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A recursive formulation of collocation in terms of canonical polynomials. (English) Zbl 0797.65056
For boundary value problems of linear differential operators with polynomial coefficients two related but analytically different techniques are discussed: the collocation method and Ortiz’s recursive formulation of the tau method. First the authors show that it is possible to simulate with the tau method collocation approximants for any desired degree in terms of shifted canonical polynomials.
In the linear case a collocation approximants of order \(N\) by this approach is shown to require \(O(N)\) arithmetic operations while obtaining the same approximants by the direct approach involves \(O(N^ 3)\). Several linear and nonlinear numerical examples are given to illustrate the new approach.

MSC:
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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[1] Ascher, U. M., Mattheij, R. M. M., Russell, R. D.: Numerical solution of boundary value problems for ordinary differential equations. New Jersey: Prentice Hall 1988. · Zbl 0671.65063
[2] Collatz, L.: The numerical treatment of differential equations, 2nd edn. Berlin Göttingen Heidelberg: Springer 1960. · Zbl 0086.32601
[3] El-Daou, M. K., Namasivayam, S., Ortiz, E. L.: Differential equations with piecewise approximate coefficients: discrete and continuous estimation for initial and boundary value problems. Comput. Math. Appl.24, 33–47 (1992). · Zbl 0763.65053 · doi:10.1016/0898-1221(92)90005-3
[4] El-Daou, M. K., Ortiz, E. L.: A recursive formulation of Galerkin’s method based on the Tau Method. Res. Rep. Imperial College, London 1992. · Zbl 0760.65088
[5] El-Daou, M. K., Ortiz, E. L.: Biorthonormality, permanence and canonical polynomials of approximation methods for differential equations. Res. Rep. Imperial College, London 1992. · Zbl 0760.65088
[6] El-Daou, M. K., Ortiz, E. L.: Error analysis of the Tau Method: dependence of the error on the degree and on the length of the interval of approximation. Comput. Math. Appl.25, 33–45 (1992). · Zbl 0772.65054 · doi:10.1016/0898-1221(93)90310-R
[7] El-Daou, M. K., Ortiz, E. L.: Error analysis of the step by step Tau Method: sharp estimates with application to collocaton. Res. Rep. Imperial College, London 1992.
[8] El-Daou, M. K., Ortiz, E. L.: A note on Accurate estimations of the local truncation error of polynomial methods for differential equations, Appl. Math. Lett.5, 69–72 (1992). · Zbl 0760.65088 · doi:10.1016/0893-9659(92)90017-4
[9] El-Daou, M. K., Ortiz, E. L., Samara, L. H.: A unified approach to the Tau Method and Chebyshev series expansions techniques. Comput. Math. Appl.25, 73–82 (1992). · Zbl 0777.65051 · doi:10.1016/0898-1221(93)90145-L
[10] Funaro, D.: Computing the inverse of the Chebyshev collocation derivative. SIAM J. Sci. Stat. Comput.9, 1050–1057 (1988). · Zbl 0662.65024 · doi:10.1137/0909071
[11] Lanczos, C.: Trigonometric interpolation of empirical and analytical functions. J. Math. Phys.17, 123–199 (1938). · Zbl 0020.01301
[12] Lanczos, C.: Applied analysis. Englewood Cliffs: Prentice-Hall 1956. · Zbl 0111.12403
[13] Namasivayam, S., Ortiz, E. L.: Error analysis of the Tau Method: dependence of the approximation error on the choice of perturbation term, Comput. Maths Appl.25, 89–104 (1992). · Zbl 0769.65045 · doi:10.1016/0898-1221(93)90215-H
[14] Nørsett, S. P.: Collocation and perturbed collocation methods. In: Numerical analysis, Dundee 1979 (Watson, G. A., ed.), pp. 119–132. Berlin Heidelberg New York: Springer 1980 (Lecture Notes in Mathematics, Vol. 773).
[15] Nørsett, S. P.: Splines and collocation for ordinary initial value problems. In: Approximation theory and splines functions (Singh, P. et al., eds), pp. 397–417. Dordrecht: Reidel Publishing Co. 397–417, 1984. · Zbl 0559.34010
[16] Morris, A. G., Horner, T. S.: Chebyshev polynomials in the numerical solution of differential equations. Math. Comp.31, 881–891 (1977). · Zbl 0386.65040 · doi:10.1090/S0025-5718-1977-0443359-7
[17] Onumanyi, P., Ortiz, E. L.: Numerical solution of stiff and singularly perturbed boundary value problems with a segmented-adaptive formulation of the Tau Method. Math. Comp.43, 189–203 (1984). · Zbl 0574.65091 · doi:10.1090/S0025-5718-1984-0744930-9
[18] Ortega, J. M., Rheinboldt, W. C.: Iterative solution of nonlinear equations in several variables. New York: Academic Press 1970. · Zbl 0241.65046
[19] Ortiz, E. L.: The Tau method. SIAM J. Numer. Anal.6, 480–92 (1969). · Zbl 0195.45701 · doi:10.1137/0706044
[20] Ortiz, E. L.: Step by step Tau Method: piecewise polynomial approximations. Comput. Math. Appl.1, 381–392 (1975). · Zbl 0356.65006 · doi:10.1016/0898-1221(75)90040-1
[21] Ortiz, E. L.: On the numerical solution of nonlinear and functional differential equations with the Tau Method. In: Numerical treatment of differential equations in applications (Ansorge, R. R., Törnig, W., eds.), pp. 358–365. Berlin Heidelberg New York: Springer 1978.
[22] Ortiz, E. L., Pham Ngoc, D. A.: On the convergence of the Tau Method for nonlinear differential equations of Riccati’s type. Nonlin. Anal. Theory Meth. Appl.9, 53–60 (1985). · Zbl 0551.65050 · doi:10.1016/0362-546X(85)90052-5
[23] Ortiz, E. L., Samara, H.: An operational approach to the Tau method for the numerical solution of nonlinear differential equations. Computing27, 15–25 (1981). · Zbl 0449.65053 · doi:10.1007/BF02243435
[24] Vainikko, G.: On the stability and convergence of the collocation method. Differen. Uravnen.1, 244–254 (1965).
[25] Wright, K.: Some relationships between implicit Runge-Kutta, collocation and Lanczos \(\tau\)-methods and their stability properties. BIT10, 217–227 (1970). · Zbl 0208.41602 · doi:10.1007/BF01936868
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