Discrete approximations of differential equations via trigonometric interpolation

Abstract

To approximate solutions of a linear differential equation, we project, via trigonometric interpolation, its solution space onto a finite-dimensional space of trigonometric polynomials and construct a matrix representation of the differential operator associated with the equation. We compute the ranks of the matrix representations of a certain class of linear differential operators. Our numerical tests show high accuracy and fast convergence of the method applied to several boundary and eigenvalue problems.MSC 34K28, 65N22, 15A03, 34B05, 34B09, PACS 0002 IntroductionIn this paper, we use trigonometric interpolation to approximate solutions of a differential equation Au = f , whose differential operator A with domain D(A) is a formal polynomial of operators 1, x, d dx , and f ∈ Range(A). A solution u is projected onto the space T L n of L-periodic trigonometric polynomials of degree n. The projection T [u], defined as a trigonometric interpolant of u, is identified with a vectorû of its values at N = 2n+1 partition points via an isomorphism ψ, where n ∈ N. The operator A is represented by an N × N matrix A defined implicitly by Av = ψT [AT [v]] for all v ∈ D(A). The original equation is approximated by a system of linear equations Aû =f , wheref = ψT [f ].

New Properties of the Zeros of Krall Polynomials

“…be the pseudospectral matrix representation of the operator D defined by (3). Then for all integer m, n such that 0 ≤ m ≤ N − 1 and 1 ≤ n ≤ N the following algebraic relations hold:…”

Generalized Pseudospectral Method and Zeros of Orthogonal Polynomials

“…Note that, in general, the definitions of these generalized matrix representations hold for any linear differential operator D, which may or may not satisfy property (2), and for n 0 being a (positive or negative) integer such that N + n 0 ≥ 1. The definition of the standard pseudospectral N × N matrix representatioñ A c of A, see [30,31,7,25], is motivated by a search for an exact discretization of a differential equation…”

“…Note that the superscript 'c' in the notation of the matrixà c stands for "collocation" in the spectral collocation method for numerical solving of differential equations, also known as the pseudospectral method [25]. It is not difficult to conclude that the matrixà c is given componentwise bỹ [20,21,26,27,31,30,7]. The last definition implies that a vector u (N ) ∈ R N solves the linear system…”

“…The definition of the standard pseudospectral N × N matrix representation Ãc of A, see [30,31,7,25], is motivated by a search for an exact discretization of a differential equation…”

“…where 1 ≤ k, j ≤ N , see [20,21,26,27,31,30,7]. The last definition implies that a vector u (N ) ∈ R N solves the linear system…”