Related papers: D'Alembert sums for vibrating bar with viscous end…
Fractional wave equation arises in different type of physical problems such as the vibrating strings, propagation of electro-magnetic waves, and for many other systems. The exact analytical solution of the fractional differential equation…
We investigate longitudinal vibrations of a bar subjected to viscous boundary conditions at each end, and an internal damper at an arbitrary point along the bar's length. The system is described by four independent parameters and exhibits a…
This paper presents the generalized Fourier series solution for the longitudinal vibrations of a bar subjected to viscous boundary conditions at each end. The model of the system produces a non-selfadjoint eigenvalue problem which does not…
We propose an analytical framework to model the effect of single and multiple mechanical surface oscillators on the dynamics of vertically polarized elastic waves propagating in a semi-infinite medium. The formulation extends the canonical…
This paper deals with the inverse problem of recovering an arbitrary number of fractional damping terms in a wave equation. We develop several approaches on uniqueness and reconstruction, some of them relying on Tauberian theorems on the…
We consider the bi-Laplacian eigenvalue problem for the modes of vibration of a thin elastic plate with a discrete set of clamped points. A high-order boundary integral equation method is developed for efficient numerical determination of…
In this paper we analyze a finite element method for solving a quadratic eigenvalue problem derived from the acoustic vibration problem for a heterogeneous dissipative fluid. The problem is shown to be equivalent to the spectral problem for…
In this work we explore the fidelity of numerical approximations to the analytic spectra of hyperbolic partial differential equation systems with variable coefficients. We are particularly interested in the ability of discrete methods to…
A non-local modified gravity model with an analytic function of the d'Alembert operator is considered. This model has been recently proposed as a possible way of resolving the singularities problem in cosmology. We present an exact bouncing…
In this paper, we introduce a novel semi-analytical method for solving a broad class of initial value problems involving differential, integro-differential, and delay equations, including those with fractional and variable-order…
An alternative way of visualizing electromagnetic waves in matter and of deriving the Finite Difference Time Domain method (FDTD) for simulating Maxwell's equations for one dimensional systems is presented. The method uses d'Alembert's…
We present a new algorithm which is named the Dynamical Functional Particle Method, DFPM. It is based on the idea of formulating a finite dimensional damped dynamical system whose stationary points are the solution to the original…
We explicitly construct the analogue of the \v{d}Alembert solution to the 1+1 wave equation in an hyperboloidal setting. This hyperboloidal \v{d}Alembert solution is used, in turn, to gain intuition into the behaviour of solutions to the…
Given a truncated perturbation expansion of a physical quantity, one can, under certain circumstances, obtain lower or upper bounds (or both) to the sum of the full perturbation series by using the Borel transform and a variational…
A multidimensional extremal problem in the idempotent algebra setting is considered which consists in minimizing a nonlinear functional defined on a finite-dimensional semimodule over an idempotent semifield. The problem integrates two…
We investigate the evolution of dispersive waves governed by linear wave equations, where a finite duration source is applied. The resulting wave may be viewed as the superposition of modes before the source is turned on and after it is…
In this paper, we study numerical methods for the solution of partial differential equations on evolving surfaces. The evolving hypersurface in $\Bbb{R}^d$ defines a $d$-dimensional space-time manifold in the space-time continuum…
The displacement field for three dimensional dynamic elasticity problems in the frequency domain can be decomposed into a sum of a longitudinal and a transversal part known as a Helmholtz decomposition. The Cartesian components of both the…
We study damped wave propagation problems phrased as abstract evolution equations in Hilbert spaces. Under some general assumptions, including a natural compatibility condition for initial values, we establish exponential decay estimates…
The mathematical model of a real flexible elastic system with distributed and discrete parameters is considered. It is a partial differential equation with non-classical boundary conditions. Complexity of the boundary conditions results in…