## Abstract

The dynamic system of a structure utilized with visco-elastic dampers can be modeled by fractional differential equations. All the resulted systems of fractional differential equations can be represented in a state space and can be transformed into a system of multi-term fractional differential equations of order 1. Considering the presence of indeterministic exogenous force like earthquake we need powerful, convergent and reliable numerical methods to simulate the response of this dynamical systems. Therefore, spline collocations method has been proposed and studied for solving system of multi-term fractional differential equations of order 1. A rigorous mathematical analysis is provided to show the efficiency and effectiveness of the method. To this end, we apply a functional analysis framework to obtain convergence and superconvergence properties of the proposed methods on the graded mesh. Some numerical experiments are provided to confirm the theoretical results. Finally, this method is used for simulating the response of a 4-story building under El Centro earthquake excitation.

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Dadkhah, E., Shiri, B., Ghaffarzadeh, H. *et al.* Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods.
*J. Appl. Math. Comput.* **63**, 29–57 (2020). https://doi.org/10.1007/s12190-019-01307-5

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DOI: https://doi.org/10.1007/s12190-019-01307-5

### Keywords

- Visco-elastic dampers
- Kelvin–Voigt model
- Spline collocation method
- \(H_\infty \) norm
- Large scale structures
- q-term fractional differential system
- Caputo-fractional derivative
- Bagley–Torvik equation
- Collocation methods
- Convergent order