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, initial-boundary value problems (IBVPs) for a semi-infinite string with a tuned-mass-damper (TMD) system attached at one end are studied. While previous studies have focused primarily on the linear behavior of springs, we extend the analysis to include cubic nonlinearity. Four types of TMD system are considered, that is, a dashpot-linear spring, a mass-dashpot-linear spring, a dashpot-nonlinear spring, and a mass-dashpot-nonlinear spring, to assess wave reflections under these configurations. A key contribution of this research is that, rather than assuming predefined forms for the reflected wave, we derive the reflection shapes directly from calculations, offering new insights into wave dynamics. The D’Alembert formula is used to describe the general solution of the wave equation, accounting for the string’s initial velocity and displacement. For the nonlinear cases, the Multiple Scales Perturbation (MSP) method is used to approximate solutions. Our results demonstrate that the mass, spring, and damper coefficients strongly influence the wave reflections. Additionally, adjusting the damping coefficient to small values reveals a completely different behavior of the reflected waves compared to large values. Numerical simulations using a fourth-order Runge-Kutta (RK4) method and a central finite difference scheme support the analytical results. Energy dissipation is studied for all scenarios, confirming that the solutions remain bounded over large timescales.

Localized waves in an Euler–Bernoulli beam on a weakly nonlinear elastic foundation, which has a specific weak structural damping, and which is under the action of a moving (constant speed) concentrated load, are studied. An asymptotic approach describing the effect of loading and damping on localized solutions (that is, on kinks and pulses), is developed. To find approximations for these kink- and pulse-like solutions, variational methods with heuristic test functions are applied. Some new effects are presented. In particular, for solitary waves we observe a resonance, when the velocity of the load and the pulse are equal, and the amplitude of the pulse increases. Also, this resonance leads to a delta-shape disturbance in the wave velocity, and leads to a jump in the pulse coordinate. Another interesting result is that for kinks there are special resonances, where the kink location experiences a jump.

The vibrations of electrically and symmetrically actuated micro-rectangular plates are considered. Coulomb and geometric nonlinearities are taken into account, as well as small linear damping. The Berger model and the Kantorovich procedure are used, which allows one to reduce the original PDE to a nonlinear ODE. Based on this ODE, the dynamic pull-in phenomenon is investigated, i.e., the transition from the oscillatory regime to an attracting one is studied. An effective numerical algorithm to determine the voltage for which the system collapses is proposed. Furthermore, the bounds for the regions of parametric instability can be computed. Applications with only an AC voltage and various variants with combined AC and DC voltages were considered. The study was carried out for a wide range of AC voltage frequency variations, that is, from zero to twice the natural frequency of the linear plate. In all cases, the instabilities that arise are parametric in nature. Along with the study of the original nonlinear equation, its simplified versions are also considered. Linearization of the Coulomb nonlinearity leads to a Mathieu–Duffing equation with double parametric forcing. Neglecting the geometric nonlinearity, a Mathieu equation with double parametric forcing is obtained. An interesting conclusion of the numerical experiments is that linearization of the original nonlinear equation and its replacement by the Mathieu equation allow one to determine quite accurately the dynamic pull-in of a microplate upon which a symmetric electrostatic actuation is applied. Since the theory for Mathieu equation is well developed, it is possible to analytically determine the frequency of the parametric resonances of the system. This is important, since in the vicinity of the corresponding AC voltage frequencies, sharp decreases in the dynamic pull-in values occur.

The effect of small internal and dashpot damping on a trapped mode of a 1D-waveguide, that is, a semi-infinite string on a Winkler elastic foundation, has been investigated. At the edge of the string a mass–spring–damper system is attached. The string is assumed to have an internal damping. Four models for the internal damping are considered: air damping, Kelvin–Voigt damping, local Kelvin–Voigt damping, and damping related to time hysteresis. Depending on the internal damping and the parameters in the formulated problem, it will be shown that the amplitude of a trapped mode of the string can decrease or increase with time.

In this paper, we develop a vibrating string model to describe the oscillations within a bio-mimetic movable pulley actuator, where transmission point disturbances can induce resonances, and so jeopardise system performance. The dynamics of longitudinal axial vibrations are formulated by a wave equation on a slowly time-varying spatial domain with a moving mass and a small harmonic disturbance at the mass-point. Due to the slow spatial domain variation, a singular perturbation problem arises. Utilizing the semigroup method, we establish the existence and uniqueness of the system’s solution. Through an averaging technique and an interior layer analysis, we construct formal asymptotic approximations for the solution. Our findings reveal that, for specific disturbance frequencies, many oscillation modes jump up from small order ε amplitudes to those of order ε. To suppress the resonances, we introduce viscous damping of varying orders. By employing multiple time-scales perturbation methods, we demonstrate that different orders of the viscous damping produce distinct anti-resonance results. Lastly, numerical simulations validate both the accuracy of our analytical results and the efficacy of the anti-resonance strategies employed.

The vibrations of electrically actuated micro- and nano rectangular plates, described by strongly nonlinear PDEs, are considered. The geometric nonlinearity is taken into account within the Berger model. One of the essentially nonlinear effects is the pull-in phenomenon, i.e., the transition from the oscillatory regime to the attracting one. A simple and physically justified algorithm for determining the voltage for which the system collapses is proposed. The algorithm is based on the detection of the voltage that leads to the merging of stable (center) and unstable (saddle) equilibrium points. The model and algorithm were validated by comparing them with other existing results in the literature, which were obtained by using the Galerkin method and FEM. The closeness of these results confirmed the adequacy of the adopted model and the high accuracy of the algorithm used in this paper. The study was conducted for a wide range of frequency changes and amplitude ratios of DC and AC voltages. Along with the determination of the pull-in voltages, the change in displacements over time was also studied. A spectral analysis was performed, which allows us to analyze the relationships between the input frequencies and the response spectra. The presence of the AC can lead to a dramatic decrease of the pull-in values. This is caused by resonances. The resonances arising in the system have a dual character. This can be either a nonlinear resonance caused by force excitation or a parametric resonance. A separate study was conducted to determine the nature of the emerging resonances. This provides useful information in practical situations. Knowing the resonant frequencies allows us to avoid them in operating electromechanical systems. This can be done by changing the frequency of the alternating voltage or by changing the ratio of the parameters of the system itself. The amplitude of the nonlinear resonance caused by force excitation can be reduced by introducing linear or nonlinear damping. Knowledge of resonant frequencies allows us to design new more effective electromechanical systems.

Localized waves in an Euler–Bernoulli beam on a weakly nonlinear elastic foundation, which has a specific weakly nonlinear structural damping, and is under the action of a moving (constant speed) concentrated load, are studied. A structural nonlinear damping, and a damping in the elastic foundation are taking into account. New approximation formulas, which describe in one expression both linear and nonlinear damping characteristics depending on the model parameters, are found. The localized approximations of the solutions of the weakly nonlinear problem have been obtained by using the Ritz method and perturbation techniques. Relatively simple formulas obtained for the asymptotic localized solution, show that the nonlinearity in the elastic foundation leads to smaller amplitudes in the displacements than in case of a linear foundation. The nonlinearity also leads to a smaller value of the critical velocity in the considered system compared to the linear case. The coefficient introduced in the weakly nonlinear elastic foundation also leads to a larger cut-off frequency in the system as compared to the linear case. Another property and effect of the nonlinearity in the elastic foundation is that at a “velocity resonance” the amplitudes of displacement remain finite even when damping (viscosity) is absent. The critical velocity for the weakly nonlinear problem has also been obtained, and the stability of the localized solution, which depends on the system parameters, is investigated.

In this paper, the vibrations of a string are considered. At one end of the string, a smooth obstacle is placed and the other end of the string is attached to a fixed point. The contact between the string and the obstacle varies in time, and leads to a linear, moving boundary value problem for the string vibrations. By applying a boundary fixing transformation, the problem is transformed from a linear problem with a moving boundary, to a nonlinear problem with fixed boundaries. It is assumed that the vibrations around the stationary position of the string are small. Explicit approximations of the solution are obtained by using a multiple time-scales perturbation method. Depending on the parameters in the problem, it turns out that three different cases for the obstacle boundary condition have to be considered, that is, Dirichlet, or Neumann, or Robin type of boundary conditions. To avoid an infinite-dimensional system of ordinary differential equations that occurs in the analysis of the modal interactions of the string vibrations, characteristic coordinates are used together with a multiple time-scales approach to analyze the string dynamics in terms of traveling waves in opposite directions. A comparison between a direct numerical integration of the PDE problem and the results obtained by using the aforementioned perturbation approach shows an excellent agreement in the results.

In this paper, a classical Stefan problem with a prescribed and small time-dependent temperature at the boundary is studied. By using a multiple time-scales perturbation method, it is shown analytically how the moving boundary profile is influenced by the prescribed temperature at the boundary and the initial conditions. Only a few exact solutions are available for this type of problems and it turns out that the constructed approximations agree very well with these exact solutions. In particular, approximations of solutions for this type of problems, with periodic and decaying temperatures at the boundary, are constructed. Furthermore, these approximations are valid on a long time scale, and seems to be not available in the literature.

In this paper, polynomial equations with real coefficients and in one variable were considered which contained a small, positive but specified and fixed parameter ε₀ ≠ 0. By using the classical asymptotic method, roots of the polynomial equations have been constructed in the literature, which were proved to be valid for sufficiently small ε-values (or equivalently for ε → 0). In this paper, it was assumed that for some or all roots of a polynomial equation, the first few terms in a Taylor or Laurent series in a small parameter depending on ε exist and can be constructed. We also assumed that at least two approximations x₁ (ε) and x₂ (ε) for the real roots exist and can be constructed. For a complex root, we assumed that at least two real approximations a₁ (ε) and a₂ (ε) for the real part of this root, and that at least two real approximations b₁ (ε) and b₂ (ε) for the imaginary part of this root, exist and can be constructed. Usually it was not clear whether for ε = ε₀ the approximations were valid or not. It was shown in this paper how the classical asymptotic method in combination with the bisection method could be used to prove how accurate the constructed approximations of the roots were for a given interval in ε (usually including the specified and fixed value ε₀ ≠ 0). The method was illustrated by studying a polynomial equation of degree five with a small but fixed parameter ε₀ = 0.1. It was shown how (absolute and relative) error estimates for the real and imaginary parts of the roots could be obtained for all values of the small parameter in the interval (0, ε₀ ].

In this paper, the dynamics of a compressed Euler-Bernoulli beam on a Winkler elastic foundation under the action of an external nonlinear force, which models a wind force, is studied. The beam is assumed to be long, and the lower part of its spectrum is prescribed. An asymptotic method is proposed to find the parameters of the beam, in order to have this prescribed lower part of the spectrum. All these parameters are necessary to guarantee the stability of the beam and to avoid resonances between the low frequency modes. These modes have special spatial supports that exclude a direct interaction between them. It is shown that the Galerkin system describing the time evolution can be decomposed into a system of almost independent equations which describes n independent nonlinear oscillators. Each oscillator has its own phase and frequency. It is shown that interaction between oscillators can exist only through high frequency modes.

In this paper, we present a new approach on how the multiple time-scales perturbation method can be applied to differential-delay equations such that approximations of the solutions can be obtained which are accurate on long time-scales. It will be shown how approximations can be constructed which branch off from solutions of differential-delay equations at the unperturbed level (and not from solutions of ordinary differential equations at the unperturbed level as in the classical approach in the literature). This implies that infinitely many roots of the characteristic equation for the unperturbed differential-delay equation are taken into account and that the approximations satisfy initial conditions which are given on a time-interval (determined by the delay). Simple and more advanced examples are treated in detail to show how the method based on differential and difference operators can be applied.

In this paper the wave propagation dynamics of a Lotka-Volterra type of model with cubic competition is studied. The existence of traveling waves and the uniqueness of spreading speeds are established. It is also shown that the spreading speed is equal to the minimal speed for traveling waves. Furthermore, general conditions for the linear or nonlinear selection of the spreading speed are obtained by using the comparison principle and the decay characteristics for traveling waves. By constructing upper solutions, explicit conditions to determine the linear selection of the spreading speed are derived.

In this paper, a classical Stefan problem is studied. It is assumed that a small, time-dependent heat influx is present at the boundary, and that the initial values are small. By using a multiple timescales perturbation approach, it is shown analytically (most likely for the first time in the literature) how the moving interface and its stability are influenced by the time-dependent heat influx at the boundary and by the initial conditions. Accurate approximations of the solution of the problem are constructed, which are valid on long timescales. The constructed approximations turn out to agree very well with solutions of problems for which similarity solutions are available (in numerical form).

In this paper the dynamics of a weakly nonlinear elastic string on a Winkler elastic foundation is studied. The foundation may be spatially heterogeneous. At one end of the string a mass-spring system is attached, and the other end of the string is fixed. The string is assumed to be long, and the lower part of the spectrum of the string is prescribed. It is shown that localized modes exist and that the dynamics of the string for large times is determined by these localized modes. The frequencies of these localized modes can be controlled by special choices for the spatial heterogeneities in the elastic foundation. Analytical and numerical results are presented to illustrate the findings.

In this paper, we study transverse and longitudinal oscillations and resonances in a hoisting system induced by boundary disturbances. The dynamics can be described by an initial-boundary value problem for a coupled system of nonlinear wave equations on a slowly time-varying spatial domain. It will be shown how the boundary excitations and the nonlinear terms influence transverse and longitudinal vibrations of the system. Firstly, due to the slow variation of the cable length, a singular perturbation problem arises. By using an interior layer analysis, many resonance manifolds are detected. Secondly, it will be shown that resonances in the system are caused not only by boundary disturbances but also by nonlinear interactions. Based on these observations, a three-timescales perturbation method is used to approximate the solution of the initial-boundary value problem analytically. It turns out that for special frequencies in the boundary excitations and for certain parameter values of the longitudinal stiffness and the conveyance mass, many oscillation modes jump up from small to large amplitudes in the transverse and longitudinal directions. Finally, numerical simulations are presented to verify the obtained analytical results.

In this paper an initial–boundary value problem on a bounded, fixed interval is considered for a one-dimensional and forced string equation subjected to a Dirichlet boundary condition at one end of the string and a Robin boundary condition with a slowly varying time-dependent coefficient at the other end of the string. This problem may serve as a simplified model describing transverse or longitudinal vibrations as well as resonances in axially moving cables for which the length changes in time. By introducing an adapted version of the method of separation of variables, by using averaging and singular perturbation techniques, and by finally using a three time-scales perturbation method, resonances in the problem are detected and accurate, analytical approximations of the solutions of the problem are constructed. It will turn out that small order ɛ excitations can lead to order ɛ responses when the frequency of the external force satisfies certain conditions. Finally, numerical simulations are presented, which are in full agreement with the obtained analytical results.