Algorithmic methods for Fuchsian systems of linear partial differential equations

Toshinori OAKU

doi:10.2969/jmsj/04720297

Toshinori OAKU

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JOURNAL FREE ACCESS

1995 Volume 47 Issue 2 Pages 297-328

DOI https://doi.org/10.2969/jmsj/04720297

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Published: 1995 Received: August 06, 1993 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: TITLE Details: Wrong : Algorithmic methods for Fuchisian systems of linear partial differential equations
Right : Algorithmic methods for Fuchsian systems of linear partial differential equations

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math., 26 (1973), 455-475.
2) J. E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979.
3) J. Briancon, Weierstrass préparé à la Hironaka, Astérisque, 7-8 (1973), 67-73.
4) B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math., 4 (1970), 374-383.
5) B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, Lecture Notes in Comput. Sci., 72, Springer, Berlin-Heidelberg-New York, 1979, pp. 3-21.
6) F. Castro, Calculs effectifs pour les idéaux d'opérateurs différentiels, Travaux en Cours, 24, Herman, Paris, 1987, pp. 1-19.
7) D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, (UndergraduateTexts in Math.), Springer, Berlin-Heidelberg-New York, 1992.
8) A. Galligo, Some algorithmic questions on ideals of differential operators, Lecture Notes in Comput. Sci., 204, Springer, Berlin-Heidelberg-New York, 1985, pp. 413-421.
9) M. Kashiwara, Systems of Microdifferential Equations, Progr. Math., 34, Birkhäuser, Boston-Basel-Stuttgart, 1983.
10) M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Lecture Notes in Math., 1016, Springer, Berlin-Heidelberg-New York, 1983, pp. 134-142.
11) M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations, III, Publ. Res. Inst. Math. Sci., Kyoto Univ., 17 (1981), 813-979.
12) M. Kashiwara and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106 (1977), 145-200.
13) Y. Laurent, and T. Monteiro Fernandes, Systèmes différentiels fuchsiens le long d'une sous-variété, Publ. Res. Inst. Math. Sci., Kyoto Univ., 24 (1988), 397-431.
14) Y. Laurent and P. Schapira, Images inverses des modules différentiels, Compositio Math., 61 (1987), 229-251.
15) T. Monteiro Fernandes, Formulation des valeurs au bord pour les systemès regulier, Compositio Math., 81 (1992), 121-142.
16) M. Noro and T. Takeshima, Risa/Asir-a computer algebra system, Proceedings of International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 1992, pp. 387-396.
17) M. Noumi, Wronskian determinants and the Grobner representation of a linear differential equation, Algebraic Analysis, (eds. M. Kashiwara and T. Kawai), Academic Press, Boston, 1988, pp. 549-569.
18) T. Oaku, Removable singularities of solutions of linear partial differential equations, J. Fac. Sci. Univ. Tokyo, 33 (1986), 403-428.
19) T. Oaku, Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. Indust. Appl. Math., 11 (1994), 485-497.
20) T. Oaku and T. Shimoyama, A Gröbner basis method for modules over rings of differential operators, to appear in J. Symbolic Comput..
21) T. Oshima, A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. Res. Inst. Math. Sci., 19 (1983), 1203-1230.
22) H. Tahara, Fuchsian type equations and Fuchsian hyperbolic equations, Japan. J. Math., 5 (1979), 245-347.
23) N. Takayama, Gröbner basis and the problem of contiguous relations, Japan J. Appl. Math., 6 (1989), 147-160.
24) N. Takayama, An algorithm of constructing the integral of a module-an infinite dimensional analog of Gröbner basis, Proceedings of International Symposium on Symbolic and Algebraic Computation, (eds. S. Watanabe and M. Nagata), ACM Press, New York, 1990, pp. 206-211.
25) N. Takayama, Propagation of singularities of solutions of the Euler-Darboux equation and a global structure of the space of holonomic solutions I, Funkcial. Ekvac., 35(1992), 343-403.
26) N. Takayama, Computational algebraic analysis and connection formula, Kokyuroku, Res. Inst. Math. Sci., Kyoto Univ., 811 (1992), 82-97.

Right : [1] M. S. Baouendi and C. Goulaouic, Cauchy problems with characteristic initial hypersurface, Comm. Pure Appl. Math., 26 (1973), 455-475.
[2] J. E. Björk, Rings of Differential Operators, North-Holland, Amsterdam, 1979.
[3] J. Briancon, Weierstrass préparé à la Hironaka, Astérisque, 7-8 (1973), 67-73.
[4] B. Buchberger, Ein algorithmisches Kriterium für die Lösbarkeit eines algebraischen Gleichungssystems, Aequationes Math., 4 (1970), 374-383.
[5] B. Buchberger, A criterion for detecting unnecessary reductions in the construction of Gröbner bases, Lecture Notes in Comput. Sci., 72, Springer, Berlin-Heidelberg-New York, 1979, pp. 3-21.
[6] F. Castro, Calculs effectifs pour les idéaux d'opérateurs différentiels, Travaux en Cours, 24, Herman, Paris, 1987, pp. 1-19.
[7] D. Cox, J. Little and D. O'Shea, Ideals, Varieties, and Algorithms, (Undergraduate Texts in Math.), Springer, Berlin-Heidelberg-New York, 1992.
[8] A. Galligo, Some algorithmic questions on ideals of differential operators, Lecture Notes in Comput. Sci., 204, Springer, Berlin-Heidelberg-New York, 1985, pp. 413-421.
[9] M. Kashiwara, Systems of Microdifferential Equations, Progr. Math., 34, Birkhäuser, Boston-Basel-Stuttgart, 1983.
[10] M. Kashiwara, Vanishing cycle sheaves and holonomic systems of differential equations, Lecture Notes in Math., 1016, Springer, Berlin-Heidelberg-New York, 1983, pp. 134-142.
[11] M. Kashiwara and T. Kawai, On holonomic systems of microdifferential equations, III, Publ. Res. Inst. Math. Sci., Kyoto Univ., 17 (1981), 813-979.
[12] M. Kashiwara and T. Oshima, Systems of differential equations with regular singularities and their boundary value problems, Ann. of Math., 106 (1977), 145-200.
[13] Y. Laurent, and T. Monteiro Fernandes, Systèmes différentiels fuchsiens le long d'une sous-variété, Publ. Res. Inst. Math. Sci., Kyoto Univ., 24 (1988), 397-431.
[14] Y. Laurent and P. Schapira, Images inverses des modules différentiels, Compositio Math., 61 (1987), 229-251.
[15] T. Monteiro Fernandes, Formulation des valeurs au bord pour les systemès regulier, Compositio Math., 81 (1992), 121-142.
[16] M. Noro and T. Takeshima, Risa/Asir-a computer algebra system, Proceedings of International Symposium on Symbolic and Algebraic Computation, ACM Press, New York, 1992, pp. 387-396.
[17] M. Noumi, Wronskian determinants and the Gröobner representation of a linear differential equation, Algebraic Analysis, (eds. M. Kashiwara and T. Kawai), Academic Press, Boston, 1988, pp. 549-569.
[18] T. Oaku, Removable singularities of solutions of linear partial differential equations, J. Fac. Sci. Univ. Tokyo, 33 (1986), 403-428.
[19] T. Oaku, Computation of the characteristic variety and the singular locus of a system of differential equations with polynomial coefficients, Japan J. Indust. Appl. Math., 11 (1994), 485-497.
[20] T. Oaku and T. Shimoyama, A Gröbner basis method for modules over rings of differential operators, to appear in J. Symbolic Comput..
[21] T. Oshima, A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. Res. Inst. Math. Sci., 19 (1983), 1203-1230.
[22] H. Tahara, Fuchsian type equations and Fuchsian hyperbolic equations, Japan. J. Math., 5 (1979), 245-347.
[23] N. Takayama, Gröbner basis and the problem of contiguous relations, Japan J. Appl. Math., 6 (1989), 147-160.
[24] N. Takayama, An algorithm of constructing the integral of a module-an infinite dimensional analog of Gröbner basis, Proceedings of International Symposium on Symbolic and Algebraic Computation, (eds. S. Watanabe and M. Nagata), ACM Press, New York, 1990, pp. 206-211.
[25] N. Takayama, Propagation of singularities of solutions of the Euler-Darboux equation and a global structure of the space of holonomic solutions I, Funkcial. Ekvac., 35(1992), 343-403.
[26] N. Takayama, Computational algebraic analysis and connection formula, Kôkyûroku, Res. Inst. Math. Sci., Kyoto Univ., 811 (1992), 82-97.

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