Study on the Applications of Semi-Analytical Method for the Construction of Numerical Solutions of the Burgers' Equation

Year 2022, Volume 5, Issue 3, 82 - 88, 30.09.2022

Mine BABAOĞLU

https://doi.org/10.32323/ujma.1173595

Abstract

In the present paper explores, the Burgers' Equation which is the considerable partial differential equation that emerges in nonlinear science. Also, Homotopy Analysis Method (HAM) has been implemented to Burgers' equation with given initial conditions. The efficieny of the proposed method is analyzed by using some illustrative problems. We are compared approximate solutions acquired via HAM with the exact solutions. As a result of comparisons, it is demonstrated that the gained solutions are in excellent agreement. Additionally, 2D-3D graphs and tables of attained results are drawn by means of a ready-made package program. The recent obtained results denote that HAM is extremely efficient technique. Nonlinear partial differential equations can be solved with the help of presented method.

Keywords

Burgers' Equation, Homotopy Analysis Method, Auxiliary parameter, Approximate solution

References

• [1] S. J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Champan & Hall/CRC Press, Boca Raton, 2003.
• [2] A. M. Wazwaz, Balkema Publishers, Partial Differential Equations: Methods and Applications, The Netherlands, 2002.
• [3] S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147(2) (2004), 499-513.
• [4] J. H. He, Comparison of Homotopy perturbation method and Homotopy analysis method, Appl. Math. Comput., 156(2) (2004), 527-539.
• [5] S. J. Liao, Comparison between the Homotopy analysis method and Homotopy perturbation method, Appl. Math. Comput., 169(2) (2005), 1186-1194.
• [6] M. Naim, Y. Sabbar, A. Zeb, Stability characterization of a fractional-order viral system with the non-cytolytic immune assumption, Mathematical Modelling and Numerical Simulation with Applications, 2(3) (2022), 164-176.
• [7] W. Wu, S. J. Liao, Solving solitary waves with discontinuity by means of the Homotopy analysis method, Chaos, Solitons & Fractals, 26 (2005), 177-185.
• [8] Z. Hammouch, M. Yavuz, N. O¨ zdemir, Numerical solutions and synchronization of a variable-order fractional chaotic system, Mathematical Modelling and Numerical Simulation with Applications, 1(1) (2021), 11-23.
• [9] S. Abbasbandy, The application of Homotopy analysis method to solve a generalized Hirota-Satsuma coupled KdV equation, Phys. Lett. A, 361 (2007), 478-483.
• [10] H. M. Baskonus, J. L. Garc´ıa Guirao, A. Kumar, F. S. Vidal Causanilles, G. Rodriguez Bermudez, Instability modulation properties of the (2 + 1)-dimensional Kundu-Mukherjee-Naskar model in travelling wave solutions, Mod. Phys. Lett. B, 35(13) (2021), 2150217.
• [11] B. Zogheib, E. Tohidi, H. M. Baskonus, C. Cattani, Method of lines for multi-dimensional coupled viscous Burgers’ equations via nodal Jacobi spectral collocation method, Phys. Scr., 96 (2021), 124011.
• [12] J. Nee, J. Duan, Limit set of trajectories of the coupled viscous Burgers’ equations, Appl. Math. Lett., 11(1) (1998), 57-61.
• [13] S. J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai, 1992.
• [14] A. M. Lyapunov, General Problem of the Stability of Motion (English translation), Taylor & Francis, London, 1992.
• [15] P. Veeresha, M. Yavuz, C. Baishya, A computational approach for shallow water forced Korteweg-De Vries equation on critical flow over a hole with three fractional operators, Int. J. Optim. Control: Theor. Appl., 11(3) (2021), 52-67.
• [16] Md. Fayz-Al-Asad, T. Oreyeni, M. Yavuz, P. O. Olanrewaju, Analytic simulation of MHD boundary layer flow of a chemically reacting upper-convected Maxwell fluid past a vertical surface subjected to double stratifications with variable properties, Eur. Phys. J. Plus, 137(7) (2022), 1-11.
• [17] M. Yavuz, Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8(1) (2018), 1-7.
• [18] P. Veeresha, A numerical approach to the coupled atmospheric ocean model using a fractional operator, Mathematical Modelling and Numerical Simulation with Applications, 1(1) (2021), 1-10.
• [19] S. Pak, Solitary wave solutions for the RLW equation by He’s semi inverse method, International Journal of Nonlinear Sciences and Numerical Simulation, 10(4) (2009), 505-508.
• [20] M. Yavuz, European option pricing models described by fractional operators with classical and generalized Mittag-Leffler kernels, Numer. Methods Partial Differ. Equ., 38(3) (2020), 434-456.
• [21] M. Yavuz, T. A. Sulaiman, F. Usta, H. Bulut, Analysis and numerical computations of the fractional regularized long-wave equation with damping term, Math. Methods Appl. Sci., 44(9) (2020), 7538-7555.
• [22] A. Yokus¸, M. Yavuz, Novel comparison of numerical and analytical methods for fractional Burger–Fisher equation, Discrete Contin. Dyn. Syst. -S, 14(7) (2021), 2591-2606.