Related papers: Spectral element methods for boundary-value proble…
Based on the Fourier extension, we propose an oversampling collocation method for solving the elliptic partial differential equations with variable coefficients over arbitrary irregular domains. This method only uses the function values on…
Motivated by Fredholm theory, we develop a framework to establish the convergence of spectral methods for operator equations $\mathcal L u = f$. The framework posits the existence of a left-Fredholm regulator for $\mathcal L$ and the…
We extend the use of piecewise orthogonal collocation to computing periodic solutions of renewal equations, which are particularly important in modeling population dynamics. We prove convergence through a rigorous error analysis. Finally,…
In this work, we exploit the capability of virtual element methods in accommodating approximation spaces featuring high-order continuity to numerically approximate differential problems of the form $\Delta^p u =f$, $p\ge1$. More…
In this article, we study numerical approximation of eigenvalue problems of the Schr\"{o}dinger operator $\displaystyle -\Delta u + \frac{c^2}{|x|^2}u$. There are three stages in our investigation: We start from a ball of any dimension, in…
Starting with some fundamental concepts, in this article we present the essential aspects of spectral methods and their applications to the numerical solution of Partial Differential Equations (PDEs). We start by using Lagrange and…
A spectral decomposition method is used to obtain solutions to a class of nonlinear differential equations. We extend this approach to the analysis of the fractional form of these equations and demonstrate the method by applying it to the…
In this paper, we design and analyze a novel spectral method for the subdiffusion equation. As it has been known, the solutions of this equation are usually singular near the initial time. Consequently, direct application of the traditional…
We demonstrate an application of the spectral method as a numerical approximation for solving Hyperbolic PDEs. In this method a finite basis is used for approximating the solutions. In particular, we demonstrate a set of such solutions for…
We develop an efficient $hp$-finite element method for piecewise-smooth differential equations with periodic boundary conditions, using orthogonal polynomials defined on circular arcs. The operators derived from this basis are banded and…
We consider the 2D quasi-periodic scattering problem in optics, which has been modelled by a boundary value problem governed by Helmholtz equation with transparent boundary conditions. A spectral collocation method and a tensor product…
A framework for Chebyshev spectral collocation methods for the numerical solution of functional and delay differential equations (FDEs and DDEs) is described. The framework combines interpolation via the barycentric resampling matrix with a…
In this paper we show that we can use a modified version of the h-p spectral element method proposed in \cite{duttora1,duttom,duttora2,tomarth} to solve elliptic problems with general boundary conditions to exponential accuracy on polygonal…
To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving…
This work investigates two physics-based models that simulate the non-linear partial differential algebraic equations describing an electric double layer supercapacitor. In one model the linear dependence between electrolyte concentration…
In this paper we consider the numerical solution of fractional differential equations. In particular, we study a step-by-step graded mesh procedure based on an expansion of the vector field using orthonormal Jacobi polynomials. Under mild…
In recent years we provided numerical methods based on pseudospectral collocation for computing the Floquet multipliers of different types of delay equations, with the goal of studying the stability of their periodic solutions. The latest…
A new approach for integration of the initial value problem for ordinary differential equations is suggested. The algorithm is based on approximation of the solution by a system of functions that contains orthogonal exponential polynomials.
This work is concerned with spectral collocation methods for fractional PDEs in unbounded domains. The method consists of expanding the solution with proper global basis functions and imposing collocation conditions on the Gauss-Hermite…
We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for the numerical solution of partial differential equations. We start with a detailed explanation of the method for the…