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

On the Alekseev-Gröbner formula in Banach spaces

Arnulf Jentzen 1, , Felix Lindner 2, and  Primož Pušnik 1,,

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  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.

Keywords: Alekseev-Gröbner formula, Banach space, ordinary differential equation, perturbation theory, infinite-dimensional analysis.
Mathematics Subject Classification: Primary: 34A99; Secondary: 65L99.
Citation: Arnulf Jentzen, Felix Lindner, Primož Pušnik. On the Alekseev-Gröbner formula in Banach spaces. Discrete & 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.   Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[10]

A. Jentzen, D. 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.  Google Scholar

[11]

G. Ladas, G. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.   Google Scholar

[2]

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

[3]

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

[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. Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar
[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.  Google Scholar

[10]

A. Jentzen, D. 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.  Google Scholar

[11]

G. Ladas, G. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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