August  2019, 24(8): 4475-4511. doi: 10.3934/dcdsb.2019128

On the Alekseev-Gröbner formula in Banach spaces

1. 

Seminar for Applied Mathematics, Department of Mathematics, ETH Zürich, Zürich, Switzerland

2. 

Institute of Mathematics, Faculty of Mathematics and Natural Sciences, University of Kassel, Kassel, Germany

* Corresponding author: Primož Pušnik

This paper is dedicated to Peter Kloeden on the occasion of his 70th birthday

Received  November 2018 Revised  March 2019 Published  August 2019 Early access  June 2019

Fund Project: This work was partially supported by the SNSF-Research project 200021_156603 "Numerical approximations of nonlinear stochastic ordinary and partial differential equations".

The Alekseev-Gröbner formula is a well known tool in numerical analysis for describing the effect that a perturbation of an ordinary differential equation (ODE) has on its solution. In this article we provide an extension of the Alekseev-Gröbner formula for Banach space valued ODEs under, loosely speaking, mild conditions on the perturbation of the considered ODEs.

Citation: Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4475-4511. doi: 10.3934/dcdsb.2019128
References:
[1]

V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary differential equations, Vestnik Moskov. Univ. Ser. I Mat. Meh., 2 (1961), 28-36. 

[2]

R. Coleman, Calculus on Normed Vector Spaces, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-3894-6.

[3]

A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Second ed., Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-05792-7.

[4]

B. Driver, Analysis Tools with Applications, Preprint, 2003. Available online: http://www.math.ucsd.edu/~ bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf.

[5]

L. C. Evans, Partial Differential Equations, Second ed., Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[6]

W. Gröbner, Die Lie-Reihen und ihre Anwendungen, Mathematische Monographien, 3, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960.

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I, Second ed., Springer Series in Computational Mathematics, 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. doi: 10.1007%2F978-3-540-78862-1.

[8] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Second ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511995569.
[9]

A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112.  doi: 10.1006/jcom.1993.1007.

[10]

A. JentzenD. Salimova and T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, J. Math. Anal. Appl., 469 (2019), 661-704.  doi: 10.1016/j.jmaa.2018.09.032.

[11]

G. LadasG. Ladde and V. Lakshmikantham, On some fundamental properties of solutions of differential equations in Banach spaces, Ann. Mat. Pura Appl. (4), 95 (1973), 255-267.  doi: 10.1007/BF02410719.

[12]

J. Niesen, A priori estimates for the global error committed by Runge-Kutta methods for a nonlinear oscillator, LMS J. Comput. Math., 6 (2003), 18-28.  doi: 10.1112/S1461157000000358.

[13]

C. Prévøt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007%2F978-3-540-70781-3.

[14]

W. Rudin, Principles of Mathematical Analysis, Third ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.

show all references

This paper is dedicated to Peter Kloeden on the occasion of his 70th birthday

References:
[1]

V. M. Alekseev, An estimate for the perturbations of the solutions of ordinary differential equations, Vestnik Moskov. Univ. Ser. I Mat. Meh., 2 (1961), 28-36. 

[2]

R. Coleman, Calculus on Normed Vector Spaces, Universitext, Springer, New York, 2012. doi: 10.1007/978-1-4614-3894-6.

[3]

A. Deitmar and S. Echterhoff, Principles of Harmonic Analysis, Second ed., Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-05792-7.

[4]

B. Driver, Analysis Tools with Applications, Preprint, 2003. Available online: http://www.math.ucsd.edu/~ bdriver/231-02-03/Lecture_Notes/PDE-Anal-Book/analpde1.pdf.

[5]

L. C. Evans, Partial Differential Equations, Second ed., Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[6]

W. Gröbner, Die Lie-Reihen und ihre Anwendungen, Mathematische Monographien, 3, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960.

[7]

E. Hairer, S. P. Nørsett and G. Wanner, Solving Ordinary Differential Equations. I, Second ed., Springer Series in Computational Mathematics, 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. doi: 10.1007%2F978-3-540-78862-1.

[8] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Second ed., Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2009.  doi: 10.1017/CBO9780511995569.
[9]

A. Iserles and G. Söderlind, Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), 97-112.  doi: 10.1006/jcom.1993.1007.

[10]

A. JentzenD. Salimova and T. Welti, Strong convergence for explicit space-time discrete numerical approximation methods for stochastic Burgers equations, J. Math. Anal. Appl., 469 (2019), 661-704.  doi: 10.1016/j.jmaa.2018.09.032.

[11]

G. LadasG. Ladde and V. Lakshmikantham, On some fundamental properties of solutions of differential equations in Banach spaces, Ann. Mat. Pura Appl. (4), 95 (1973), 255-267.  doi: 10.1007/BF02410719.

[12]

J. Niesen, A priori estimates for the global error committed by Runge-Kutta methods for a nonlinear oscillator, LMS J. Comput. Math., 6 (2003), 18-28.  doi: 10.1112/S1461157000000358.

[13]

C. Prévøt and M. Röckner, A Concise Course on Stochastic Partial Differential Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin, 2007. doi: 10.1007%2F978-3-540-70781-3.

[14]

W. Rudin, Principles of Mathematical Analysis, Third ed., International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York-Auckland-Düsseldorf, 1976.

[1]

Savin Treanţă. On a class of differential quasi-variational-hemivariational inequalities in infinite-dimensional Banach spaces. Evolution Equations and Control Theory, 2022, 11 (3) : 827-836. doi: 10.3934/eect.2021027

[2]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[3]

Ismara Álvarez-Barrientos, Mijail Borges-Quintana, Miguel Angel Borges-Trenard, Daniel Panario. Computing Gröbner bases associated with lattices. Advances in Mathematics of Communications, 2016, 10 (4) : 851-860. doi: 10.3934/amc.2016045

[4]

Masashi Wakaiki, Hideki Sano. Stability analysis of infinite-dimensional event-triggered and self-triggered control systems with Lipschitz perturbations. Mathematical Control and Related Fields, 2022, 12 (1) : 245-273. doi: 10.3934/mcrf.2021021

[5]

Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012

[6]

Sergey V Lototsky, Henry Schellhorn, Ran Zhao. An infinite-dimensional model of liquidity in financial markets. Probability, Uncertainty and Quantitative Risk, 2021, 6 (2) : 117-138. doi: 10.3934/puqr.2021006

[7]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945

[8]

Brendan Weickert. Infinite-dimensional complex dynamics: A quantum random walk. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 517-524. doi: 10.3934/dcds.2001.7.517

[9]

Björn Augner, Birgit Jacob. Stability and stabilization of infinite-dimensional linear port-Hamiltonian systems. Evolution Equations and Control Theory, 2014, 3 (2) : 207-229. doi: 10.3934/eect.2014.3.207

[10]

Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems and Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567

[11]

Radu Ioan Boţ, Sorin-Mihai Grad. On linear vector optimization duality in infinite-dimensional spaces. Numerical Algebra, Control and Optimization, 2011, 1 (3) : 407-415. doi: 10.3934/naco.2011.1.407

[12]

Kening Lu, Alexandra Neamţu, Björn Schmalfuss. On the Oseledets-splitting for infinite-dimensional random dynamical systems. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1219-1242. doi: 10.3934/dcdsb.2018149

[13]

Chris Guiver, Mark R. Opmeer. Bounded real and positive real balanced truncation for infinite-dimensional systems. Mathematical Control and Related Fields, 2013, 3 (1) : 83-119. doi: 10.3934/mcrf.2013.3.83

[14]

Qing Xu. Backward stochastic Schrödinger and infinite-dimensional Hamiltonian equations. Discrete and Continuous Dynamical Systems, 2015, 35 (11) : 5379-5412. doi: 10.3934/dcds.2015.35.5379

[15]

Pengfei Wang, Mengyi Zhang, Huan Su. Input-to-state stability of infinite-dimensional stochastic nonlinear systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 821-836. doi: 10.3934/dcdsb.2021066

[16]

David Cheban, Zhenxin Liu. Averaging principle on infinite intervals for stochastic ordinary differential equations. Electronic Research Archive, 2021, 29 (4) : 2791-2817. doi: 10.3934/era.2021014

[17]

Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232

[18]

Viorel Barbu, Gabriela Marinoschi. An identification problem for a linear evolution equation in a banach space. Discrete and Continuous Dynamical Systems - S, 2020, 13 (5) : 1429-1440. doi: 10.3934/dcdss.2020081

[19]

Hannes Bartz, Antonia Wachter-Zeh. Efficient decoding of interleaved subspace and Gabidulin codes beyond their unique decoding radius using Gröbner bases. Advances in Mathematics of Communications, 2018, 12 (4) : 773-804. doi: 10.3934/amc.2018046

[20]

Angelo Favini, Yakov Yakubov. Regular boundary value problems for ordinary differential-operator equations of higher order in UMD Banach spaces. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 595-614. doi: 10.3934/dcdss.2011.4.595

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (206)
  • HTML views (175)
  • Cited by (0)

Other articles
by authors

[Back to Top]