In this work, we consider a flexible Euler-Bernoulli beam with variable coefficients clamped at one end subjected to a force control  in rotation and velocity rotation. From the weak formulation, we construct spaces in order to better define the weak solution. Using Faedo-Galerkin techniques as well as a compactness result on intermediate spaces (see [9]), we demonstrate the existence of a weak solution of the considered problem passing to the limit. Another result is given by demonstrating the uniqueness of the solution. We finally study the regularity of this solution.

existence, uniqueness, weak solution, beam equation.

Received: March 30, 2021; Accepted: April 16, 2021; Published: June 15, 2021

How to cite this article: Bomisso Gossrin Jean-Marc and Touré Kidjégbo Augustin, Existence and uniqueness of weak solution for a flexible Euler-Bernoulli beam with variable coefficients, Far East Journal of Applied Mathematics 110(1) (2021), 65-80. DOI: 10.17654/AM110010065

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

References

[1] Francisco Chinesta, Roland Keunings and Adrien Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations, Springer Briefs in Applied Sciences and Technology, Library of Congress Control Number: 2013951633, 2014.
[2] Elias Cueto, David Gonzalez and Iciar Alfaro, Proper Generalized Decompositions An Introduction to Computer Implementation with Matlab, Springer Briefs in Applied Sciences and Technology, Library of Congress Control Number: 2016933200, 2016.
[3] Francisco Chinesta and Elias Cueto, PGD-based Modeling of Materials, Structures and Processes, ESAFORM Book Series on Material Forming, Library of Congress Control Number: 2014937278, 2014.
[4] A. Dumon, C. Allery and A. Ammar, Proper general decomposition (PGD) for the resolution of Navier-Stokes equations, J. Comput. Phys. 230 (2011), 1387-1407.
[5] David Gonzalez, Elias Cueto, Francisco Chinesta, Pedro Diez and Antonio Huerta, Streamline upwind/Petrov-Galerkin-based stabilization of proper generalized decompositions for high-dimensional advection-diffusion equations, Internat. J. Numer. Methods Engrg. 94 (2013), 1216-1232.