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Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients

Günther Hörmann and Ljubica Oparnica (UGent)
(2007) APPLICABLE ANALYSIS. 86(11). p.1347-1363
Author
Organization
Abstract
We study existence and uniqueness of distributional solutions w to the ordinary differential equation d(2)/dx(2)(a(x).(d(2)w(x)/dx(2)) + P(x)(d(2)w(x)/dx(2)) = g(x) with discontinuous coefficients and right-hand side. For example, if a and w" are non-smooth the product a . w" has no obvious meaning. When interpreted on the most general level of the hierarchy of distributional products discussed by Oberguggenberger, M. [1992, Multiplication of distributions and applications to partial differential equations (Harlow: Longman Scientific & Technical)], it turns out that existence of a solution w forces it to beat least continuously differentiable. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability.
Keywords
ordinary differential equations with discontinuous coefficients, distributional solutions, multiplication of distributions, GENERALIZED SOLUTIONS, HYPERBOLIC SYSTEMS

Citation

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MLA
Hörmann, Günther, and Ljubica Oparnica. “Distributional Solution Concepts for the Euler–Bernoulli Beam Equation with Discontinuous Coefficients.” APPLICABLE ANALYSIS, vol. 86, no. 11, 2007, pp. 1347–63, doi:10.1080/00036810701595944.
APA
Hörmann, G., & Oparnica, L. (2007). Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients. APPLICABLE ANALYSIS, 86(11), 1347–1363. https://doi.org/10.1080/00036810701595944
Chicago author-date
Hörmann, Günther, and Ljubica Oparnica. 2007. “Distributional Solution Concepts for the Euler–Bernoulli Beam Equation with Discontinuous Coefficients.” APPLICABLE ANALYSIS 86 (11): 1347–63. https://doi.org/10.1080/00036810701595944.
Chicago author-date (all authors)
Hörmann, Günther, and Ljubica Oparnica. 2007. “Distributional Solution Concepts for the Euler–Bernoulli Beam Equation with Discontinuous Coefficients.” APPLICABLE ANALYSIS 86 (11): 1347–1363. doi:10.1080/00036810701595944.
Vancouver
1.
Hörmann G, Oparnica L. Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients. APPLICABLE ANALYSIS. 2007;86(11):1347–63.
IEEE
[1]
G. Hörmann and L. Oparnica, “Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients,” APPLICABLE ANALYSIS, vol. 86, no. 11, pp. 1347–1363, 2007.
@article{8644762,
  abstract     = {{We study existence and uniqueness of distributional solutions w to the ordinary differential equation d(2)/dx(2)(a(x).(d(2)w(x)/dx(2)) + P(x)(d(2)w(x)/dx(2)) = g(x) with discontinuous coefficients and right-hand side. For example, if a and w" are non-smooth the product a . w" has no obvious meaning. When interpreted on the most general level of the hierarchy of distributional products discussed by Oberguggenberger, M. [1992, Multiplication of distributions and applications to partial differential equations (Harlow: Longman Scientific & Technical)], it turns out that existence of a solution w forces it to beat least continuously differentiable. Curiously, the choice of the distributional product concept is thus incompatible with the possibility of having a discontinuous displacement function as a solution. We also give conditions for unique solvability.}},
  author       = {{Hörmann, Günther and Oparnica, Ljubica}},
  issn         = {{0003-6811}},
  journal      = {{APPLICABLE ANALYSIS}},
  keywords     = {{ordinary differential equations with discontinuous coefficients,distributional solutions,multiplication of distributions,GENERALIZED SOLUTIONS,HYPERBOLIC SYSTEMS}},
  language     = {{eng}},
  number       = {{11}},
  pages        = {{1347--1363}},
  title        = {{Distributional solution concepts for the Euler–Bernoulli beam equation with discontinuous coefficients}},
  url          = {{http://doi.org/10.1080/00036810701595944}},
  volume       = {{86}},
  year         = {{2007}},
}
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