In this paper, we introduce a multiple time scales (MTS) framework for partial difference equations (PΔEs). Such a framework is underdeveloped for fully discrete systems. We investigate a classical initial-boundary value problem for a PDE using a standard finite difference discretisation. For a nonlinear example, we additionally apply a nonstandard discretisation. Operators for fast and slow iteration scales are introduced, and secularity conditions governing the slow evolution of modal amplitudes are derived via discrete modal projection. Quantities such as natural frequencies and the stability of periodic solutions are analysed by comparing continuous and discrete MTS approximations. We prove the asymptotic validity of approximations in a Hilbert space setting. For standard discretisations, we derive the bounds on the mode numbers that can be accurately represented by given spatial and temporal resolutions. Beyond these bounds, the discrete natural frequencies can lead to spurious modal interactions, causing the approximations to fail at O(ε) accuracy over iteration scales of O1ε. For both standard and nonstandard schemes, we obtain qualitatively consistent solutions. Notably, the nonstandard discretisation yields solutions that exactly match the continuous PDE in the limit ε=0 and closely approximate the continuous MTS expansion for 0<ε≪1.

In this paper, we investigate an initial-boundary value problem for a linear Euler-Bernoulli beam equation governing the dynamics of pipes conveying fluid. The fluid flow velocity inside the pipe is assumed to have a small amplitude and to be time-varying, that is, V(t)=ε(V0+V1sin(Ωt)), where a two time-scales perturbation method is applied to construct approximations of the solutions. We explore fluid velocity pulsation frequencies Ω that lead to resonance in the pipe system. Depending on the order of bending stiffness, we study beam, stretched-beam and string models, each displaying distinct resonance behaviour. For special values of the frequency Ω and the bending stiffness, resonance frequencies can coincide, leading to internal resonances that exhibit more complex dynamical behaviour. We investigate both pure and detuned internal resonance scenarios, highlighting how early truncation in the number of oscillation modes leads to erroneous approximations and incorrect stability conclusions.