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

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.


65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
Full Text: DOI EuDML


[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 · doi:10.1016/j.apnum.2007.07.001
[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 · doi:10.1016/j.amc.2006.10.008
[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 · doi:10.1016/S0378-4754(01)00421-9
[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 · doi:10.1137/0915089
[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 · doi:10.1016/S0168-9274(96)00033-5
[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 · doi:10.1016/S0096-3003(02)00290-4
[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 · doi:10.1007/s00365-006-0645-4
[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 · doi:10.1016/j.mcm.2006.10.008
[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 · doi:10.1016/j.jfranklin.2005.11.002
[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 · doi:10.1080/002077201300080910
[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 · doi:10.1080/00207160008805011
[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 · doi:10.1137/0916006
[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 · doi:10.1137/S0036142999361608
[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 · doi:10.1016/0893-9659(92)90014-Z
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