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]**Yudell L. Luke,*The special functions and their approximations, Vol. I*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0241700****[4]**Günter Meinardus,*Approximation of functions: Theory and numerical methods*, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR**0217482****[5]**Eduardo L. Ortiz,*The tau method*, SIAM J. Numer. Anal.**6**(1969), 480–492. MR**0258287**, https://doi.org/10.1137/0706044**[6]**E. L. Ortiz,*On the numerical solution of nonlinear and functional differential equations with the tau method*, Numerical treatment of differential equations in applications (Proc. Meeting, Math. Res. Center, Oberwolfach, 1977) Lecture Notes in Math., vol. 679, Springer, Berlin, 1978, pp. 127–139. MR**515576****[7]**E. L. Ortiz,*Step by step Tau method. I. Piecewise polynomial approximations*, Computers and mathematics with applications, Pergamon, Oxford, 1976, pp. 381–392. MR**0464550****[8]**Hilmi Samara,*Resolución numérica de ecuaciones diferenciales*, Cuadernos del Instituto de Matemática “Beppo Levi” [Notes of the Beppo Levi Institute of Mathematics], vol. 10, Universidad Nacional de Rosario, Facultad de Ciencias Exactas e Ingeniera, Rosario, 1979 (Spanish). Formulación operacional del método tau. [Operational formulation of the tau method]; Notes edited by Graciela G. Garguichevich, Mirta B. Stampella and Aída Taiana and revised by E. L. Ortiz. MR**645356**

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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