Abstract
In this research article, we present Galerkin weighted residual (WRM) technique to find the numerically approximated eigenvalues of the sixth order linear Sturm–Liouville problems (SLP) and Bénard layer problems. In the current method, Bernstein polynomials are being employed as the basis functions and precise matrix formulation is derived for solving eigenvalue problems. Numerical examples with homogeneous boundary conditions are considered to verify the efficiency and implementation of the proposed method. The numerical results offered in this paper are also compared with those investigated by other numerical/analytical methods and the computed eigenvalues are in good agreement.
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References
Chandrasekhar, S.: Hydrodynamic and Hydro Magnetic Stability. Clarendon Press, Oxford (1961). [Reprinted: Dover Books, New York (1981)]
Baldwin, P.: Asymptotic estimates of the eigenvalues of a sixth-order boundary-value problem obtained by using global phase-integral methods. Philos. Trans. Royal Soc. Lond. Ser. A 322(1566), 281–305 (1987)
Twizell, E.H., Boutayeb, B.: Numerical methods for the solution of special and general sixth-order boundary-value problems, with applications to Benard-Layer eigenvalue problems. Proc. R. Soc. A 431, 433–450 (1990)
Wang, Y., Zhao, Y.B., Wei, G.W.: A note on numerical solution of high order differential equations. J. Comput. Appl. Math. 159, 387–398 (2003)
Lesnic, D., Attili, B.: An efficient method for sixth order Sturm–Liouville problems. Int. Jpn. Sci. Technol. 2, 109–114 (2007)
Siyyam, H.I., Syam, M.I.: An efficient technique for finding the eigenvalue of sixth order Sturm–Liouville problems. Appl. Math. Sci. 5, 2425–2436 (2011)
Gheorghiu, C.I., Dragomirescu, F.I.: Spectral methods in linear stability. Application to thermal convection with variable gravity field. Appl. Numer. Math. 59, 1290–1302 (2009)
Bhatti, M.I., Bracken, P.: Solutions of differential equations in a Bernstein polynomial basis. J. Comput. Appl. Math. 205, 272–280 (2007)
Straughan, B.: The Energy Method, Stability, and Nonlinear Convection. Springer, Berlin (2003)
Shen, J., Tang. T.: Spectral and High Order Methods with Application, Mathematics Monoqraph Series 3. Science Press, Beijing (2006)
Finlayson, B.A.: The Method of Weighted Residual and Variational Principles with Application in Fluid Mechanics, Heat and Mass Transfer, vol. 87. Academic press, New York (1972)
Qian, W., Riedel, M.D., Rosenberg, I.: Uniform approximation and Bernstein polynomials with coefficients in the unit interval. Eur. J. Comb. 32, 448–463 (2011)
Wu, T.Y., Liu, G.R.: Application of generalized differential quadrature rule to sixth order differential equation. Commun. Numer. Methods Eng. 16, 777–784 (2000)
Gutierrez, R.H., Laura, P.A.A.: Vibrations of nonuniform rings studied by means of differential quadrature method. J. Sound Vib. 185(3), 507–513 (1995)
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The authors are grateful to the learned referees for their valued comments and suggestions to enhance the quality and improvement of the first version of this manuscript. The second author is indebted to Professor Dr. Amal Krishna Halder, Department of Mathematics, University of Dhaka, for his invaluable suggestions and kind assistance.
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Islam, M.S., Farzana, H. & Bhowmik, S.K. Numerical Solutions of Sixth Order Eigenvalue Problems Using Galerkin Weighted Residual Method. Differ Equ Dyn Syst 25, 187–205 (2017). https://doi.org/10.1007/s12591-016-0323-9
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DOI: https://doi.org/10.1007/s12591-016-0323-9