Al-Bahir Journal for Engineering and Pure Sciences


The emergence of the Burgers-Huxley equation (which involves the famous Burgers equation and the Huxley equation) to predict response systems, dispersion moves, and nerve charge transmission in traffic patterns, sound, turbulent conditions theory, hydrodynamics has attracted the attention of scientists to provide reliable and efficient solutions to the problem. The present work employed the Tanh method to solve the Burgers-Huxley nonlinear partial differential equations. In contrast to previous results with complicated and laborious solution characteristics, this method is accurate, efficient, and requires little computational work. In showing this, we solved four Burgers-Huxley case study problems using the Tanh approach and obtained the exact solution. The solutions of the four cases were presented graphically. In addition, the findings demonstrate that the Tanh method is an effective and robust approach for constructing the exact solution of nonlinear differential equations.


[1] R, Loyinmi A, Miller JC, Sharkey KJ. Approximating quasi-stationary behaviour in network-based SIS dynamics. Bull Math Biol 2022;84:1-32.

[2] Idowu KO, Akinwande TG, Fayemi I, Adam UM, Loyinmi AC. Laplace homotopy perturbation method (LHPM) for solving systems of N-dimensional non-linear partial differential equation. 2023. https://doi.org/10.55810/2313-0083.1031.

[3] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, et al. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and appli- cations. Comput Methods Appl Mech Eng 2020;362:112790.

[4] Loyinmi AC, Erinle-Ibrahim LM, Adeyemi AE. The new iterative method (NIM) for solving telegraphic equation. J. Niger. Assoc. Math. Phys. 2017;43:31-6.

[5] Loyinmi AC, Lawal OW, Sottin DO. Reduced differential transform method for solving partial integro-differential equation. J. Niger. Assoc. Math. Phys. 2017;43:37-42.

[6] Loyinmi AC, Lawal OW. The asymptotic solution for the steady variable-viscosity free convection flow on a porous plate. J. Niger. Assoc. Math. Phys. 2011;19:273-6.

[7] Mohanty RK, Sharma S. High-accuracy quasi-variable mesh method for the system of 1D quasi-linear parabolic partial differential equations based on off-step spline in compres- sion approximations. Adv Differ Equ 2017;2017:1-30.

[8] Akinfe KT, Loyinmi AC. The implementation of an improved differential transform scheme on the schrodinger equation governing wave-particle duality in quantum physics and optics. 2021. Available SSRN 4098920.

[9] Akinfe TK, Loyinmi AC. An improved differential transform scheme implementation on the generalized AlleneCahn equation governing oil pollution dynamics in oceanog- raphy. Partial Differ. Equations Appl. Math. 2022;6:100416. https://doi.org/10.1016/j.padiff.2022.100416.

[10] Kudryashov NA. Exact solutions of the equation for surface waves in a convecting fluid. Appl Math Comput 2019;344: 97-106.

[11] Rezazadeh H, Tariq H, Eslami M, Mirzazadeh M, Zhou Q. New exact solutions of nonlinear conformable time-fracional Phi-4 equation. Chin J Phys 2018;56:2805-16.

[12] Tang B, Wang X, Wei L, Zhang X. Exact solutions of fractional heat-like and wave-like equations with variable coefficients. Int J Numer Methods Heat Fluid Flow 2014;24:455-67.

[13] Erinle-Ibrahim LM, Adewole AI, Loyinmi C, Sodeinde OK. AN optimization scheme using linear programming in a production line of rite foods limited ososa. FUDMA J. Sci. 2020;4:502-10.

[14] Odulaja DO, Erinle-Ibrahim LM, Loyinmi AC. Numerical computation and series solution for mathematical model of Fig. 4. Solution plot for case 4 at a 1, b ¼ 1. HIV/AIDS Computation and series solution for mathetical model of HIV/AIDS, online). Scienpress Ltd; 2013.

[5] Morenikeji E-IL, Babajide AO, Oluwatobi IK. Application of homotopy perturbation method to the mathematical modelling of temperature rise during microwave hyper- thermia. FUDMA J. Sci. 2021;5:273-82.

[6] Babajide AO, Oluwatobi IK. On the elzaki substitution and homotopy pertubation methods for solving partial differen- tial equation involving mixed partial derivatives. FUDMA J. Sci. 2021;5:159-68.

[7] Erinle-Ibrahim Latifat M, Oluwatobi IK, Sulola Abigail I. Mathematical modelling of the transmission dynamics of malaria infection with optimal control. Kathmandu Univ J Sci Eng Technol 2021;15.

[8] Akinfe KT, Loyinmi AC. Stability analysis and semi-analytic solution to a SEIR-SEI Malaria transmission model using He's variational iteration method. 2020.

[9] Lawal OW, Loyimi AC. Application of new iterative method for solving linear and nonlinear initial boundary value prob- lems with non local conditions. Sci World J 2019;14:100-4.

[10] Lawal OW, Loyimi AC. Erinle-ibrahim, algorithm for solving a generalized hirota-satsuma coupled KDV equation using homotopy perturbation transform method. Sci World J 2018;13. www.scienceworldjournal.org.

[15] Lawal OO, Loyinmi AC, Sowunmi OS. Homotopy pertur- bation algorithm using laplace transform for linear and nonlinear ordinary delay differential equation. J. Niger. Assoc. Math. Phys. 2017;41:27-34.

[16] Alesemi M, Iqbal N, Hamoud AA. The analysis of fractional- order proportional delay physical models via a novel trans- form. Complexity 2022;2022. https://doi.org/10.1155/2022/2431533.

[17] Suleman M, Elzaki T, Wu Q, Anjum N, Rahman JU. New application of Elzaki projected differential transform method. J Comput Theor Nanosci 2017;14:631-9.

[18] Mohanty RK, Sharma S. A new high-accuracy method based on off-step cubic polynomial approximations for the solution of coupled Burgers' equations and BurgerseHuxley equation. Eng Comput 2021;37:3049-66.

[19] Yasmin H, Iqbal N. A comparative study of the fractional- order nonlinear system of physical models via analytical methods. Math Probl Eng 2022;2022. https://doi.org/10.1155/2022/7488996.

[20] Akinfe TK, Loyinmi AC. A solitary wave solution to the generalized Burgers-Fisher’s equation using an improved differential transform method: a hybrid scheme approach. Heliyon 2021;7:-07001.

[21] Loyinmi AC, Akinfe TK. An algorithm for solving the BurgerseHuxley equation using the Elzaki transform. SN Appl Sci 2020;2:1-17.

[22] Loyinmi AC, Akinfe TK. Exact solutions to the family of Fisher's reaction-diffusion equation using Elzaki homotopy transformation perturbation method. Eng. Reports. 2020;2: -12084.

[23] Nourazar SS, Soori M, Nazari-Golshan A, Nourazar SS, Soori M, Nazari-Golshan A. On the exact solution of bur- gers-huxley equation using the homotopy perturbation method. J Appl Math Phys 2015;3:285-94. https://doi.org/10.4236/JAMP.2015.33042.

[24] Alomari AK, Noorani MSM, Nazar R. Solutions of heat-like and wave-like equations with variable coefficients by means of the homotopy analysis method. Chin Phys Lett 2008;25: 589-92. https://doi.org/10.1088/0256-307x/25/2/064.

[25] Darvishi MT, Kheybari S, Khani F. Spectral collocation method and Darvishi's preconditionings to solve the gener- alized Burgers-Huxley equation. Commun Nonlinear Sci Numer Simul 2008;13:2091-103. https://doi.org/10.1016/J.CNSNS.2007.05.023.

[26] Zhu M, Zhu M. Solving the burgers-huxley equation by G’/ G expansion method. J Appl Math Phys 2016;4:1371-7. https://doi.org/10.4236/JAMP.2016.47146.

[27] Molabahrami A, Khani F. The homotopy analysis method to solve the BurgerseHuxley equation. Nonlinear Anal R World Appl 2009;10:589-600. https://doi.org/10.1016/J.NONRWA.2007.10.014.

[34] Lawal OW, Loyinmi AC, Ayeni BO. Laplace homotopy perturbation method for solving coupled system of linear and nonlinear partial differential equationvol. 46; 2019. p. 83-91.

[35] Lawal OW, Loyinmi AC, Hassan AR. Finite difference solu- tion for magnetohydrodynamics thin film flow of a third- grade fluid down inclined plane with ohmic heating. Abacus 2019;46:92-7.

[36] Lawal OW, Loyinmi AC, Aruba DA. Approximate solutions of higher dimensional linear and nonlinear initial boundary valued problems using new iterative method, J Niger. Assoc Math Phys 2017;41:35-40.

[37] Agbomola JO, Loyinmi AC. Modelling the impact of some control strategies on the transmission dynamics of Ebola virus in human-bat population: an optimal control analysis. Heliyon 2022;8:-12121.

[38] Loyinmi AC, Idowu KO. Semi-analytic approach to solving rosenau-hyman and korteweg-de vries equations using integral transform. Tanzan J Sci 2023;49:26-40.

[39] Erinle-Ibrahim LM, Idowu KO. Mathematical modelling of pneumonia dynamics of children under the age of five. Abacus (Mathematics Sci. Ser. 2021;48.

[40] Idowu KO, Loyinmi AC. Impact of contaminated surfaces on the transmission dynamics of corona virus disease (Covid- 19). Biomed. J. Sci. Tech. Res. 2023;51:42280e94. https://doi.org/10.26717/BJSTR.2023.51.008046.

[41] Oluwatobi IK, Chris LA. Qualitative analysis of the trans- mission dynamics and optimal control of covid-19 (preprint). 2023.

[42] Malfliet W, Hereman W. The tanh method: I. Exact solutions of nonlinear evolution and wave equations. Phys Scripta 1996;54:563.

[43] Loyinmi AC, Oredein AI. The unsteady variable viscosity free convection flow on a porous plate. J. Niger. Assoc. Math. Phys. 2011;19:229 232.

[44] Lawal OW, Loyinmi AC. Magnetic and porosity effect on MHD flow of a dusty visco-elastic fluid through horizontal plates with heat transfer. J Nig Assoc Math Phys 2012;21:95-104.

[45] Agbomola J, Loyinmi A. A mathematical model for the dynamical behavior of ebola virus transmission in human- bat population: implication of immediate discharge of recovered individuals. 2022.

[46] Loyinmi AC, Oredein AI, Prince SU. Homotopy Adoman decomposition method for solving linear and nonlinear partial differential equations. Tasued J. Pure Appl. Sci. 2018;1:254-60.

[47] Lawal OW, Loyinmi AC. The effect of Magnetic field on MID Viscoelastic flow and heat transfer over a stretching sheet. Pioneer J. Adv. Appl. Math. 2011;3:83-90.

[48] Lawal OW, Loyinmi AC. Oscillating flow on a visco-elastic fluid under exponential pressure gradient with heat transfer. Pioneer J. Adv. Appl. Math. 2011;3:33-82.

[49] Akinfe KT. A reliable analytic technique for the modified proto- typical KelvineVoigt viscoelastic fluid model by means of the hyperbolic tangent function. Partial Differ. Equations Appl. Math. 2023;7:100523. https://doi.org/10.1016/J.PADIFF.2023.100523.

[50] Erinle-Ibrahim LM, Adebimpe O, Lawal WO, Agbomola JO. A mathematical model and sensitivity analysis of Lassa fever with relapse and reinfection rate. Tanzan J Sci 2022;48:414-26.

[51] Chin PWM. The analysis of the solution of the Burgers - Huxley equation using the Galerkin method. 2023. p. 2787-807. https://doi.org/10.1002/num.22987.

[52] Temam R, Temam R. Infinite-dimensional dynamical sys- tems in mechanics and physics. xvi 500pp., 応用数理 Springer 1988;1(1991):350-1.

[53] Evans LC. Partial differential equations graduate studies in mathematicsvol. 19. (American Mathematical Society: Prov- idence, Rhode Island); 1998.

[54] Yefimova OY, Kudryashov NA. Exact solutions of the Bur- gers-Huxley equation. J Appl Math Mech 2004;3:413-20.

[55] Tomasiello S. Numerical solutions of the BurgerseHuxley ation by the IDQ method. Int J Comput Math 2010;87:129-40.