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On the eventual stability of asymptotically autonomous systems with constraints
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 |
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.
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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. |
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