Nonstandard finite difference schemes for general linear delay differential systems

Abstract

This paper deals with the construction of non‐standard finite difference methods for coupled linear delay differential systems in the general case of non‐commuting matrix coefficients. Based on an expression for the exact solution of the continuous initial value vector delay problem, a family of non‐standard numerical methods of increasing orders is defined. Numerical examples show that the new proposed numerical methods preserve the stability properties of the exact solutions. This work extends previous results for delay systems with commuting matrix coefficients, allowing the use of the new numerical non‐standard methods in more general problems.

Exact numerical solutions and high order nonstandard difference schemes for a second order delay differential equation

“…We also recall that, from Lemma 2 in Castro et al [4], the infinite sums in () are absolutely convergent, and that the matrices defined in () satisfy $$$$ {Q}_1^{\prime }(t)=A{Q}_1(t)+A+B,\kern2em {Q}_k^{\prime }(t)=A{Q}_k(t)+B{Q}_{k-1}(t),k\ge 2. $$$$ …”

“…From Lemma 1, considering a regular mesh of amplitude $$$ h=\tau /N $$$, for some integer $$$ N\ge 1 $$$, an exact numerical scheme can be immediately derived [4, Theorem 2]. Writing $$$ {t}_n\equiv nh $$$ and $$$ {X}_n\equiv X\left({t}_n\right) $$$, for $$$ n\ge -N $$$, an exact numerical solution is given by $$$ {X}_n=F\left({t}_n\right) $$$, for $$$ -N\le n\le 0 $$$, and, for $$$ \left(m-1\right)\tau \le nh< m\tau $$$ and $$$ m\ge 1 $$$, by …”

“…Although the exact numerical solution given by Theorem 1 avoids the infinite sums present in the solution of the general problem ()–() given in Theorem 2 in Castro et al [4], recalled in (), it still includes an integral term, $$$ {R}_m(h) $$$, which in general needs to be numerically approximated, except for particular initial functions $$$ F(t) $$$ that could allow an exact computation. Therefore, in our next Theorem, we propose two families of nonstandard methods of as high precision as needed, by the same approach previously developed in García et al [3] for coupled systems with commuting coefficients, computing in the first $$$ M $$$ intervals the exact solution, and then discarding the integral term in the following intervals, either computing the full sum in () or truncating it up to the $$$ M $$$ term.…”

“…Castro et al [4] considered problem ()–() in the general case of non‐commuting matrix coefficients $$$ A $$$ and $$$ B $$$. They provided an expression for the exact numerical solution of ()–() involving infinite series, which were truncated to construct NSFD methods of any desire order.…”

“…It was suggested in Castro et al [4] that the expressions given there could be used to obtain closed form exact numerical solutions for particular problems, depending on the structure of the specific coefficient matrices, and in particular that they could provide the basis to tackle higher order scalar problems by reducing them to first order vector problems, as the resulting coefficient matrices would be in general non‐commuting. In the present work, we show that this strategy can be successfully applied to the second order initial‐value DDE problem $$$$ {x}^{\prime \prime }(t)= ax(t)+ bx\left(t-\tau \right),\kern2em t>0, $$$$ $$$$ x(t)=f(t),\kern4.00em -\tau \le t\le 0, $$$$ where $$$ a,b\in \mathrm{\mathbb{R}} $$$, $$$ \tau >0 $$$, and $$$ f:\left[-\tau, 0\right]\to \mathrm{\mathbb{R}} $$$ is a continuous function.…”