Fourier-like methods for equations with separable variables

Danuta Przeworska-Rolewicz

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2009)

  • Volume: 29, Issue: 1, page 19-42
  • ISSN: 1509-9407

Abstract

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It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that the results concerning the Fourier method are proved under weaker assumptions than these obtained in [6] (cf. also [7, 8, 11]).

How to cite

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Danuta Przeworska-Rolewicz. "Fourier-like methods for equations with separable variables." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 29.1 (2009): 19-42. <http://eudml.org/doc/271161>.

@article{DanutaPrzeworska2009,
abstract = {It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that the results concerning the Fourier method are proved under weaker assumptions than these obtained in [6] (cf. also [7, 8, 11]).},
author = {Danuta Przeworska-Rolewicz},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {algebraic analysis; commutative algebra with unit; Leibniz condition; logarithmic mapping; antilogarithmic mapping; right invertible operator; sine mapping; cosine mapping; initial value problem; boundary value problem; Fourier method},
language = {eng},
number = {1},
pages = {19-42},
title = {Fourier-like methods for equations with separable variables},
url = {http://eudml.org/doc/271161},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Danuta Przeworska-Rolewicz
TI - Fourier-like methods for equations with separable variables
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2009
VL - 29
IS - 1
SP - 19
EP - 42
AB - It is well known that a power of a right invertible operator is again right invertible, as well as a polynomial in a right invertible operator under appropriate assumptions. However, a linear combination of right invertible operators (in particular, their sum and/or difference) in general is not right invertible. It will be shown how to solve equations with linear combinations of right invertible operators in commutative algebras using properties of logarithmic and antilogarithmic mappings. The used method is, in a sense, a kind of the variables separation method. We shall obtain also an analogue of the classical Fourier method for partial differential equations. Note that the results concerning the Fourier method are proved under weaker assumptions than these obtained in [6] (cf. also [7, 8, 11]).
LA - eng
KW - algebraic analysis; commutative algebra with unit; Leibniz condition; logarithmic mapping; antilogarithmic mapping; right invertible operator; sine mapping; cosine mapping; initial value problem; boundary value problem; Fourier method
UR - http://eudml.org/doc/271161
ER -

References

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  1. [1] A. Favini, Sum of operators' method in abstract equations, in: Partial Differential Equations, Progr. Nonlinear Differential Equations Appl. 22 (Burkhäuser, Boston, 1996) 132-146. Zbl0854.34058
  2. [2] V. Lovass-Nagy and D.L. Powers, On under- and over-determined initial value problems, Int. J. Control 19 (1974), 653-656. 
  3. [3] P. Multarzyński, On some right invertible operators in differential spaces, Demonstratio Math. 37 (4) (2004), 905-920. Zbl1083.58007
  4. [4] P. Multarzyński, On divided difference operators in function algebras, Demonstratio Math. 41 (2) (2008), 273-289. Zbl1196.12006
  5. [5] W. Pogorzelski, Integral Equations and Their Applications, 1-st Polish ed. Vol. I-1953, Vol. II-1958, Vol. III-1960; PWN-Polish Scientific Publishers, Warszawa; English ed. Pergamon Press and PWN-Polish Scientific Publishers, Oxford-Warszawa, 1966. 
  6. [6] D. Przeworska-Rolewicz, Remarks on boundary value problems and Fourier method for right invertible operators, Math. Nachrichten 72 (1976), 109-117. 
  7. [7] D. Przeworska-Rolewicz, Algebraic Analysis, D. Reidel and PWN-Polish Scientific Publishers, Dordrecht-Warszawa, 1988. 
  8. [8] D. Przeworska-Rolewicz, Logarithms and Antilogarithms. An Algebraic Analysis Approach. With Appendix by Z. Binderman, Kluwer Academic Publishers, Dordrecht, 1998. 
  9. [9] D. Przeworska-Rolewicz, Linear combinations of right invertible operators in commutative algebras with logarithms, Demonstratio Math. 31 (4) (1998), 887-898. Zbl0944.47003
  10. [10] D. Przeworska-Rolewicz, Postmodern Logarithmo-technia, Computers and Mathematics with Applications 41 (2001), 1143-1154. Zbl0987.05014
  11. [11] D. Przeworska-Rolewicz, Non-Leibniz algebras with logarithms do not have the trigonometric identity, in: Algebraic Analysis and Related Topics, Proc. Intern. Conf. Warszawa, September 21-25, 1999. Banach Center Publications, 53. Inst. of Math., Polish Acad. of Sci., Warszawa, 2000, 177-189. Zbl0973.13011
  12. [12] D. Przeworska-Rolewicz, Algebraic Analysis in structures with Kaplansky-Jacobson property, Studia Math. 168 (2) (2005), 165-186. Zbl1070.20067
  13. [13] D. Przeworska-Rolewicz, Some summations formulae in commutative Leibniz algebras with logarithms, Control and Cybernetics 36 (3) (2007), 841-857. Zbl1167.47005
  14. [14] D. Przeworska-Rolewicz, Sylvester inertia law in commutative Leibniz algebras with logarithms, Demonstratio Math. 40 (2007), 659-669. Zbl1143.47060
  15. [15] D. Przeworska-Rolewicz, Nonlinear separable equations in linear spaces and commutative Leibniz algebras. Preprint 691, Institute of Mathematics, Polish Academy of Sciences, Warszawa, September 2008; http//www.impan.pl/Preprints/p691; Annales Polon. Math. (to appear). Zbl1185.47080
  16. [16] D. Przeworska-Rolewicz, Fourier-like methods for equations with separable variables. Preprint 693, Institute of Mathematics, Polish Academy of Sciences, Warszawa, October 2008; http//www.impan.pl/Preprints/p693. 
  17. [17] G. Virsik, Right inverses of vector fields, J. Austral. Math. Soc. (Series A), 58 (1995), 411-420. Zbl0828.47001

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