Caputo fractionally damped oscillator:viscoelastic memory and numerical error control in physics teaching

Abstract

A mass supported by a viscoelastic material may show small oscillations while still
taking a long time to return to equilibrium. This behavior motivates the study of an oscillator
with Caputo fractional damping, whose damping force depends on the history of motion. Starting
from a mechanical support example, we introduce the model, explain its parameters and discuss
the known solution in terms of a bivariate Mittag–Leffler function. The response combines an
oscillatory transient with a residual displacement that follows a power-law decay. We evaluate
this response by comparing a truncated series with numerical Laplace inversion. On the tested
grid, forty series layers yield a maximum difference of approximately 6.03 × 10−18 from the
inversion result. We also present an upper bound on the truncation error over a whole interval,
distinguishing this guarantee from agreement at selected points. The application relates the results
to a displacement tolerance and provides the basis for undergraduate activities. The paper brings
together physical interpretation, numerical checks and the material needed to reproduce the results;
the longer mathematical arguments are given in the appendices