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On the naturality of the iteration and recursive specifications. (English) Zbl 0784.68053

Summary: In [J. L. N. Freire and M. F. Aguado, Int. J. Comput. Math. 35, No. 1-4, 7-14 (1990; Zbl 0699.68107)] the naturality of level-of- sentence control structures, concatenation, conditional, and iterative recursion is given. In the present article, we emphasize, from a categorical semantics point of view following [E. G. Manes and M. A. Arbib, Algebraic approaches to program semantics, Springer Verlag, New York (1986; Zbl 0599.68008)], the iteration and some of its relations with general recursion at the unit level.

MSC:

68Q55 Semantics in the theory of computing
18C10 Theories (e.g., algebraic theories), structure, and semantics
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[1] Lunardi, A.; Sinestrari, E., \(C^α\)-regularity for non-autonomous linear integrodifferential equations of parabolic type, J. Differential Equations, 63, 88-116 (1986) · Zbl 0596.45019
[2] Chen, C.; Thomée, V.; Wahlbin, L. B., Finite element approximation of a parabolic integrodifferential equation with a weakly singular kernel, Math. Comp., 58, 587-602 (1992) · Zbl 0766.65120
[3] Thomée, V.; Zhang, N.-Y., Error estimates for semi-discrete finite element methods for parabolic integrodifferential equations, Math. Comp., 53, 121-139 (1989)
[4] Sloan, I. H.; Thomée, V., Time discretization of an integrodifferential equation of parabolic type, SIAM J. Numer. Anal., 26, 1052-1061 (1986) · Zbl 0608.65096
[5] LeRoux, M.-N.; Thomée, V., Numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal., 26, 1291-1309 (1989) · Zbl 0701.65091
[6] Lubich, C., Discretized fractional calculus, SIAM J. Math. Anal., 17, 704-719 (1986) · Zbl 0624.65015
[7] Lubich, L., Fractional linear multistep methods for Abel-Volterra integral equations of the second kind, Math. Comp., 45, 463-469 (1985) · Zbl 0584.65090
[8] Sanz-Serna, J. M., A numerical method for a partial integro-differential equation, SIAM J. Numer. Anal., 25, 319-327 (1988) · Zbl 0643.65098
[9] López-Marcos, J. C., A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal., 27, 20-31 (1990) · Zbl 0693.65097
[10] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[11] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0691.35001
[12] Lubich, C., Convolution quadrature and discretized operational calculus I, Numer. Math., 52, 129-145 (1988) · Zbl 0637.65016
[13] Mitrinovic, D. S., Analytic Inequalities (1970), Springer-Verlag: Springer-Verlag New York · Zbl 0199.38101
[14] Thomée, V., Galerkin finite element methods for parabolic problems, Springer Lecture Notes in Mathematics, 1054 (1984)
[15] Da, Xu, On the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel II: Nonsmooth initial data, Appl. Math. Comp., 58, 29-60 (1993) · Zbl 0782.65161
[16] Lin, Y.; Thomée, V.; Wahlbin, L. B., Ritz-Volterra projections to finite element spaces and applications to integro-differential and related equations, Technical Report (1989), MSI
[17] Henrici, P., Fast Fourier methods in computational complex analysis, SIAM Review, 21, 481-527 (1979) · Zbl 0416.65022
[18] Camino, P., A numerical method for a partial differential equation with memory, (Proceedings of the Ninth CEDYA (Congreso de Ecuaciones Diferenciales y Aplicaciones) (1987), Universidad de Valladolid: Universidad de Valladolid Spain), 107-112
[19] (Erdelyi, A., Higher Transcendetal Functions I (1953), McGraw-Hill: McGraw-Hill New York)
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