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The Bernstein polynomial basis sees significant use owing to its unique properties, particularly in the field of optimal control. However, the basis is known to have a slow rate of convergence to the function it approximates. With this in…
In this paper, we propose a novel and efficient differential quadrature element based on Lagrange interpolation to solve a sixth order partial differential equations encountered in non-classical beam theories. These non-classical theories…
This paper introduces an efficient high-order numerical method for solving the 1D stationary Schr\"odinger equation in the highly oscillatory regime. Building upon the ideas from [Arnold, Ben Abdallah, Negulescu, SIAM J. Numer. Anal.,…
In this paper, a well-conditioned collocation method is constructed for solving general $p$-th order linear differential equations with various types of boundary conditions. Based on a suitable Birkhoff interpolation, we obtain a new set of…
We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution…
In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can…
This overview is devoted to splitting methods, a class of numerical integrators intended for differential equations that can be subdivided into different problems easier to solve than the original system. Closely connected with this class…
The model of laminated wave turbulence puts forth a novel computational problem - construction of fast algorithms for finding exact solutions of Diophantine equations in integers of order $10^{12}$ and more. The equations to be solved in…
We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in…
Solutions of Fredholm integral equations of the second kind with oscillatory kernels likely exhibit oscillation. Standard numerical methods applied to solving equations of this type have poor numerical performance due to the influence of…
We reconsider the variational integration of optimal control problems for mechanical systems based on a direct discretization of the Lagrange-d'Alembert principle. This approach yields discrete dynamical constraints which by construction…
This contribution presents an integration method based on the Simpson quadrature. The integrator is designed for finite-dimensional nonlinear mechanical systems that derive from variational principles. The action is discretized using…
A novel efficient and high accuracy numerical method for the time-fractional differential equations (TFDEs) is proposed in this work. We show the equivalence between TFDEs and the integer-order extended parametric differential equations…
We consider the numerical integration of the Gross-Pitaevskii equation with a potential trap given by a time-dependent harmonic potential or a small perturbation thereof. Splitting methods are frequently used with Fourier techniques since…
Contact interfaces with dry friction are frequently used in turbomachinery. Dry friction damping produced by the sliding surfaces of these interfaces reduces the amplitude of bladed-disk vibration. The relative displacements at these…
This contribution investigates the connection between isogeometric analysis and integral equation methods for full-wave electromagnetic problems up to the low-frequency limit. The proposed spline-based integral equation method allows for an…
We consider a recently introduced formulation for fluid-structure interaction problems which makes use of a distributed Lagrange multiplier in the spirit of the fictitious domain method. In this paper we focus on time integration methods of…
In the paper, we propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for solving Stokes and Navier-Stokes equations. We start with a detailed explanation of the method for the…
We present a stable and convergent method for solving initial value problems based on the use of differentiation matrices obtained by Lagrange interpolation. This implicit multistep-like method is easy-to-use and performs pretty well in the…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…