UPSI Digital Repository (UDRep)
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UPSI Digital Repository (UDRep)
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| Abstract : Perpustakaan Tuanku Bainun |
| This study explores the application of the weighted average method for solving the Burger-Fisher equation, a nonlinear partial differential equation (PDE) of significant interest in various scientific disciplines. Nonlinear PDEs, such as the Burger-Fisher equation, are fundamental in describing complex physical, biological, and engineering phenomena but pose challenges for both analytical and numerical solutions. The weighted average method, known for its ability to converge rapidly to exact solutions, offers a promising approach for tackling such equations. By discretizing both spatial and temporal derivatives using a combination of forward, backward, and central differences, the method approximates solutions with high accuracy and stability. Conducting convergence and stability analyses, this study elucidates the computational requirements and performance characteristics of the weighted average method. Utilizing mathematical software like MATLAB and MAPLE, the method's implementation involves solving a tridiagonal matrix system at each time step. Comparison between numerical solutions obtained using the method and exact solutions demonstrates the method's accuracy, with negligible errors observed. Visual representations further illustrate the close agreement between the numerical and exact solutions, validating the method's reliability for practical applications. The study's findings underscore the practical utility of the weighted average method in solving the Burger-Fisher equation and similar nonlinear PDEs. Its ability to accurately approximate solutions while maintaining stability highlights its efficacy as a computational tool for addressing complex mathematical problems across diverse scientific and engineering fields. The study contributes to advancing the understanding and application of numerical methods for nonlinear PDEs, offering valuable insights for researchers and practitioners seeking precise and reliable solutions to complex mathematical models. Overall, the study emphasizes the importance of fine-tuning numerical parameters and leveraging computational resources to achieve optimal accuracy when utilizing the weighted average method for solving nonlinear PDEs.
Keywords: Burger-Fisher equation, Weighted Average Method, Nonlinear Partial Differential equation |
| References |
Iyanda, F.K., Akanbi, B.K. and Muazu, N. (2021). Numerical solution for nonlinear Burgers equation with source term. American International Jouranal of Sciences and Engineering Research, 4(1), 53-65 Jaiswal, S., Chapra, M., Das, S. (2019). Numerical solution of nonlinear partial differential equation for porous medium using operational matrices. Mathematics and Computers in Simulation. Volume 160, pp 138-154 \ Jiwari, R., Mittal, R.C., and Sharma, K.K (2013). A numerical scheme based on weighted average differential quadrature method for the numerical solution of Burgers equation. Applied Mathematics and Computation. Volume 219, Issue 12 Mendoza, J. and Muriel, C. (2021). New exact solutions for a generalized Burger-Fisher equation. Chaos, Solitions & Fractals, Volume 152 Moghim, M., Fatemeh, S.A. Hejazi (2007). Variation iteration method for solving generalized Burger-Fisher and Burger equations. Chaos, Solitions Fractals, pp 1756-1761 Mukundan, V. and Awasthi, A. (2016). Linearized implicit numerical method for Burger equation. Nonlinear Engineering, 5(4), pp 219-234. Olayiwola, M. (2014). An improved algorithm for the solution of generalized Burger-Fisher equation. Applied Mathematics, 5, pp 1609-1614 Perez, R.Y.M. (2021). Bifurcation and dynamics in hyperbolic Burger-Fisher equation. Master Thesis, Universidad Nacional Autonoma De Mexico Pirdawood, M.A. and Sabawi, Y.A. (2021). High-order solution of generalized Burger-Fisher equation using compact finite differences and DIRK methods. J. Phys.:Conf.Ser. 1999 012088. O. W Lawal, A. C Loyinmi, The effect of magnetic field on MHD viscoelastic flow and heat transfer over a stretching sheet. Pioneer J. Adv. Appl. Math. 3(2011), 83-90. Li, H. (2016). The numerical analysis of the weighted average method for heat conduction equation in one dimension. J. Comp. Methods in Sci. and Eng. A. C Loyinmi, A. L Ijaola, Fractional order model of dynamical behavior and qualitative analysis of Anthrax with infected vector and saturation. Preprints (2024). 2024030632. https://doi.org/10.20944/preprints202403.0632.v1 Aksan, E.N., Ozdes, A. and Ozis, T. (2006). A numerical solution of Burgers equation based on least squares approximation, Applied Mathematics and Computation. Volume 176, Issue 1, 270-279 L. M. Erinle-Ibrahim, A. I. Adewole, A. C. Loyinmi, O. K. Sodeinde, An optimization scheme using linear programming in a production line of rites food limited, Ososa. FUDMA J. Sci., 4 (2020) 502-510. O. K. Idowu, A. C. Loyinmi, Qualitative analysis of the transmission dynamics and optimal control of covid-19. EDUCATUM Journal of Science, Mathematics and Technology. 10(1) (2023) 54-70. https://doi.org/10.37134/ejsmt.vol10.1.7.2023 Cao, H.H., Liu, L.B., Zhang, Y. and Fu, S.M. (2011). A fourth-order method of convection-diffusion equations with Neumann boundary conditions. Applied Mathematics and Computation. 217(22). A. C. Loyinmi, S. O. Gbodogbe, Mathematical modeling and control strategies for Nipah virus transmission incorporating Bat – to – pig –to – human pathway. EDUCATUM Journal of Science, Mathematics and Technology. 11(1) (2024). 54-80. https://doi.org/10.37134/ejsmt.vol11.1.7.2024 Chen, X.Y (2007). Numerical methods for Burger-Fisher equation. Master Thesis, University of Aeronautics and Astronautics, China. A. C. Loyinmi, S. O. Gbodogbe, K. O. Idowu, On the interaction of the human immune system with foreign body: mathematical modeling approach. Kathmandu University Journal of Science, Engineering and Technology. 17(2023), No 2. 1-17. https://journals.ku.edu.np/kuset/article/view/137 K. O. Idowu, A. C. Loyinmi, Impact of contaminated surfaces on the transmission dynamics of corona virus disease (Covid-19). Biomed J. Sci. Tech. Res., 51(2023) 42280- 90. https://doi.org/10.26717/BJSTR 2023.51008046 Crank-Nicolson Method. Encyclopaedia of Mathematics. URL: http://encyclopaediaofmath.org/index.php?title=Crank-Nicolson_method$oldid=55358 K. O. Idowu, A. C. Loyinmi, The analytical solution of non-linear Burgers- Huxley equations using the Tanh method. Al-Bahir journal for Engineering and Pure Sciences, 3(2023), 68 – 77. https://doi.ord/10.55810/2312-5721.1038 Kocacoban, D., Koc, A.B., Kurnaz, A. and Keskin, Y. (2011). “A better approximation to the solution of Burger-Fisher equation”, in proceedings of the World Congress of Engineering (WCE 11), Vol.1, London, UK O. W. Lawal, A. C. Loyinmi, S. O. Sowumi, Homotopy perturbation algorithm using Laplace transform for linear and nonlinear ordinary delayed differential equation. J. Niger. Assoc. Math. Phys., 41(2017), 27- 34. Wazzan, L. (2009). A modified tanh-coth method for solving the general Burgers-Fisher and Kuramoto-Sivashinsky equations. Communication in Nonlinear Science and Numerical Simulation, Volume 14, Issue 6, pp 2642-2652. O. W. Lawal, A. C. Loyinmi, Application of new iterative method for solving linear and nonlinear initial boundary value problems with non-local conditions. Science World Journal, 14 (2019), 100-104. Jiang, L.U., Yu-Gui, G. and Shu-Jang, X. (2007). Some new exact solutions to the Burger- Fisher equation and generalized Burger-Fisher equation. China, Phys.Sci and IOP publishing Ltd. O.W. Lawal, A. C. Loyinmi, A. R. Hassan, Finite difference solution for Magneto hydrodynamics thin film flow of a third grade fluid down inclined plane with ohmic heating. J. Math. Assoc. Niger. 46 (2019), 92- 97. Wazwaz, A.M (2009). Nonlinear partial differential equations and solitary wave theory. Nonlinear Physical Science. Springer, Berlin. O. W. Lawal, A. C. Loyinmi, Magnetic and porosity effect on MHD flow of a dusty visco-elastic fluid through horizontal plates with heat transfer. J. Niger. Assoc. Math. Phys., 21(2012), 95- 104. Wazwaz, A.M (2005). The tanh method for generalized forms of nonlinear heat conduction and Burger-Fisher equation. Appl. Math. Comput., 169, pp 321-338. O. W. Lawal, A. C. Loyinmi, L. M. Erinle-Ibrahim, Algorithm for solving a generalized Hirota-Satsuma coupled KDV equation using homotopy perturbation transformed method. Science World Journal, 13 (2018), 23-28. Vinay, C., Ashish, A., and Simon, J. (2016). Numerical treatment of Burger-Fisher equation. Procedia Technology, Volume 26, pp 1217-1225. Mickens, R.E, Gumel, A.B (2002). Construction and analysis of a non-standard finite difference scheme for the Burger-Fisher equation. J.Sound Vib. 257(4), pp791-797. J. O. Agbomola, A. C. Loyinmi, Modelling the impact of some control strategies on the transmission dynamics of Ebola virus in human-bat population: An optimal control analysis. Heliyon, 8:e12121. https://doi.org/10.1016/j.heliyon.2022.e12121 Vinay, C., Ashish, A. and Simon, J. (2015). A numerical treatment of Fisher equation. Procedia Engineering, Volume 127. O. W. Lawal, A. C. Loyinmi, O. B. Ayeni, Laplace homotopy perturbation method for solving coupled system of linear and nonlinear partial differential equation. J. Math. Assoc. Niger., 46 (2019), 83-91. Thakar, S. and Wani, S. (2013). Weighted average method for one dimensional nonlinear Burgers equation. International Journal of Physical, Chemical and Mathematical Sciences. 2, pp 16-27. S. Lot, O. W. Lawal, A. C. Loyinmi, Magnetic field’s effect on two phase flow of Jeffery and non- Jeffery fluid with partial slip and heat transfer in an inclined medium. Al – Bahir Journal for Engineering and Pure Sciences, 4(1) (2024), 71-79. https://doi.org/10.55810/2313-0083.1054 Ismail, N.A., Kamal, R. and Rabboh, A.A.A. (2004). Adomian decomposition method for BurgerHuxley and Burger-Fisher equations. Applied Mathematics and Computation, Volume159, Issue 1, pp 291-301. A. C. Loyinmi, T. K. Akinfe, An algorithm for solving the Burgers-Huxley equation using the Elzaki transform. SN Applied Sciences. 2 (2020) 1-17. https://doi.org/10.1007/s42452- 019-1652-3. Javidi, M. (2006). Spectral collocation method for the solution of the generalized Burger- Fisher equation. Appl. Math. Comput. 174(1), pp 45-352. Ismail, H.N.A and Rabboh, A.A.A (2004). A restrictive Pade approximation for the solution of the generalized Fisher and Burger-Fisher equation. Appl. Math. Comput.,154, pp 203-210. A. C. Loyinmi, T. K. Akinfe, Exact solution to the family of Fisher’s reaction-diffusion equations using Elzaki homotopy transformation perturbation method. Eng. Reports, 2:e12084. https://doi.org/10.1002/eng2.12084 Marquez, A.P., de la Rosa, R., Garrido, T.M., Gandaris, M.L. (2023). Conservation laws and exact solutions for time-delayed Burgers-Fisher equations. Mathematics K. O. Idowu, T. G. Akinwande, I. Fayemi, U. M. Adam, A. C. Loyinmi, Laplace homotopy perturbation method (LHPM) for solving system of N-dimensional non-linear partial differential equation. Al-Bahir Journal for Engineering and Pure Sciences, 3(2023), 11-27. https://doi.org.10.55810/2313-0083.1031 Ravneet, K.S., Kumar, S. and Kukreja, V.K. (2021). Numerical approximation of generalized Burger-Fisher and generalized Burger-Huxley equation by compact finite difference method. Advances in Mathematical Physics. Vol.2021 A. C. Loyinmi, K. O. Idowu, Semi –analytical approach to solving Rosenau-Hyman and Korteweg-de Vries equations using integral transform. Tanzanian Journal of Science. 49 (2023), 26-40. https://doi.org/10.4314/tjs.v49il.3 J. Agbomola, A. Loyinmi, A mathematical model for the dynamical behavior of Ebola transmission in human-bat population: implication of immediate discharge of recovered individuals. Preprints 2022. : https://doi.org/10.21203/rs.3.rs-1399224/v1 Taigbemu, A.E. (1999). Burgers equation. In: The Green Element Method. Springer, Boston. Pp 195-216 A. C. Loyinmi, O. W. Lawal, The asymptotic solution for the steady variable-viscosity free convection flow on a porous plate. J. Niger. Assoc. Math. Phys., 19 (2011) 273- 276. Selvaraj, R., Swaminathan, V., Devi, A.D. and Krishnakumar, K. (2020). Exact solutions of time fractional generalized Burgers-Fisher equation using generalized Kudryashov method. A. C. Loyinmi , L. M. Erinle-Ibrahim, A. E. Adeyemi, The new iterative method (NIM) for solving telegraphic equation. J. Niger. Assoc. Math. Phys., 43 (2017) 31- 36. Soori, M. (2018). “The variational iteration method and the homotopy perturbation method to the exact solution of the generalized Burger-Fisher equation”. Calculus of variations and partial differential equation. Vol.5, no.8, pp 19-26 O. W. Lawal, A. C. Loyinmi, D. A. Arubi, Approximate solutions of higher dimensional linear and nonlinear initial boundary problems using new iterative method. J. Niger. Assoc. Math. Phys. 41(2017), 35- 40. A. C. Loyinmi, O. W. Lawal, D. O. Sottin, Reduced differential transform method for solving partial integro-differential equation. J. Niger. Assoc. Math. Phys., 43 (2017), 37-42. Kaya, D.S.M; Sayed, El (2004). A numerical simulation and explicit solutions of the generalized Burger-Fisher equation. Appl. Math. Comput., 152, pp 403-413. A. C. Loyinmi, A. I. Oredein, S. U. Prince, Homotopy adomian decomposition method for solving linear and nonlinear partial differential equations. Tasued J. Pure Appl. Sci. 1(2018), 254-260. Kumar, S., and Saha, R.S. (2021). Numerical treatment for Burger-Fisher and generalized Burger- Fisher equation. Math. Sci. 15, pp 21-28. A. C. Loyinmi, T. K. Akinfe, A. A. Ojo, Qualitative analysis and dynamical behavior of a Lassa haemorrhagic fever model with exposed rodents and saturated incidence rate. Scientific African, 14 (2021); e01028. https://doi.org/10.1016/j.sciaf.2021.e01028. C. E. Overton, R. R. Wilkinson, A. Loyinmi, J. C. Miller, K. J. Sharkey, Approximating quasi-stationary behaviour in network-based SIS dynamics. Bulletin of Mathematical Biology, 84 (2022) 1–32. Hajinezhad, H. (2022). A numerical approximation for the one dimensional Burger-Fisher equation. Asian Research Journal of Mathematics. 18(5), pp 22-30 A. C. Loyinmi, A. I. Oredein, The unsteady variable viscosity free convection flow on a porous plate. J. Niger. Assoc. Math. Phys. 19 (2011), 229-232. https://www.ajol.info/index.php/jonamp/article/view/91459. Malik, S.A., Qureshi, I.M., Amir, M. Malik, A.N., Haq, I. (2015). Numerical solution to generalized Burgers-Fisher equation using exp-function method hybridized with heuristic computation. PLoS ONE 10(3) O. W, Lawal, A. C Loyinmi, Oscillating flow on a visco-elastic fluid under exponential pressure gradient with heat transfer. Pioneer J. Adv. Appl. Math. 3(2011), 33-82. Javidi, M., Golbabai, A. (2009). A spectral domain decomposition approach for the generalized Burger-Fisher equation. Chaos Solitions Fract. 39, pp 385-392.
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