Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Approximate diagonalization in differential systems and an effective algorithm for the computation of the spectral matrix

B. M. Brown, M. S. P. Eastham, D. K. R. McCormack, W. D. Evans

TL;DR

The paper develops an approximate diagonalization framework to obtain rigorous, computable asymptotics for the n-th order ODE $y^{(n)}(x)-Q(x)y(x)=0$, enabling effective computation of the Titchmarsh-Weyl spectral matrix $m_{ij}(\lambda)$ for even $n$. It introduces a sequence of near-diagonal transformations that progressively suppress perturbations, with explicit error control via $\epsilon_M$, and implements a symbolic (and partly numerical) Mathematica-based algorithm to generate and manage the resulting terms. The method is tested on $Q(x)=\lambda+x^{\alpha}$ (with $0<\alpha\le4/3$) and applied to compute spectral matrices, supported by independent checks against higher-order Airy-type results, and by comparisons with prior work (BBEMM95). The work also discusses extensions to oscillatory coefficients, Hamiltonian systems, and provably correct computations, highlighting practical benefits and limitations of the approach.

Abstract

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 Titchmarsh-Weyl M-matrix for the fourth order operator.

Approximate diagonalization in differential systems and an effective algorithm for the computation of the spectral matrix

TL;DR

The paper develops an approximate diagonalization framework to obtain rigorous, computable asymptotics for the n-th order ODE , enabling effective computation of the Titchmarsh-Weyl spectral matrix for even . It introduces a sequence of near-diagonal transformations that progressively suppress perturbations, with explicit error control via , and implements a symbolic (and partly numerical) Mathematica-based algorithm to generate and manage the resulting terms. The method is tested on (with ) and applied to compute spectral matrices, supported by independent checks against higher-order Airy-type results, and by comparisons with prior work (BBEMM95). The work also discusses extensions to oscillatory coefficients, Hamiltonian systems, and provably correct computations, highlighting practical benefits and limitations of the approach.

Abstract

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 Titchmarsh-Weyl M-matrix for the fourth order operator.

Paper Structure

This paper contains 13 sections, 1 theorem, 142 equations, 7 tables.

Table of Contents

  1. Introduction
  2. Diagonalization in differential systems
  3. Asymptotic formulae for solutions
  4. The sequence of transformations
  5. The basis of the algorithm
  6. A symbolic algorithm for the computation of solutions
  7. Generalised Titchmarsh-Weyl theory
  8. Computation of the spectral matrix
  9. Example $\alpha=1$
  10. Example $\alpha=4/3$
  11. Example $\alpha=1/2$
  12. The higher-order Airy equation
  13. Concluding remarks

Key Result

Lemma 3.1

Let $M (\geq 2)$ be an integer and let $D_M$, $R_M$ and $P$ be as in (eq:3.8) - (eq:3.10). In some interval $[X,\infty)$, let have constant sign (either $\geq 0$ or $\leq 0$), with for $j \neq k$. Also, let Then, for $1 \leq k \leq n$ and $1 \leq r \leq n$, (1.3) has solutions $y_k$ such that where, in $[X,\infty)$, the $\eta_j$$\;\;(1 \leq j \leq n)$ satisfy the estimate with $\parallel R_

Theorems & Definitions (1)

  • Lemma 3.1