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Pseudospectral penalty method for fourth order partial differential equations
|關鍵字:||擬譜法;補償法;四次微分;spectral method;penalty method;fourth order derivative||引用:|| T.F. Ma, J. Da Silva, “Iterative solutions for a beam equation with nonlinear boundary conditions of third order,” Appl. Math. and Comput., vol. 159, no. 1, pp. 11-18, Nov. 2004.  M.M. Chawla, C.P. Katti, “Finite difference methods for two point boundary value problems involving higher order differential equations,” BIT, vol. 19, no. 1, pp. 27-33, Mar. 1979.  E. Doedel, “Finite difference collocation methods for nonlinear two point boundary value problems,” SIAM Journal of Numerical Analysis, vol. 16, no. 2, pp. 173-185, Apr. 1979.  M.A. Noor, S.T. Mohyud-Din, “An efficient method for fourth order boundary value problems,” Computers and Mathematics with Applications, vol. 54, no. 7-8, pp. 1101-1111, Oct. 2007.  Fazal-i-Haq, “Numerical solution of fourth order boundary value problems using Haar wavelets,” Applied Mathematical Sciences, Vol. 5, no. 63, pp. 3131 - 3146, 2011.  G. Strang, Computational science and engineering. Blacksburg: Wiley and Sons, 2007.  K.N.S. Kasi Viswanadham, Y.S. Raju, “Cubic b-spline collocation method for fourth order boundary value problems,” International Journal of Nonlinear Science, vol. 14, No.3, pp. 336-344, 2012.  B. Hossain, S. Islam, “Numerical solutions of general fourth order two point boundary value problems by Galerkin method with Legendre polynomials,” Dhaka Univ. J. Sci., vol. 62, no. 2, pp. 103-108, July 2014.  J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral methods for time dependent problems. UK: Cambridge Univeraity Press, 2007.  C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang Jr, Spectral methods in fluid dynamics, Springer Series in Computational Physics, Springer-Verlag, New York, 1988.  D. Funaro, D. Gottlieb, “Convergence results for pseudospectral approximations of hyperbolic systems by a penalty type boundary treatment,” Math. Comput., vol. 57, no. 196, pp. 585-596, Oct. 1991.  M.H. Carpenter, D. Gottlieb, “Spectral methods on arbitrary grids,” Journal of Computational Physics, vol. 129, no. 1, pp. 74-86, Nov. 1996.  J.S. Hesthaven, “Spectral penalty methods,” Appl. Numer. Math., vol. 33, no. 1-4, pp. 23-41, May 2000.  M. Y. Hussaini, C.L. Streett, T.A. Zang, “Spectral methods for partial differential equations,” NASA Langley Research Center, United States, 1984.  D. Funaro, D. Gottlieb, “A new method of imposing boundary conditions in pseudospectral approximations of hyperbolic equations,” Math. of Comput., vol. 51, no. 184, pp. 599-613, Oct. 1998.  B.C. Smith, Eigenvalues and eigenvectors of the discrete laplacian, 2012.  S.H. Friedberg, A.J. Insel, L.E. Spence, Linear Algebra, 4th ed. Pearson, pp. 373, 2003.  A.J. Laub, Matrix analysis for Scientists and Engineers. SIAM, 2005.  R.E. Lynch, J.R. Rice, D.H. Thomas, “Direct solution of partial difference equations by tensor product methods,” Number. Math., vol. 6, pp. 185-199, Dec. 1964.  Tzyy-Leng Horng, Chun-Hao Teng, “An error minimized pseudospectral penalty direct Poisson solver,” Journal of Computational Physics, vol. 231, no.6, pp. 2498-2509, Mar. 2010.||摘要:||
In this paper, we discuss how to approximate the solution of a fourth derivative boundary value problem with common boundary conditions. We construct an invertible fourth order derivative operator by using pseudospectral method, penalty method and some matrix properties. Then we give some examples to test the convergence rate of the proposed method.
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