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Equivalent Systems for Differential Equations with Polynomially Distributed Delay

Roland Pulch

TL;DR

This paper tackles delay differential equations with polynomially distributed delays by deriving an exact equivalent system that uses only two discrete delays, enabling standard DDE techniques to be applied; it also introduces a Gaussian-quadrature discretisation to approximate the distributed delay as a DDE with multiple discrete delays. The authors establish the auxiliary-function framework $x_i(t)=\int_a^b y(t-\tau) \tau^i d\tau$ and show how to replace the integral with $\sum_{i=0}^n \alpha_i x_i(t)$, leading to a coupled system whose stability is analyzed; a scaling $\tilde{t}=t/b$ is proposed to mitigate error amplification. Generalisations to systems and to multiple distributed delays are provided, along with a quadrature-based alternative that preserves the distributed-delay dynamics in approximate form. Numerical experiments on a nonlinear test model (SIR-type) compare both approaches, showing qualitative agreement and a convergence of quadrature-based solutions to the exact distributed-delay solution as the number of nodes grows, with time-step accuracy playing a crucial role in the observed error growth.

Abstract

We consider delay differential equations with a polynomially distributed delay. We derive an equivalent system of delay differential equations, which includes just two discrete delays. The stability of the equivalent system and its stationary solutions are investigated. Alternatively, a Gaussian quadrature generates a discretisation of the integral, which describes the distributed delay in the original delay differential equation. This technique yields an approximate differential equation with multiple discrete delays. We present results of numerical computations, where initial value problems of the differential equations are solved. Therein, the two approaches are compared.

Equivalent Systems for Differential Equations with Polynomially Distributed Delay

TL;DR

This paper tackles delay differential equations with polynomially distributed delays by deriving an exact equivalent system that uses only two discrete delays, enabling standard DDE techniques to be applied; it also introduces a Gaussian-quadrature discretisation to approximate the distributed delay as a DDE with multiple discrete delays. The authors establish the auxiliary-function framework and show how to replace the integral with , leading to a coupled system whose stability is analyzed; a scaling is proposed to mitigate error amplification. Generalisations to systems and to multiple distributed delays are provided, along with a quadrature-based alternative that preserves the distributed-delay dynamics in approximate form. Numerical experiments on a nonlinear test model (SIR-type) compare both approaches, showing qualitative agreement and a convergence of quadrature-based solutions to the exact distributed-delay solution as the number of nodes grows, with time-step accuracy playing a crucial role in the observed error growth.

Abstract

We consider delay differential equations with a polynomially distributed delay. We derive an equivalent system of delay differential equations, which includes just two discrete delays. The stability of the equivalent system and its stationary solutions are investigated. Alternatively, a Gaussian quadrature generates a discretisation of the integral, which describes the distributed delay in the original delay differential equation. This technique yields an approximate differential equation with multiple discrete delays. We present results of numerical computations, where initial value problems of the differential equations are solved. Therein, the two approaches are compared.
Paper Structure (13 sections, 3 theorems, 43 equations, 4 figures, 1 table)

This paper contains 13 sections, 3 theorems, 43 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem definition
  3. Equivalent systems
  4. Preliminaries
  5. Derivation of systems
  6. Stability
  7. Scaling
  8. Generalisations to systems
  9. Systems with one distributed delay
  10. Systems with multiple distributed delays
  11. Approximate DDEs from quadrature
  12. Numerical results
  13. Conclusions

Key Result

Theorem 1

Let $y : [-b,\infty) \rightarrow \mathbbm{R}$ be continuous. We define for $i \ge 0$ and $t \ge 0$. It follows that $x$ is continuously differentiable and for $i \ge 0$ and $t > 0$, without loss of generality let $x_{-1} \equiv 0$.

Figures (4)

  • Figure 1: Numerical solution of DDE system with two discrete delays in case (i).
  • Figure 2: Numerical solution of DDE system with two discrete delays in case (ii).
  • Figure 3: Numerical solution of DDE systems including a single delay.
  • Figure 4: Maximum difference between solutions of equivalent system and solutions of approximate systems using Gaussian quadrature with different numbers of nodes.

Theorems & Definitions (9)

  • Remark 1
  • Definition 1
  • Theorem 1
  • Remark 2
  • Definition 2
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Remark 4