A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix

A Well-Conditioned Collocation Method Using a Pseudospectral Integration Matrix

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 polynomial basis functions that results in a collocation scheme with two important features: the condition number of the linear system is independent of the n...

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order μ ∈ (0, 1) to compute that of any order k + μ with in...

A Fast and Well-Conditioned Spectral Method

A spectral method is developed for the direct solution of linear ordinary differential equations with variable coefficients. The method leads to matrices which are almost banded, and a numerical solver is presented that takes O(m2n) operations, where m is the number of Chebyshev points needed to resolve the coefficients of the differential operator and n is the number of Chebyshev coefficients ...

A Numerical Integration Method by Using Generalized Series of Functions

Particle-without-Particle: a practical pseudospectral collocation method for numerical differential equations with distributional sources

Differential equations with distributional sources—in particular, involving delta distributions and/ or derivatives thereof—have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but the singular (“point-like”) modeling of the sources in these problems typically introduces n...