American Institute of Mathematical Sciences

Advanced
January  2021, 20(1): 359-368. doi: 10.3934/cpaa.2020270

Isomorphism between one-dimensional and multidimensional finite difference operators

Anton A. Kutsenko

Jacobs University, Campus Ring 1, 28759 Bremen, Germany, Saint-Petersburg State University, Universitetskaya nab. 7/9, 199034 St. Petersburg, Russia

Received  July 2020 Revised  September 2020 Published  January 2021 Early access  November 2020

Fund Project: This paper is a contribution to the project M3 of the Collaborative Research Centre TRR 181 "Energy Transfer in Atmosphere and Ocean" funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Projektnummer 274762653. This work is also supported by the RFBR (RFFI) grant No. 19-01-00094

Finite difference operators are widely used for the approximation of continuous ones. It is well known that the analysis of continuous differential operators may strongly depend on their dimensions. We will show that the finite difference operators generate the same algebra, regardless of their dimension.

Keywords: Representation of finite difference operators, UHF algebras, ODE and PDE.
Mathematics Subject Classification: Primary: 35P99, 16G99; Secondary: 39A05, 39A14.
Citation: Anton A. Kutsenko. Isomorphism between one-dimensional and multidimensional finite difference operators. Communications on Pure and Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270
References:
[1]

V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra, 217 (2013), 495-529.  doi: 10.1016/j.jpaa.2012.06.024.

[2]

K. Davidson, C*-Algebras by Example, American Mathematical Society and Fields Institute, 1997. doi: 10.1090/fim/006.

[3]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, Rhode Island, 2010.

[4]

J. G. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc., 95 (1960), 318-340.  doi: 10.2307/1993294.

[5]

L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522-540.  doi: 10.1016/j.jpaa.2007.06.008.

[6]

L. Guo, G. Regensburger and M. Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra, 218 (2014), 456-473.  doi: 10.1016/j.jpaa.2013.06.015.

[7]

R. Hoegh-Krohn and T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus, J. Reine Angew. Math., 328 (1981), 1-8.  doi: 10.1515/crll.1981.328.1.

[8] M. Rordam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000. 
[9]

M. Rosenkranz, A new symbolic method for solving linear two-point boundary value problems on the level of operators, J. Symb. Comput, 39 (2005), 171-199.  doi: 10.1016/j.jsc.2004.09.004.

[10]

M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012.

[11]

H. S. Yin, A simple proof of the classification of rational rotation C*-algebras, P. Am. Math. Soc., 98 (1986), 469-470.  doi: 10.2307/2046204.

show all references

References:
[1]

V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra, 217 (2013), 495-529.  doi: 10.1016/j.jpaa.2012.06.024.

[2]

K. Davidson, C*-Algebras by Example, American Mathematical Society and Fields Institute, 1997. doi: 10.1090/fim/006.

[3]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, American Mathematical Society, Providence, Rhode Island, 2010.

[4]

J. G. Glimm, On a certain class of operator algebras, Trans. Am. Math. Soc., 95 (1960), 318-340.  doi: 10.2307/1993294.

[5]

L. Guo and W. Keigher, On differential Rota-Baxter algebras, J. Pure Appl. Algebra, 212 (2008), 522-540.  doi: 10.1016/j.jpaa.2007.06.008.

[6]

L. Guo, G. Regensburger and M. Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra, 218 (2014), 456-473.  doi: 10.1016/j.jpaa.2013.06.015.

[7]

R. Hoegh-Krohn and T. Skjelbred, Classification of C*-algebras admitting ergodic actions of the two-dimensional torus, J. Reine Angew. Math., 328 (1981), 1-8.  doi: 10.1515/crll.1981.328.1.

[8] M. Rordam, F. Larsen and N. J. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, 2000. 
[9]

M. Rosenkranz, A new symbolic method for solving linear two-point boundary value problems on the level of operators, J. Symb. Comput, 39 (2005), 171-199.  doi: 10.1016/j.jsc.2004.09.004.

[10]

M. Tenenbaum and H. Pollard, Ordinary Differential Equations, Dover Publications Inc., 2012.

[11]

H. S. Yin, A simple proof of the classification of rational rotation C*-algebras, P. Am. Math. Soc., 98 (1986), 469-470.  doi: 10.2307/2046204.

Figure 1.  Two first partitions for the unitary transform $ {\mathcal U}_{2,1}^{-1} $ between $ L^2_{2,1} $ and $ L^2_{1,1} $ are shown. The characteristic functions of squares and intervals with the same "blue" and "red" numbers are transformed into each other under the action of $ {\mathcal U}_{2,1} $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The unitary transform $ {\mathcal U}_{2,1}^{-1} $, see Fig. 1, applied to the function $ z(x,y) = 1+\sin(\pi(x^2+y^2)) $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Marita Holtmannspötter, Arnd Rösch, Boris Vexler. A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear PDE-ODE system. Mathematical Control and Related Fields, 2021, 11 (3) : 601-624. doi: 10.3934/mcrf.2021014
[2]

Augusto VisintiN. On the variational representation of monotone operators. Discrete and Continuous Dynamical Systems - S, 2017, 10 (4) : 909-918. doi: 10.3934/dcdss.2017046
[3]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637
[4]

Bernd Ammann, Robert Lauter and Victor Nistor. Algebras of pseudodifferential operators on complete manifolds. Electronic Research Announcements, 2003, 9: 80-87.

[5]

Adam Bobrowski, Katarzyna Morawska. From a PDE model to an ODE model of dynamics of synaptic depression. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2313-2327. doi: 10.3934/dcdsb.2012.17.2313
[6]

Sabine Eisenhofer, Messoud A. Efendiev, Mitsuharu Ôtani, Sabine Schulz, Hans Zischka. On an ODE-PDE coupling model of the mitochondrial swelling process. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1031-1057. doi: 10.3934/dcdsb.2015.20.1031
[7]

Vanessa Baumgärtner, Simone Göttlich, Stephan Knapp. Feedback stabilization for a coupled PDE-ODE production system. Mathematical Control and Related Fields, 2020, 10 (2) : 405-424. doi: 10.3934/mcrf.2020003
[8]

Cristian Dobre. Mathematical properties of the regular *-representation of matrix $*$-algebras with applications to semidefinite programming. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 367-378. doi: 10.3934/naco.2013.3.367
[9]

Chady Ghnatios, Guangtao Xu, Adrien Leygue, Michel Visonneau, Francisco Chinesta, Alain Cimetiere. On the space separated representation when addressing the solution of PDE in complex domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 475-500. doi: 10.3934/dcdss.2016008
[10]

Kanghui Guo and Demetrio Labate. Sparse shearlet representation of Fourier integral operators. Electronic Research Announcements, 2007, 14: 7-19. doi: 10.3934/era.2007.14.7
[11]

Claire david@lmm.jussieu.fr David, Pierre Sagaut. Theoretical optimization of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 286-293. doi: 10.3934/proc.2007.2007.286
[12]

Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054
[13]

Youshan Tao, J. Ignacio Tello. Nonlinear stability of a heterogeneous state in a PDE-ODE model for acid-mediated tumor invasion. Mathematical Biosciences & Engineering, 2016, 13 (1) : 193-207. doi: 10.3934/mbe.2016.13.193
[14]

Ciro D'Apice, Peter I. Kogut, Rosanna Manzo. On relaxation of state constrained optimal control problem for a PDE-ODE model of supply chains. Networks and Heterogeneous Media, 2014, 9 (3) : 501-518. doi: 10.3934/nhm.2014.9.501
[15]

Maria Laura Delle Monache, Paola Goatin. A front tracking method for a strongly coupled PDE-ODE system with moving density constraints in traffic flow. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 435-447. doi: 10.3934/dcdss.2014.7.435
[16]

Matthias Gerdts, Sven-Joachim Kimmerle. Numerical optimal control of a coupled ODE-PDE model of a truck with a fluid basin. Conference Publications, 2015, 2015 (special) : 515-524. doi: 10.3934/proc.2015.0515
[17]

Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations and Control Theory, 2022, 11 (1) : 199-224. doi: 10.3934/eect.2020108
[18]

Pavel Krejčí, Harbir Lamba, Sergey Melnik, Dmitrii Rachinskii. Kurzweil integral representation of interacting Prandtl-Ishlinskii operators. Discrete and Continuous Dynamical Systems - B, 2015, 20 (9) : 2949-2965. doi: 10.3934/dcdsb.2015.20.2949
[19]

Catarina Carvalho, Victor Nistor, Yu Qiao. Fredholm criteria for pseudodifferential operators and induced representations of groupoid algebras. Electronic Research Announcements, 2017, 24: 68-77. doi: 10.3934/era.2017.24.008
[20]

Emma Hoarau, Claire david@lmm.jussieu.fr David, Pierre Sagaut, Thiên-Hiêp Lê. Lie group study of finite difference schemes. Conference Publications, 2007, 2007 (Special) : 495-505. doi: 10.3934/proc.2007.2007.495

2021 Impact Factor: 1.273

Tools

Metrics

  • PDF downloads (199)
  • HTML views (62)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy