Öz
This paper presents three distinct approximate methods for solving Blasius Equation. The first method can be regarded as an improvement to a series solution of Blasius by means of Padè approximation. The second method is a famous type of weighted residual technique which is called Galerkin method after the famous Russian engineer and mathematician Boris Galerkin. The last method is a simple discrete, numerical technique. Additionally, in order to show the power of the last method, the Thomas-Fermi problem is solved using the same technique. Results obtained by all three methods are highly accurate in comparison with the Howarth’s solution and Bender’s solution.
Anahtar Kelimeler
Blasius equation perturbation technique Padè approximation weighted residual method Galerkin method Thomas Fermi equation
Kaynakça
- White F.M., Viscous Fluid Flow, Second Edition, McGraw Hill, Inc., p. 104, (1991).
- Schlichting, H., et al., Boudary Layer Theory, Springer, Newyork, (2000).
- Blasius H., Grenzschichten in Flu¨ssigkeiten mit kleiner Reibung, Z Math Phys., 56, 1–37, (1908).
- Datta B.K., Analitic solution for THE Blasius equation, Indian Jounal of Pure and Applied Mathematics, 34(2), 237-240, (2003).
- He, J.H., A simple perturbation approach to Blasius equation, Appl. Math. Comput., 140(2-3), 217–222, (2003).
- He, J.H., Approximate analitical solution of Blasius’ equation, Communications in Nonlinear Science & Numerical Simulation, 13(4), (1998).
- Wazwaz, A.M., The variational iteration method for solving two forms of Blasius equation on a half infinite domain, Appl. Math. Comput., 188(1), 485-491, (2007).
- Aiyesimi, Y.M. and Niyi, O.O., Computational analysis of the non-linear boundary layer flow over a flat plate using Variational Iterative Method (VIM), American Journal of Computational and Applied Mathematics, 1(2), 94-97, (2011).
- Fazio, R., Numerical transformation methods: Blasius problem and its variants. Appl. Math. Comput., 215(4), 1513–1521, (2009).
- Asaithambi, A., Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients, J. Comput. Appl. Math., 176(1), 203–214, (2005).
- Akgül A., A novel method for the solution of blasius equation in semi-infinite domains, An International Journal of Optimization and Control: Theories & Applications 7(2), 225-233, (2017).
- Howarth, L., Laminar boundary layers. In Handbuch der Physik (herausgegeben von S. Fl¨ugge), Bd. 8 1, Strmungsmechanik I (Mitherausgeber C. Truesdell), pages 264 350. Springer-Verlag, Berlin-Gottingen-Heidelberg, (1959).
- Parand, K., Dehghan, M., and Pirkhedri, A., Sinc collocation method for solving the Blasius equation, Phys. Lett. A, 373(44), 4060–4065, (2009).
- Yao, B., and Chen, J., A new analytical solution branch for the Blasius equation with a shrinking sheet, Appl. Math. Comput., 215(3), 1146–1153, (2009).
- Liao, S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, Internat. J. Non-Linear Mech., 34(4), 759–778, (1999).
- Gheorghiu, C.I., Laguerre collocation solutions to boundary layer type problems, Numer. Algor. 64, 385–401, (2012).
- Liao, S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, Internat. J. Non-Linear Mech., 34(4), 759–778, (1999).
- Lin, J., A new approximate iteration solution of Blasius equation, Commun. Nonlinear Sci. Numer. Simul., 4(2), 91–99, (1999).
- Yu, L.T., and Chen, C.K., The solution of the Blasius equation by the differential transformation method, Math. Comput. Modelling, 28(1), 101–111, (1998).
- Peker, H.A., Karaolu, O., and Oturan, G. The differential transformation method and Pade approximant for a form of Blasius equation, Math. Comput. Appl., 16(2), 507–513, (2011).
- Abbasbandy, S., A numerical solution of Blasius equation by Adomians decomposition method and comparison with homotopy perturbation method, Chaos, Solutions and Fractals, 3, 257-260, (2007).
- Wang L., A new algorithm for solving classical Blasius equation, Applied Mathematics and Computation, 157, 1–9, (2004).
- Tajvidi T., Razzaghi M., Dehghan M., Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate, Chaos, Solutions and Fractals, 35, 59–66, (2008).
- Baker, G. A. Jr. The theory and application of the pade approximant method. In Advances in Theoretical Physics, 1 (Ed. K. A. Brueckner). New York: Academic Press
- Finlayson, B.A., The Method of Weighted Residuals and Variational Principles With Applications in Fluid Mechanics. Academic Press, New York and London, (1972),
- Liao, S., Beyond Perturbation Introduction to The Homotopy Analysis Method, Part I, Chapman & Hall/CRC, (2004).
- Bender, C.M. and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, p. 25, (1978).
- Kobayashi, S., et al., Some coefficients of the TFD function, J. Phys. Soc. Jpn. 10, 759–765, (1955).
Öz
Bu makalede Blasius Denklemi’ni çözmek için üç farklı yaklaşık yöntem sunmaktadır. İlk yöntem Blasius’un seri çözümünün Padè yaklaşımı yardımı ile iyileştirilmesi olarak değerlendirilebilir. İkinci yöntem ünlü Rus mühendis ve matematikçi Boris Galerkin’e izafeten Galekin Metodu olarak adlandırılan bir ağırlıklı artık yöntemdir. Son yöntem ise basit, ayrık bir sayısal tekniktir. Ek olarak son yöntemin gücünü göstermek adına Thomas-Fermi Problemi de aynı teknik ile çözülmüştür. Her üç yöntem, sonuçlar Howarth’ın ve Bender’in çözümü ile kıyaslandığında, oldukça başarılı sonuç vermektedir.
Anahtar Kelimeler
Blasius denklemi pertürbasyon tekniği Padè yaklaşımı ağırlıklı artık yöntemler Galerkin metodu Thomas Fermi denklemi
Kaynakça
- White F.M., Viscous Fluid Flow, Second Edition, McGraw Hill, Inc., p. 104, (1991).
- Schlichting, H., et al., Boudary Layer Theory, Springer, Newyork, (2000).
- Blasius H., Grenzschichten in Flu¨ssigkeiten mit kleiner Reibung, Z Math Phys., 56, 1–37, (1908).
- Datta B.K., Analitic solution for THE Blasius equation, Indian Jounal of Pure and Applied Mathematics, 34(2), 237-240, (2003).
- He, J.H., A simple perturbation approach to Blasius equation, Appl. Math. Comput., 140(2-3), 217–222, (2003).
- He, J.H., Approximate analitical solution of Blasius’ equation, Communications in Nonlinear Science & Numerical Simulation, 13(4), (1998).
- Wazwaz, A.M., The variational iteration method for solving two forms of Blasius equation on a half infinite domain, Appl. Math. Comput., 188(1), 485-491, (2007).
- Aiyesimi, Y.M. and Niyi, O.O., Computational analysis of the non-linear boundary layer flow over a flat plate using Variational Iterative Method (VIM), American Journal of Computational and Applied Mathematics, 1(2), 94-97, (2011).
- Fazio, R., Numerical transformation methods: Blasius problem and its variants. Appl. Math. Comput., 215(4), 1513–1521, (2009).
- Asaithambi, A., Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients, J. Comput. Appl. Math., 176(1), 203–214, (2005).
- Akgül A., A novel method for the solution of blasius equation in semi-infinite domains, An International Journal of Optimization and Control: Theories & Applications 7(2), 225-233, (2017).
- Howarth, L., Laminar boundary layers. In Handbuch der Physik (herausgegeben von S. Fl¨ugge), Bd. 8 1, Strmungsmechanik I (Mitherausgeber C. Truesdell), pages 264 350. Springer-Verlag, Berlin-Gottingen-Heidelberg, (1959).
- Parand, K., Dehghan, M., and Pirkhedri, A., Sinc collocation method for solving the Blasius equation, Phys. Lett. A, 373(44), 4060–4065, (2009).
- Yao, B., and Chen, J., A new analytical solution branch for the Blasius equation with a shrinking sheet, Appl. Math. Comput., 215(3), 1146–1153, (2009).
- Liao, S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, Internat. J. Non-Linear Mech., 34(4), 759–778, (1999).
- Gheorghiu, C.I., Laguerre collocation solutions to boundary layer type problems, Numer. Algor. 64, 385–401, (2012).
- Liao, S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, Internat. J. Non-Linear Mech., 34(4), 759–778, (1999).
- Lin, J., A new approximate iteration solution of Blasius equation, Commun. Nonlinear Sci. Numer. Simul., 4(2), 91–99, (1999).
- Yu, L.T., and Chen, C.K., The solution of the Blasius equation by the differential transformation method, Math. Comput. Modelling, 28(1), 101–111, (1998).
- Peker, H.A., Karaolu, O., and Oturan, G. The differential transformation method and Pade approximant for a form of Blasius equation, Math. Comput. Appl., 16(2), 507–513, (2011).
- Abbasbandy, S., A numerical solution of Blasius equation by Adomians decomposition method and comparison with homotopy perturbation method, Chaos, Solutions and Fractals, 3, 257-260, (2007).
- Wang L., A new algorithm for solving classical Blasius equation, Applied Mathematics and Computation, 157, 1–9, (2004).
- Tajvidi T., Razzaghi M., Dehghan M., Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate, Chaos, Solutions and Fractals, 35, 59–66, (2008).
- Baker, G. A. Jr. The theory and application of the pade approximant method. In Advances in Theoretical Physics, 1 (Ed. K. A. Brueckner). New York: Academic Press
- Finlayson, B.A., The Method of Weighted Residuals and Variational Principles With Applications in Fluid Mechanics. Academic Press, New York and London, (1972),
- Liao, S., Beyond Perturbation Introduction to The Homotopy Analysis Method, Part I, Chapman & Hall/CRC, (2004).
- Bender, C.M. and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers. New York: McGraw-Hill, p. 25, (1978).
- Kobayashi, S., et al., Some coefficients of the TFD function, J. Phys. Soc. Jpn. 10, 759–765, (1955).
Ayrıntılar
| Birincil Dil | İngilizce |
|---|---|
| Bölüm | Araştırma Makalesi |
| Yazarlar | |
| Yayımlanma Tarihi | 29 Ekim 2018 |
| Gönderilme Tarihi | 24 Temmuz 2018 |
| Yayımlandığı Sayı | Yıl 2018 Cilt: 20 Sayı: 3 |