Numerical solution of systems of ordinary differential equations with the Tau method: an error analysis

Authors:
J. H. Freilich and E. L. Ortiz

Journal:
Math. Comp. **39** (1982), 467-479

MSC:
Primary 65L05

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669640-6

MathSciNet review:
669640

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Abstract | References | Similar Articles | Additional Information

Abstract: The recursive formulation of the Tau method is extended to the case of systems of ordinary differential equations, and an error analysis is given.

Upper and lower error bounds are given in one of the examples considered. The asymptotic behavior of the error compares in this case with that of the best approximant by algebraic polynomials for each of the components of the vector solution.

**[1]**J. H. Freilich & E. L. Ortiz,*Simultaneous Approximation of a Function and its Derivative with the Tau Method*, Conf. on Numerical Analysis, Dundee, 1975 and Imperial College, NAS Res. Report, 1975, pp. 1-45.**[2]**C. Lanczos, "Trigonometric interpolation of empirical and analytical functions,"*J. Math. Phys.*, v. 17, 1938, pp. 123-199.**[3]**Y. L. Luke,*The Special Functions and Their Approximations*, Vols. I and II, Academic Press, New York, 1969. MR**0241700 (39:3039)****[4]**G. Meinardus,*Approximation of Functions*:*Theory and Numerical Methods*, Springer-Verlag, Berlin, 1967. MR**0217482 (36:571)****[5]**E. L. Ortiz, "The Tau method,"*SIAM J. Numer. Anal.*, v. 6, 1969, pp. 480-492. MR**0258287 (41:2934)****[6]**E. L. Ortiz, "On the numerical solution of nonlinear and functional differential equations with the Tau method," in*Numerical Treatment of Differential Equations in Applications*(R. Ansorge and W. Törnig, Eds.), Springer-Verlag, Berlin and New York, 1978, pp. 127-139. MR**515576 (80c:65180)****[7]**E. L. Ortiz, "Step by step Tau method,"*Comput. Math. Appl.*, v. 1, 1975, pp. 381-392. MR**0464550 (57:4480)****[8]**E. L. Ortiz & H. Samara, "A new operational approach to the numerical solution of differential equations in terms of polynomials," in*Innovative Numerical Analysis for the Engineering Sciences*(R. Shaw and W. Pilkey, Eds.), The University Press of Virginia, Charlottesville, Va., 1980, pp. 643-652. MR**645356 (83i:65067)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669640-6

Keywords:
Initial value problems,
boundary value problems,
systems of ordinary differential equations,
simultaneous approximation of functions,
Tau method,
collocation methods

Article copyright:
© Copyright 1982
American Mathematical Society