# Fourier-like methods for equations with separable variables

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

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## 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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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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