# A Closed-form Approximation for Impulse Response of Fractionally Damped Oscillators

## TL;DR

This paper introduces a computationally efficient and robust closed-form approximation for the impulse response of fractionally damped oscillators, overcoming numerical instabilities of the infinite series representation.

## Contribution

The authors develop a novel closed-form approximation for the impulse response of fractional oscillators, improving computational stability and efficiency over the traditional series approach.

## Key findings

- Approximation reduces computational instability.
- Maintains reasonable accuracy in practical applications.
- Enhances efficiency for engineering use.

## Abstract

We consider a fractionally damped oscillator, where the damping term is expressed by the Caputo fractional derivative of order $\beta\in (0,1).$ The impulse response of this oscillator can be expressed in terms of the bivariate Mittag-Leffler function consisting of a double infinite series. Although this series is uniformly convergent, its numerical implementation suffers from computational instabilities. In this contribution, we propose an approximate closed-form solution that avoids these numerical pitfalls while maintaining a reasonable accuracy. The resulting approximation is computationally efficient and robust, making it suitable for practical engineering applications.