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Guimarães, Osvaldo; Piqueira, José Roberto C.; Netto, Marcio Lobo

Direct computation of operational matrices for polynomial bases. (English) Zbl 1198.65121

Math. Probl. Eng. 2010, Article ID 139198, 12 p. (2010).
Summary: Several numerical methods for boundary value problems use integral and differential operational matrices, expressed in polynomial bases in a Hilbert space of functions. This work presents a sequence of matrix operations allowing a direct computation of operational matrices for polynomial bases, orthogonal or not, starting with any previously known reference matrix. Furthermore, it shows how to obtain the reference matrix for a chosen polynomial base. The results presented here can be applied not only for integration and differentiation, but also for any linear operation.

Cited in 8 Documents

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations

Keywords:

boundary value problems; operational matrices; polynomial bases; integration; differentiation; linear operation
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\textit{O. Guimarães} et al., Math. Probl. Eng. 2010, Article ID 139198, 12 p. (2010; Zbl 1198.65121)
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References:

[1] E. H. Doha and A. H. Bhrawy, “Efficient spectral-Galerkin algorithms for direct solution of fourth-order differential equations using Jacobi polynomials,” Applied Numerical Mathematics, vol. 58, no. 8, pp. 1224-1244, 2008. · Zbl 1152.65112
[2] E. Babolian and F. Fattahzadeh, “Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration,” Applied Mathematics and Computation, vol. 188, no. 1, pp. 417-426, 2007. · Zbl 1117.65178
[3] E. M. E. Elbarbary, “Legendre expansion method for the solution of the second- and fourth-order elliptic equations,” Mathematics and Computers in Simulation, vol. 59, no. 5, pp. 389-399, 2002. · Zbl 1004.65120
[4] J. Shen, “Efficient spectral-Galerkin method. I. Direct solvers of second- and fourth-order equations using Legendre polynomials,” SIAM Journal on Scientific Computing, vol. 15, no. 6, pp. 1489-1505, 1994. · Zbl 0811.65097
[5] D. J. Higham, “Runge-Kutta type methods for orthogonal integration,” Applied Numerical Mathematics, vol. 22, no. 1-3, pp. 217-223, 1996. · Zbl 0866.65047
[6] C. Ke\csan, “Taylor polynomial solutions of linear differential equations,” Applied Mathematics and Computation, vol. 142, no. 1, pp. 155-165, 2003. · Zbl 1034.34009
[7] J. Obermaier and R. Szwarc, “Orthogonal polynomials of discrete variable and boundedness of Dirichlet kernel,” Constructive Approximation, vol. 27, no. 1, pp. 1-13, 2008. · Zbl 1132.41308
[8] I. Sadek, T. Abualrub, and M. Abukhaled, “A computational method for solving optimal control of a system of parallel beams using Legendre wavelets,” Mathematical and Computer Modelling, vol. 45, no. 9-10, pp. 1253-1264, 2007. · Zbl 1117.49026
[9] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, Academic Press, San Diego, Calif, USA, 4th edition, 1995. · Zbl 0970.00005
[10] F. Khellat and S. A. Yousefi, “The linear Legendre mother wavelets operational matrix of integration and its application,” Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 343, no. 2, pp. 181-190, 2006. · Zbl 1127.65105
[11] M. Razzaghi and S. Yousefi, “The Legendre wavelets operational matrix of integration,” International Journal of Systems Science, vol. 32, no. 4, pp. 495-502, 2001. · Zbl 1006.65151
[12] M. Sezer and C. Ke\csan, “Polynomial solutions of certain differential equations,” International Journal of Computer Mathematics, vol. 76, no. 1, pp. 93-104, 2000. · Zbl 0973.65055
[13] J. Shen, “Efficient spectral-Galerkin method. II. Direct solvers of second-and fourth-order equations using Chebyshev polynomials,” SIAM Journal on Scientific Computing, vol. 16, no. 1, pp. 74-87, 1995. · Zbl 0840.65113
[14] M. G. Armentano, “Error estimates in Sobolev spaces for moving least square approximations,” SIAM Journal on Numerical Analysis, vol. 39, no. 1, pp. 38-51, 2001. · Zbl 1001.65014
[15] J. P. Boyd, “Defeating the Runge phenomenon for equispaced polynomial interpolation via Tikhonov regularization,” Applied Mathematics Letters, vol. 5, no. 6, pp. 57-59, 1992. · Zbl 0760.65007
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