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Add autonomous system of differential equations solutions, separate out FEM, in applications section |
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==Applications==
=== Systems of differential equations ===
Quadratic eigenvalue problems arise naturally in the solution of systems of second order linear differential equations without forcing:
:<math> M q''(t) +C q'(t) + K q(t) = 0 </math>
Where <math> q(t) \in \mathbb{R}^n </math>, and <math> M, C, K \in \mathbb{R}^{n\times n}</math>. If all quadratic eigenvalues of <math> Q(\lambda) = \lambda^2 M + \lambda C + K </math> are distinct, then the solution can be written as
:<math>
q(t) = \sum_{j=1}^{2n} \alpha_j x_j e^{\lambda_j t} = X e^{\Lambda t} \alpha
</math>
Where <math>\Lambda = \text{Diag}([\lambda_1, \ldots, \lambda_{2n}]) \in \mathbb{R}^{2n, 2n} </math>, <math> X = [x_1, \ldots, x_{2n}] \in \mathbb{R}^{n, 2n} </math>, and <math> \alpha = [\alpha_1, \cdots, \alpha_{2n}]^\top \in \mathbb{R}^{2n}</math>. [[Stability theory]] for linear equations can now be applied, as the behavior of a solution depends explicitly on the (quadratic) eigenvalues.
=== Finite element methods ===
A QEP can result in part of the dynamic analysis of structures [[Discretization|discretized]] by the [[finite element method]]. In this case the quadratic, <math>Q(\lambda)</math> has the form <math>Q(\lambda)=\lambda^2 M + \lambda C + K</math>, where <math>M</math> is the [[mass matrix]], <math>C</math> is the [[damping matrix]] and <math>K</math> is the [[stiffness matrix]].
Other applications include vibro-acoustics and fluid dynamics.
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