Related papers: A fast and well-conditioned spectral method
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
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.
Several recently discovered properties of multiple families of special polynomials (some orthogonal and some not) that satisfy certain differential, difference or q-difference equations are reviewed. A general method of construction of…
We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we…
This paper develops a computational method for studying stable/unstable manifolds attached to periodic orbits of differential equations. The method uses high order Chebyshev-Taylor series approximations in conjunction with the…
We introduce a novel spectral element method based on the ultraspherical spectral method and the hierarchical Poincar\'{e}-Steklov scheme for solving second-order linear partial differential equations on polygonal domains with unstructured…
We present a novel and unifying framework for constructing spectral approximations to fractional integral operators. These spectral approximations are based on transplanted Chebyshev polynomials, which are obtained by composing Chebyshev…
This paper reports on recent work to compute the asymptotic solution of a n-th order ordinary differential equation. Symbolic methods are used to compute the asymptotics over a large region. Application is made to the computation of the…
Numerical approximation of the Boltzmann equation presents a challenging problem due to its high-dimensional, nonlinear, and nonlocal collision operator. Among the deterministic methods, the Fourier-Galerkin spectral method stands out for…
We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral…
Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of…
We develop a spectrally accurate numerical method to compute solutions of a model partial differential equation used in plasma physics to describe diffusion in velocity space due to Fokker-Planck collisions. The solution is represented as a…
We develop a spectral method to solve the heat equation in a closed cylinder, achieving a quasi-optimal $\mathcal{O}(N\log N)$ complexity and high-order, spectral accuracy. The algorithm relies on a Chebyshev--Chebyshev--Fourier (CCF)…
The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
A spectral element method (SEM) is developed to solve polarized radiative transfer in multidimensional participating medium. The angular discretization is based on the discrete-ordinates approach, and the spatial discretization is conducted…
We present a spectral method for one-sided linear fractional integral equations on a closed interval that achieves exponentially fast convergence for a variety of equations, including ones with irrational order, multiple fractional orders,…
In this paper, a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients. This method is based on our previous work [10] for convection-diffusion equations, which relies on a…
Calculating the spectral function of two dimensional systems is arguably one of the most pressing challenges in modern computational condensed matter physics. While efficient techniques are available in lower dimensions, two dimensional…
This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial…