Remarks on generalized solutions of ordinary linear differential equations in the Colombeau algebra

Jan Ligęza

Mathematica Bohemica (1998)

  • Volume: 123, Issue: 3, page 301-316
  • ISSN: 0862-7959

Abstract

top
In this paper first order linear ordinary differential equations are considered. It is shown that the Cauchy problem for these systems has a unique solution in 𝒢 n ( ) , where 𝒢 ( ) denotes the Colombeau algebra.

How to cite

top

Ligęza, Jan. "Remarks on generalized solutions of ordinary linear differential equations in the Colombeau algebra." Mathematica Bohemica 123.3 (1998): 301-316. <http://eudml.org/doc/248318>.

@article{Ligęza1998,
abstract = {In this paper first order linear ordinary differential equations are considered. It is shown that the Cauchy problem for these systems has a unique solution in $ \{\mathcal \{G\}\}^n (\mathbb \{R\}) $, where $ \{\mathcal \{G\}\} (\mathbb \{R\}) $ denotes the Colombeau algebra.},
author = {Ligęza, Jan},
journal = {Mathematica Bohemica},
keywords = {generalized ordinary differential equations; Cauchy problem; distributions; Colombeau algebra; generalized ordinary differential equations; Cauchy problem; distributions; Colombeau algebra},
language = {eng},
number = {3},
pages = {301-316},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on generalized solutions of ordinary linear differential equations in the Colombeau algebra},
url = {http://eudml.org/doc/248318},
volume = {123},
year = {1998},
}

TY - JOUR
AU - Ligęza, Jan
TI - Remarks on generalized solutions of ordinary linear differential equations in the Colombeau algebra
JO - Mathematica Bohemica
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 123
IS - 3
SP - 301
EP - 316
AB - In this paper first order linear ordinary differential equations are considered. It is shown that the Cauchy problem for these systems has a unique solution in $ {\mathcal {G}}^n (\mathbb {R}) $, where $ {\mathcal {G}} (\mathbb {R}) $ denotes the Colombeau algebra.
LA - eng
KW - generalized ordinary differential equations; Cauchy problem; distributions; Colombeau algebra; generalized ordinary differential equations; Cauchy problem; distributions; Colombeau algebra
UR - http://eudml.org/doc/248318
ER -

References

top
  1. J. F. Colombeau, Elementary Introduction to New Generalized Functions, North Holland, Amsterdam, 1985. (1985) Zbl0584.46024MR0808961
  2. J. F. Colombeau, Multiplication of Distributions, Lecture Notes in Math. 1532, Springer, Berlin, 1992. (1992) Zbl0815.35002MR1222643
  3. S. G. Deo S. G. Pandit, DifferentiaІ Systems Involving Impulses, Lecture Notes in Math. 954, Springer, Berlin, 1982. (1982) MR0674119
  4. V. Doležal, Dynamics of Linear Systems, Academia, Praha, 1967. (1967) MR0220530
  5. Y. Egorov, A theory of generalized functions, Uspehi Math. Nauk 455 (1990), 3-40. (In Russian.) (1990) MR1084986
  6. A. F. Filippov, Differential Equations with Discontinuous Right Part, Nauka, Moscow, 1985. (In Russian.) (1985) MR0790682
  7. I. M. Geľfand G. E. Shilov, Generalized Functions I, Academic Press, NeW York, 1964. (1964) MR0166596
  8. T. H. Hildebrandt, On systems of linєar differential Stieltjes integral equations, Illinois J. Math. 3 (1959), 352-373. (1959) MR0105600
  9. J. Kurzweil, Generalized ordinary differential equations and continuous dependence on a parameter, Czechoslovak Math. J. 7 (1957), 418-449. (1957) Zbl0090.30002MR0111875
  10. J. Kurzweil, Linear differential equations with distributions coefficients, Bull. Acad. Polon. Sci. Ser. Math. Phys. 7 (1959), 557-560. (1959) MR0111887
  11. J. Ligęza, On distributional solutions of some systems of linear differential equations, Časopis Pěst. Mat. 102 (1977). 37-41. (1977) MR0460757
  12. J. Ligęza, Weak Solutions of Ordinary Differential Equations, Prace Nauk. Uniw. Śląsk. Katowic. 842, 1986. (1986) MR0868863
  13. J. Ligęza, Generalized solutions of ordinary linear dilferential equations in the Colombeau algebra, Math. Bohem. 118 (1993), 123-146. (1993) MR1223478
  14. M. Oberguggenberger, 10.1016/0022-247X(89)90014-0, J. Math. Anal. Appl. 142 (1989), 452-467. (1989) Zbl0705.35146MR1014590DOI10.1016/0022-247X(89)90014-0
  15. M. Pelant M. Tvrdý, Linear distributional differential equations in the space of regulated functions, Math. Bohem. 118 (1993), 379-400. (1993) MR1251883
  16. J. Persson, The Cauchy system for linear distribution differential equations, Functial Ekvac. 30 (1987), 162-168. (1987) 
  17. R. Pfaff, 10.1017/S0308210500011860, Proc. Roy. Soc. Edingburgh, Sect. A. 85 (1980), 291-298. (1980) MR0574022DOI10.1017/S0308210500011860
  18. Š. Schwabik M. Tvrdý O. Vejvoda, Differential and Integral Equations, Academia, Praha, 1979. (1979) MR0542283
  19. E. E. Rosinger, Nonlinear Partial Differential Equations, Sequential and Weak Solutions, Math. Studies 44, North-Holland, 1980. (1980) Zbl0447.35001MR0590891
  20. L. Schwartz, Sur l'impossibilité de la multiplication des distributions, C. R. Acad. Sci. Paris Sér. I Math. 239 (1954), 847-848. (1954) Zbl0056.10602MR0064324
  21. Z. Wyderka, Some problems of optimal control for linear systems with measures as coefficients, Systems Sci. 5 (1979), 425-431. (1979) Zbl0442.49002MR0572042

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.