Solution Estimates for some Weakly Nonlinear ODEs

Authors

  • D. Buske
  • J.P. Zingano

DOI:

https://doi.org/10.5540/tema.2005.06.01.0065

Abstract

We derive a few fundamental estimates for solutions u of weakly nonlinear ODE systems of the form ut = Au + B(t)u + "f(t, u) + g(t), t > t0, where A is a constant n x n matrix all of whose eigenvalues have negative real part and "f is suitably small, with B ∈ Lp(t0,∞), g ∈ Lq(t0,∞) for some 1 ≤ p, q ≤ ∞. Our analysis improves and extends some well known results obtained elsewhere for important families of equations within this class.

References

[1] P.A. Braz e Silva, “Stability of plane Couette flow: the resolvent method”, PhD Thesis, University of New Mexico, Albuquerque, NM, USA, 2002.

E.A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations”, McGraw-Hill, New York, 1955.

L.C. Evans, “Partial Differential Equations”, American Mathematics Society, Providence, 1998.

T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Ann. Math., 20 (1919), 292-296.

T. Hagstrom, H.-O. Kreiss, J. Lorenz and P. Zingano, Decay in time of incompressible flows, J. Math. Fluid Mech., 5 (2003), 231-244.

E. Hairer, S.P. Norsett and G. Wanner, “Solving Ordinary Differential Equations”, Vol. I, Springer, Berlin, 1987.

J. Hale, “Ordinary Differential Equations”, Wiley, New York, 1969.

H.O. Kreiss and J. Lorenz, Resolvent estimates and quantification of nonlinear stability, Acta Math. Sin., 16 (2000), 1-20.

A.M. Lyapunov, “The General Problem of Stability of Motion” (Russian), 1892, reprinted by Princeton Univ. Press, Princeton, 1947.

H. Poincaré, “Les Méthodes Nouvelles de la Mécanique Céleste”, Gauthier- Villars, Paris, 1893.

E.J. Routh, “A Treatise on the Stability of a given State of Motions”, Adams Prize essay, Cambridge University, Cambridge, 1877.

F. Verhulst, “Nonlinear Differential Equations and Dynamical Systems,” Springer, Berlin, 1990.

Downloads

Published

2005-06-01

How to Cite

Buske, D., & Zingano, J. (2005). Solution Estimates for some Weakly Nonlinear ODEs. Trends in Computational and Applied Mathematics, 6(1), 65–71. https://doi.org/10.5540/tema.2005.06.01.0065

Issue

Vol. 6 No. 1 (2005)

Section

Original Article

License

Copyright

Authors of articles published in the journal Trends in Computational and Applied Mathematics retain the copyright of their work. The journal uses Creative Commons Attribution (CC-BY) in published articles. The authors grant the TCAM journal the right to first publish the article.

Intellectual Property and Terms of Use

The content of the articles is the exclusive responsibility of the authors. The journal uses Creative Commons Attribution (CC-BY) in published articles. This license allows published articles to be reused without permission for any purpose as long as the original work is correctly cited.

The journal encourages Authors to self-archive their accepted manuscripts, publishing them on personal blogs, institutional repositories, and social media, as long as the full citation is included in the journal's website version.