A SEMI ANALYTIC ITERATIVE METHOD FOR SOLVING TWO FORMS OF BLASIUS EQUATION
Keywords:
Semi analytic iterative method, Blasius equation, Padé approximantsAbstract
In this paper, a semi analytic iterative method (SAIM) is presented for solving two forms of Blasius equation. Blasius equation is a third order nonlinear ordinary differential equation in the problem of the two-dimensional laminar viscous flow over half-infinite domain. In this scheme, the first solution which is in a form of convergent series solution is combined with Padé approximants to handle the boundary condition at infinity. Comparison the results obtained by SAIM with those obtained by other method such as variational iteration method and differential transform method revealed the effectiveness of the SAIM.
References
Abbasbandy, S. (2007). A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos, Solitons & Fractals, 31(1), 257-260.
Adomian, G. (1994). Solving Frontier Problems Decomposition Method. Berlin. Springer.
Ahmad, H. (2018). Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order. Nonlinear Science Letter A, 9(1), 27-35.
Aghakhani, M., Suhatril, M., Mohammadhassani, M., Daie, M., & Toghroli, A. (2015). A simple modification of homotopy perturbation method for the solution of Blasius equation in semi-infinite domains. Mathematical Problems in Engineering, 2015. Article ID 671527, 7 pages.
Altaie, S. A., Jameel, A. F., & Saaban, A. (2019). Homotopy Perturbation Method Approximate Analytical Solution of Fuzzy Partial Differential Equation. International Journal of Applied Mathematics, 49(1).
Al-Jawary, M. A., Adwan, M. I., & Radhi, G. H. (2018). Three iterative methods for solving second order nonlinear ODEs arising in physics. Journal of King Saud University-Science. (in press)
Al-Jawary, M. A., & Al-Razaq, S. G. (2016). A semi analytical iterative technique for solving duffing equations. International Journal of Pure and Applied Mathematics, 108(4), 871-885.
AL-Jawary, M. A., & Raham, R. K. (2017). A semi-analytical iterative technique for solving chemistry problems. Journal of King Saud University of Science, 29(3), 320-332.
Al-Jawary, M. A. (2017). A semi-analytical iterative method for solving nonlinear thin film flow problems. Chaos, Solitons & Fractals, 99, 52-56.
Al-Jawary, M., & Hatif, S. (2017). A semi-analytical iterative method for solving differential algebraic equations. Ain Shams Engineering Journal. 9(4),2581-2586
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. Computers & Mathematics with Applications, 76(1), 155-171.
AL-Jawary, M. A., Radhi, G. H., & Ravnik, J. (2017). Semi-analyical method for solving Fokker-Planck’s equations. Journal of the Association of Arab Universities for Basic and Applied Sciences. 24, 254-262.
Asaithambi, A. (2016). Numerical solution of the Blasius equation with Crocco-Wang transformation. Journal of Applied Fluid Mechanics, 9(5), 2595-2603.
Bakodah, H. O., Al Qarni, A. A., Banaja, M. A., Zhou, Q., Moshokoa, S. P., & Biswas, A. (2017). Bright and dark
Thirring optical solitons with improved adomian decomposition method. Optik, 130, 1115-1123.
Biazar, J., & Montazeri, R. (2019). Optimal Homotopy Asymptotic and Multistage Optimal Homotopy Asymptotic Methods for Solving System of Volterra Integral Equations of the Second Kind. Journal of Applied Mathematics, 2019 Article ID 3037273, 17 pages.
Bougoffa, L., & Wazwaz, A. M. (2015). New approximate solutions of the Blasius equation. International Journal of Numerical Methods for Heat & Fluid Flow, 25(7), 1590-1599.
Boyd, J. P. (1997). Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain. Computers in Physics, 11(3), 299-303.
Chirita, A., Ene, H., & Nicolescu, B. N. (2012). On some application of the Blasius problems. Bulletin of the Transilvania University of Brasov. Mathematics, Informatics, Physics. Series III, 5, 93-101.
Ertürk, V. S., & Momani, S. (2008). Numerical solutions of two forms of Blasius equation on a half-infinite domain. Journal of Algorithms & Computational Technology, 2(3), 359-370.
Fazio, R. (2016). A non-iterative transformation method for Blasius equation with moving wall or surface gasification. International Journal of Non-Linear Mechanics, 78, 156-159.
Fernández, F. M. (2009). On some approximate methods for nonlinear models. Applied Mathematics and Computation, 215(1), 168-174.
Hashim, I. (2006). Comments on “A new algorithm for solving classical Blasius equation” by L. Wang. Applied Mathematics and Computation, 176(2), 700-703.
Haq, F., Shah, K., Khan, A., Shahzad, M., & Rahman, G. (2017). Numerical solution of fractional order epidemic model of a vector born disease by Laplace Adomian decomposition method. Punjab University Journal of Mathematics, 49(2), 13-22.
He, J. H. (2000). Variational iteration method for autonomous ordinary differential systems. Applied Mathematics and Computation, 114(2-3), 115-123.
He, J. H. (1999). Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 178(3-4), 257-262.
He, J. H. (2003). A simple perturbation approach to Blasius equation. Applied Mathematics and Computation, 140(2-3), 217-222.
Jameel, A.F, Saaban, A., Akhadkulov, H and Alipiah, F. M. (2019). Homotopy Perturbation Method (HPM) for Solving Nonlinear Volterra Hammerstein Integral Equations. Journal of Engineering and Applied Sciences, 14: 5598-5601.
Kang, S. M., Nazeer, W., Tanveer, M., Mehmood, Q., & Rehman, K. (2015). Improvements in Newton-Rapshon method for nonlinear equations using modified Adomian decomposition method. International Journal of Mathematical Analysis, 9(39), 1919-1928.
Liao, S. (2004). On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147(2), 499-513.
Mabood, F., Lorenzini, G., Pochai, N., & Shateyi, S. (2018). Homotopy Analysis Method for Radiation and Hydrodynamic-Thermal Slips Effects on MHD Flow and Heat Transfer Impinging on Stretching Sheet. In Defect and Diffusion Forum (Vol. 388, pp. 317-327). Trans Tech Publications.
Maitama, S. (2016). Explicit solution of solitary wave equation with compact support by natural homotopy perturbation method. Journal MESA, 7(4), 625-635.
Martin, O. (2016). A modified variational iteration method for the analysis of viscoelastic beams. Applied Mathematical Modelling, 40(17-18), 7988-7995.
Najafi, E. (2018). Numerical q uasilinearization scheme for the integral equation form of the Blasius equation. Computational Methods for Differential Equations, 6(2), 141-156.
Ogunlaran, O. M., & Sagay-Yusuf, H. (2016). Adomain Sumudu transform method for the Blasius equation. British Journal of Mathematics & Computer Science, 14(3), 1.
Olumuyiwa, P., Folasade, A. A., Abiodun, O. F., Sambo, B., & Emmanuel, A. C. (2018). Multi-step Homotopy Analysis Method for Solving Malaria Model. Malaysian Journal of Applied Sciences, 3(2), 34-45.
Rafiq, M., Ahmad, H., & Mohyud-Din, S. T. (2017). Variational iteration method with an auxiliary parameter for solving Volterra’s population model. Nonlinear Sciences Letter A, 8(4), 389-396.
Sakar, M. G., & Ergören, H. (2015). Alternative variational iteration method for solving the time-fractional Fornberg–Whitham equation. Applied Mathematical Modelling, 39(14), 3972-3979.
Sajid, M., Abbas, Z., Ali, N., & Javed, T. (2015). A Hybrid Variational Iteration Method for Blasius Equation. Applications and Applied Mathematics, 10, 223-229.
Temimi, H., & Ansari, A. R. (2011a). A semi-analytical iterative technique for solving nonlinear problems. Computers & Mathematics with Applications, 61(2), 203-210.
Temimi, H., & Ansari, A. R. (2011b). A new iterative technique for solving nonlinear second order multi-point boundary value problems. Applied Mathematics and Computation, 218(4), 1457-1466.
Trujillo, J. A., Marin-Ramirez, A. M., & de Indias, C. (2018). An Asymptotic Solution to the Blasius Equation and Nonexistence of Periodic Orbits of the Blasius System. Journal of Engineering and Applied Sciences, 13(10), 3392-3395.
Wang, L. (2004). A new algorithm for solving classical Blasius equation. Applied Mathematics and Computation, 157(1), 1-9.
Wazwaz, A. M. (2007). The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Applied Mathematics and Computation, 188(1), 485- 491.
Yin, X. B., Kumar, S., & Kumar, D. (2015). A modified homotopy analysis method for solution of fractional wave equations. Advances in Mechanical Engineering, 7(12), 1-8.
Zheng, J., Han, X., Wang, Z., Li, C., & Zhang, J. (2017). A globally convergent and closed analytical solution of the Blasius equation with beneficial applications. AIP Advances, 7(6), 065311.
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