordinary differential equations, World Applied Sciences Journal, 2011, 15(12), pp. 1774–1779.
    21. T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlineardispersive systems, Philos. Trans. Roy. Soc. London 272 (1972) 47–78.
    22. J.L. Bona, V.A. Dougalis, An initial- and boundary-value problem for a model equationfor propagation of long waves, J. Math. Anal. Appl. 75 (1980), 503-522.
    23. Y.M. Chen, Remark on the global existence for the generalized Benjamin–Bona–Mahony equations in arbitrary dimension, Appl. Anal. 30, (1988), 1-15.
    24. D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25(1996) 321–330.
    25. V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753–763.
    26. V. Daftardar-Gejji, S. Bhalekar, Solving fractional diffusion-wave equations using a new iterative method, Frac. Calc. Appl. Anal. 11 (2) (2008) 193–202.
    27. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer, 1994.
    28. J.H. He, Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng. 178 (1999) 257–262.
    29. J.H. He, Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Meth. Appl. Mech. Eng. 167 (1998) 57–68.
    30. Satsuma J. Topics in soliton theory and exactly solvable nonlinear equations. Singapore: World Scientific; 1987.
    31. B. Batiha, M.S.M. Noorani, I. Hashim, Application of variational iteration method to the generalized Burgers-Huxley equation,Chaos, Solitons and Fractals 36 (2008) 660–663.
    32. Hashim I, Noorani MSM, Al-Hadidi MRS. Solving the generalized Burgers-Huxley equation using the Adomian decomposition method. Mathl Comput Model 2006;43:1404–1411.
    33. I. Hashim, M.S.M. Noorani, & B. Batiha, A note on the Adomian decomposition method for the generalized Huxley Equation, Applied Mathematics and Computations, 2006, 181:1439–1445.
    34. Wang XY, Zhu ZS, Lu YK. Solitary wave solutions of the generalised Burgers-Huxley equation. J Phys A: Math Gen 1990;23:271–274.
    35. Belal Batiha, Firas Ghanim, Solving strongly nonlinear oscillators by new numerical method, International Journal of Pure and Applied Mathematics, 2017; 116 (1) 115–124.
    36. Belal Batiha, Firas Ghanim, Numerical implementation of Daftardar-Gejji and Jafari method to the quadratic Riccati equation, Bul. Acad. S¸tiint¸e Repub. Mold. Mat., 2021, Number 3, 21–29.
    37. Batiha, B. New Solution of the Sine-Gordon Equation by the Daftardar-Gejji and Jafari Method. Symmetry 2022, 14, 57. https://
    38. doi.org/10.3390/sym14010057
    39. Belal Batiha, Firas Ghanim, O. Alayed, Ra’ed Hatamleh, Ahmed Salem Heilat, Hamzeh Zureigat, Omar Bazighifan, ”Solving Multispecies Lotka– Volterra Equations by the Daftardar-Gejji and Jafari Method”, International Journal of Mathematics and Mathematical Sciences, vol. 2022, Article ID 1839796, 2022.
    40. Sachin Bhalekar, Varsha Daftardar-Gejji, Convergence of the New Iterative Method, International Journal of Differential Equations, 2011; Vol. 2011, 1-10.
    41. A.M. Wazwaz, New travelling wave solutions of different physical structures to gBBM equation, Physics Letters A 355 (2006) 358–362.
    42. Elcin Yusufoglu, Ahmet Bekir, The variational iteration method for solitary patterns solutions of gBBM equation, Physics Letters A, Volume 367, Issue 6, 2007, 461-464.

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  • How to Cite

    Zureigat, H. (2023). Solitary Pattern Solutions of gBBM Equation using New Iterative Method (NIM). International Journal of Applied Mathematical Research, 12(1), 11-17. https://doi.org/10.14419/5j1jpw81