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Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation

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Abstract

We study a class of linear ordinary differential equations (ODEs) with distributional coefficients. These equations are defined using an intrinsic multiplicative product of Schwartz distributions which is an extension of the Hörmander product of distributions with non-intersecting singular supports (Hörmander in The analysis of linear partial differential operators I, Springer, Berlin, 1983). We provide a regularization procedure for these ODEs and prove an existence and uniqueness theorem for their solutions. We also determine the conditions for which the solutions are regular and distributional. These results are used to study the Euler–Bernoulli beam equation with discontinuous and singular coefficients. This problem was addressed in the past using intrinsic products (under some restrictive conditions) and the Colombeau formalism (in the general case). Here we present a new intrinsic formulation that is simpler and more general. As an application, the case of a non-uniform static beam displaying structural cracks is discussed in some detail.

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  1. Hilbert space methods are also very important, namely in the context of singular perturbations of Schrödinger operators [1, 2, 11, 16].

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Acknowledgements

Cristina Jorge was supported by the Ph.D. Grant SFRH/BD/85839/2012 of the Portuguese Science Foundation. N.C. Dias and J.N. Prata were supported by the Portuguese Science Foundation (FCT) under the Grant PTDC/MAT-CAL/4334/2014.

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Dias, N.C., Jorge, C. & Prata, J.N. Ordinary Differential Equations with Singular Coefficients: An Intrinsic Formulation with Applications to the Euler–Bernoulli Beam Equation. J Dyn Diff Equat 33, 593–619 (2021). https://doi.org/10.1007/s10884-020-09822-x

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