Bugers Denkleminin İterarif Diferansiyel Quadrature Çözümü için Eğri Uydurmalı Başlangıç Tahmini

Abstract

Literatürde sunulan sayısal çalışmalara göre, Burgers Denkleminin (BE) çözümü yaygın olarak dt = 0.001 ve dt = 0.0001 için yapılmıştır. Bu çalışmada BE'nin sayısal çözümü, İteratif Diferansiyel Quadrature Yöntemi (I-DQM) kullanılarak dt = 0.01 olarak gerçekleştirilmiştir. İteratif yöntemlerin yakınsama hızı ve doğruluğu başlangıç tahmini değerine bağlıdır. Her Kısmi Diferansiyel Denklem (PDE), mühendislik bakış açısından bir veya daha fazla fiziksel problemi tanımlar. Önceki iteratif çalışmalardan farklı olarak, bu çalışmada, eğri uydurma kullanılarak tartışılan problemin fiziksel doğasına uygun bir başlangıç tahmini değeri kullanılmıştır. Elde edilen sonuçların mutlak hata analizi, önceki bazı çalışmalarla karşılaştırılmak üzere yapılmıştır. Karşılaştırmaların sonucu, eğri uydurmalı başlangıç tahmini ile I-DQM kullanılarak diğer çalışmalardan daha doğru sonuçların ve daha hızlı çözümün elde edilebileceğini göstermektedir.

Keywords

• Burgers denklemi
• iteratif diferansiyel quadrature metodu
• başlangıç tahmini
• newton-raphson metodu

Curve Fitting Initial Guess for Iterative Differential Quadrature Solution of Burgers Equation

Abstract

According to presented numerical studies in the literature, the solution of Burgers Equation (BE) performed for dt=0.001 and dt=0.0001 commonly. In this study, numerical solution of BE carried out by using the Iterative Differential Quadrature Method (I-DQM), as dt=0.01. Convergence speed and accuracy of iterative methods depends on the initial guess. Every Partial Differential Equation (PDE) describes one or more than one physical problems from the perspective of the engineering view. Unlike the previous iterative studies, in this work, an initial guess value is used in accordance with the physical nature of the discussed problem by using curve fitting. Absolute error analysis of obtained results performed for comparison with some previous studies. The consequence of comparisons shows that more accurate results and faster solution than other studies could be obtained by using I-DQM with curve fitting initial guess.

Keywords

• Burgers equation
• iterative differential quadrature method
• initial guess
• newton-raphson method

References

1. [1] Bateman H., “Some Recent Researches on The Motion of Fluids”, Monthly Weather Review, 40: 163-170 (1915).
2. [2] Burgers J. M., “Mathematical examples illustrating relations occurring in the theory of turbulent fluid motion”, Trans. Roy. Neth. Acad. Sci., 17 (2): 1-5 (1939).
3. [3] Burgers J. M., “A Mathematical Model Illustrating the Theory of Turbulence”, Advances in Applied Mechanics, 1: 171-199 (1948).
4. [4] Hopf E., “The partial differential equation ut + uux = μxx”, Pure and Applied Mathematics, 3(3): 201-230 (1950).
5. [5] Cole J. D., “On a quasi-linear parabolic equation occurring in aerodynamics”, Quart. Appl. Math., 9: 225-236 (1951).
6. [6] Caldwell J. and Smith P., “Solution of Burgers' equation with a large Reynolds number”, Appl. Math. Model. 6: 381-385 (1982).
7. [7] Evans D. J. and Abdullah A. R., “The group explicit method for the solution of Burgers' equation”, Quart. Appl. Math., 30: 239-253 (1984).
8. [8] Mittal R. C. and Signnal P., “Numerical solution of Burgers' equation”, Commun. Numer. Methods Eng, 9: 397-406 (1993).

9. [9] Öziş T. and Özdeş A., “A direct variational methods applied to Burgers' equation”, J. Comput. Appl. Math, 71: 163-175 (1996).
10. [10] Kutluay S. A., Bahadir A. R. and Özdeş A., “Numerical solution of one-dimensional Burgers equation: explicit and exact-explicit finite difference methods”, J. Comput. Appl. Math, 103: 251-261 (1999).
11. [11] Kutluay S. A. and Esen A., “A linearized numerical scheme for Burgers-like equations”, Appl. Math. Comput, 156: 295-305 (2004).
12. [12] Liao W., “An implicit fourth-order compact finite difference scheme for one-dimensional Bugers equation”, Appl. Math. Comput, 206: 755-764 (2008).
13. [13] Öziş T. and Erdoğan U., “An exponentially fitted method for solving Bugers equation”, Int. J. Numer. Meth. Engng. 79: 696-705 (2009).
14. [14] Gao Q. and Zou M.Y., “An analytical solution for two and three dimensional nonlinear Burgers’ equation”, Applied Mathematical Modelling. 45: 255–270 (2017).
15. [15] Mittal R. and Jiwari R., “A differential quadrature method for numerical solutions of Burgers'‐type equations”, International Journal of Numerical Methods for Heat & Fluid Flow, 22(7): 880-895 (2012).
16. [16] Gupta S. ad Ray S., “Comparison between homotopy perturbation method and optimal homotopy asymptotic method for the soliton solutions of Boussinesq–Burger equations”, Computers and Fluids, 103: 34-41 (2014).
17. [17] Nascimento A., Silveria-Neto F. P. M. A. and Padilla E. L. M., “A comparison of Fourier pseudospectral method and finite volume method used to sol, the Burgers equation”, J Braz. Soc. Mech. Sci. Eng. 36: 737-742 (2014).
18. [18] Jiwari R., “A hybrid numerical scheme for the numerical solution of the Burgers’ equation”, Computer Physics Communications, 188: 59-67 (2015).
19. [19] Jiwari R., “A Haar wavelet quasilinearization approach for numerical simulation of Bugers equation”, Computer Physics Communications, 193: 2413-2423 (2012).
20. [20] Tamsir M., Srivastava V. K. and Jiwari R., “An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation”, Applied Mathematics and Computation. 290 111-124 (2016).
21. [21] Girgin Z., Aysal F. E., Bayrakçeken H., “Numerical Solution of the Burgers Equation by Using Iterative DQM”, 5th International Symposium on Innovative Technologies in Engineering and Science, 268-277 (2017).
22. [22] Civalek Ö, Kiracioglu O, “Free vibration analysis of Timoshenko beams by DSC method”, Int J Numer Methods Biomed Eng, 26(12): 1890–1898 (2010).
23. [23] Civalek, Ö. and Yavas A. “Large deflection static analysis of rectangular plates on two parameter elastic foundations”, International Journal of Science and Technology, 1(1): 43–50 (2006).
24. [24] Mercan K, Demir Ç. and Civalek Ö., “Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique”. Curved Layer Struct, 3(1): 82-90 (2016).
25. [25] Bellman R. and Casti J., “Differential quadrature and long-term integration”, J. Math. Anal. Appl., 34(2): 235-238 (1971).
26. [26] Bellman R., Kashef B. G. J. “Casti, Differential quadrature: A technique for the rapid solution of nonlinear partial differential equations”, Journal of Computational Physics, 40(1): 40-52 (1972).
27. [27] Bellman R., Kashef B. G, Lee E. and Vasudevan S., R., “Differential quadrature and splines”, Computers and Mathematics with Applications. 1(3-4): 371-376 (1975)
28. [28] Quan J. R. and Chang C. T., “New Insights in Solving Distributed System Equations by The Quadrature Methods – I”, Computational Chemical Engineering. 13: 779-788 (1989).
29. [29] Quan J. R. and Chang C. T., “New Insights in Solving Distributed System Equations by The Quadrature Methods – II”, Computational Chemical Engineering. 13: 1017-1024 (1989).
30. [30] Shu B. and Richards E., “High Resolution of Natural Convection in A Square Cavity by Generalized Differential Quadrature”, Proceeding of 3rd Conference on Advanced in Numerical Methods in Engineering: Theory and Application, Swansea, UK, 2: 978-985 (1990).
31. [31] Shu B., “Generalized Differential-Integral Quadrature and Application to The Simulation of Incompressible Viscous Flows Including Paralel Computation”, PhD Dissertation. Uni,rsity of Glosgow, UK., (1991).
32. [32] Doğan A., “A Galerkin finite element approach to Burgers' equation”, Appl. Math. Comput. 157: 331-346 (2004).
33. [33] Kadalbajoo M. and Awasthi A., “A numerical method based on Crank–Nicolson scheme for Burgers’ equation”, Appl. Math. Comput, 182: 1430-1442 (2006).
34. [34] Xu M., Wang R.-H., Zhang J.-H. and Fang Q., “A novel numerical scheme for solving Burgers’ equation”, Appl. Math. Comput. 217: 4473-4482 (2011).
35. [35] Başhan A., “A numerical treatment of the coupled viscous Burgers’ equation in the presence of very large Reynolds number” Physica A: Statistical Mechanics and its Applications. 545: 123755 (2020).
36. [36] Ucar Y., Yağmurlu N. M. and Başhan A. "Numerical Solutions and Stability Analysis of Modified Burgers Equation via Modified Cubic B-Spline Differential Quadrature Methods." Sigma: Journal of Engineering & Natural Sciences. 37 (1): 129-142 (2019).
37. [37] Başhan, A,. Karakoç S. B. G and Geyikli T. "B-spline differential quadrature method for the modified Burgers' equation." Çankaya Üniversitesi Bilim ve Mühendislik Dergisi. 12(1): 001–013 (2015).