Hello . I want to solve this problem by finite element method. Can I guide you on how to solve this problem? It is known that the eigenvalue of this problem depends on boundary conditions. And the boundary conditions have a differential problem in time. Thanks
Explaining the complete procedure in detail is a bit too lengthy to post here. However, solving the one-dimensional wave equation is a standard example that can be found in most FEM textbooks. I recommend Cook, R. D., Malkus, D. S., Plesha, M. E., & Witt, R. J. (2002). Concepts and applications of finite element analysis.
Chapter 11 deals with dynamics. If you have any specific doubts let us know. Good luck.
Construct the corresponding residual R=your partial differential equation=0 ,
choose your weight functions=wi bar shape functions and then
integrate for each subdomain : intgral{wi*R*dv}=0
Explaining the complete procedure in detail is a bit too lengthy to post here. However, solving the one-dimensional wave equation is a standard example that can be found in most FEM textbooks. I recommend Cook, R. D., Malkus, D. S., Plesha, M. E., & Witt, R. J. (2002). Concepts and applications of finite element analysis.
Chapter 11 deals with dynamics. If you have any specific doubts let us know. Good luck.
Amirhossein Taherkhani I found this useful for your case. Please check it out.
https://dspace.library.uu.nl/bitstream/handle/1874/366451/thesis.pdf?sequence=2
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