On numerical and analytical solutions of the generalized Burgers-Fisher equation
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Abstract
In this paper, the semi-analytic iterative and modified simple equation methods have been implemented to obtain solutions to the generalized Burgers-Fisher equation. To demonstrate the accuracy, efficacy as well as reliability of the methods in finding the exact solution of the equation, a selection of numerical examples was given and a comparison was made with other well-known methods from the literature such as variational iteration method, homotopy perturbation method and diagonally implicit Runge-Kutta method. The results have shown that between the proposed methods, the modified simple equation method is much faster, easier, more concise and straightforward for solving nonlinear partial differential equations, as it does not require the use of any symbolic computation software such as Maple or Mathematica. Additionally, the iterative procedure of the semi-analytic iterative method has merit in that each solution is an improvement of the previous iterate and as more and more iterations are taken, the solution converges to the exact solution of the equation.
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References
Javidi, M. (2006). Modified pseudospectral method for generalized Burgers-Fisher equation. Int. Math. Forum, 1(32), 1555-1564.
Arora, S., Jain, R., & Kukreja, V. K. (2022). A robust Hermite spline collocation technique to study generalized Burgers-Huxley equation, generalized Burgers-Fisher equation and modified Burgers’ equation. J. Ocean Eng. Sci. https://doi.org/10.1016/j.joes.2022.05.016
Ibrahim, N. I., & Mohammed, O. H. (2021). Semi-analytic technique for solving fractional partial differential equations with conformable derivatives. Al-Nahrain J. Sci., 24(1), 39-44. https://doi.org/10.22401/ANJS.24.1.07
Al-Jawary, M. A., & Mohammed, A. S. (2015). A semi-analytical iterative method for solving linear and nonlinear partial differential equations. Int. J. Sci. Res., 6(5), 978-982. https://doi.org/10.21275/ART20173220
Akinfe, T. K., & Loyinmi, A. C. (2021). A solitary wave solution to the generalized Burgers-Fisher’s equation using an improved differential transform method: A hybrid scheme approach. Heliyon, 7(5), 1-16. https://doi.org/10.1016/j.heliyon.2021.e07001
Ayati, Z., Moradi, M., & Mirzazadeh, M. (2015). Application of modified simple equation method to Burgers, Huxley and Burgers-Huxley equations. Iranian J. Numer. Anal. Optim., 5(2), 59-73. https://doi.org/10.22067/ijnao.v5i2.35910
Al-Jawary, M. A., Azeez, M. M., & Radhi, G. H. (2018). Analytical and numerical solutions for the nonlinear Burgers and advection-diffusion equations by using a semi-analytical iterative method. Comput. Math. Appl., 76(1), 155-171. https://doi.org/10.1016/j.camwa.2018.04.010
Kasumo, C. (2020). On exact solutions of Klein-Gordon equations using the semi analytic iterative method. Int. J. Adv. Appl. Math. Mech., 8(2), 54-63.
Selamat, M. S., Muhidir, A., Abd Razak, N. N., & Hassanuddin, Z. A. (2020). Semi analytical iterative method for solving Klein-Gordon equation. GADING J. Sci. Technol., 3(1), 10-18.
Temimi, H., & Ansari, A. R. (2011). A semi analytical iterative technique for solving nonlinear problems. Comput. Math. Appl., 61(2), 203-210. https://doi.org/10.1016/j.camwa.2010.10.042
Ramya, S., Krishnakumar, K., & Ilangovane, R. (2023). Exact solutions of time fractional generalized Burgers-Fisher equation using exp and exponential rational function methods. Int. J. Dyn. Control, 1-11. https://doi.org/10.1007/s40435-023-01267-6
Taghizadeh, N., Mirzazadeh, M., Paghaleh , A. S., & Vahidi, J. (2012). Exact solutions of nonlinear evolution equations by using the modified simple equation method. Ain Shams Eng. J., 3(3), 321-325. https://doi.org/10.1016/j.asej.2012.03.010
Pirdawood, M. A., & Sabawi, Y. A. (2021). High-order solution of generalized Burgers-Fisher equation using compact finite difference and DIRK methods. J. Phys.: Conf. Ser., 1999. https://doi.org/10.1088/1742-6596/1999/1/012088
Latif, B., Selamat, M. S., Rosli, A. N., Yusoff, A. I., & Hasan, N. M. (2020). The semi analytics iterative method for solving Newell-Whitehead-Segel equation. Math. Stat., 8(2), 89-94. https://doi.org/10.13189/ms.2020.080203
Mendoza, J., & Muriel, C. (2021). New exact solutions for a generalised Burgers-Fisher equation. Chaos, Solitons and Fractals, 152. https://doi.org/10.1016/j.chaos.2021.111360
Rashidi, M. M., Ganji, D. D., & Dinarvand, S. (2009). Explicit analytical solutions of the generalized Burger and Burger-Fisher equations by homotopy perturbation method. Numer. Methods Partial Differ. Equ.: An Int. J., 25(2), 409-417. https://doi.org/10.1002/num.20350
Zhong, M., Yang, Q. J., & Tian, S. F. (2021). The modified high-order Haar wavelet scheme with Runge-Kutta method in the generalized Burgers-Huxley equation. Modern Physics Letters B, 35(24). https://doi.org/10.1142/S0217984921504194
Khan, K., Akbar, M. A., & Ali, N. H. M. (2013). The modified simple equation method for exact and solitary wave solutions of nonlinear evolution equation: the GZK-BBM equation and right-handed noncommutative Burgers equations. Math. Phys., 2013. https://doi.org/10.1155/2013/146704
Temimi, H., & Ansari, A. R. (2015). A computational iteration method for solving nonlinear ordinary differential equations. LMS J. Comput. Math., 18(1), 730-753. https://doi.org/10.1112/S1461157015000285
Kumar, S., & Saha Ray, S. (2021). Numerical treatment for Burgers-Fisher and generalized Burgers-Fisher equations. Math. Sci., 15, 21-28. https://doi.org/10.1007/s40096-020-00356-3
Soori, M. (2018). The variational iteration method and the homotopy perturbation method to the exact solution of the generalized Burgers-Fisher equation. Calc. Var. Partial Differ. Equ., 5(8), 19-26.
Mohammadi, R. (2011). Spline solution of the generalized Burgers-Fisher equation. Applicable Analysis, 91(12), 2189-2215. https://doi.org/10.1080/00036811.2011.596479