Skip to main content

Stability of numerical solution for partial differential equations with piecewise constant arguments

Abstract

In this paper, the numerical stability of a partial differential equation with piecewise constant arguments is considered. Firstly, the θ-methods are applied to approximate the original equation. Secondly, the numerical asymptotic stability conditions are given when the mesh ratio and the corresponding parameter satisfy certain conditions. Thirdly, the conditions under which the numerical stability region contains the analytic stability region are also established. Finally, some numerical examples are given to demonstrate the theoretical results.

Introduction

Recently, differential equations with piecewise constant arguments (EPCA) have received much attention from a number of investigators [15] in such various fields as population dynamics, physics, mechanical systems, control science and economics. The theory of EPCA was initiated in 1983 and 1984 with the contributions of Cooke and Wiener [6], Shah and Wiener [7], Wiener [8], and it has been developed by many authors [913]. In 1993, Wiener, pioneer of EPCAs, recollects in the book [14] the investigation of EPCA until that moment. Later, continuous efforts have been made devoted to considering various properties of EPCA [1518].

Generally speaking, in many cases analytic solutions of EPCA are hard to achieve and we are forced to use numerical methods to approximate them. Nevertheless, compared with the qualitative investigation of EPCA, the numerical study of EPCA is very late and rare. The original work for this field should be attributed to Liu et al. [19]. We think that it is the key step toward solving EPCA by numerical methods. Next, several results about the convergence, the stability and the dissipativity of numerical solutions for EPCA have been reported [2024]. However, all of them are based on ordinary differential equations (ODEs). To the best of the author’s knowledge, only few results were presented in the case of numerical treatment of partial differential equations with piecewise constant arguments (PEPCA) [25, 26]. In these two articles, the authors investigated the numerical stability of θ-methods and Galerkin methods for a simple PEPCA, respectively. In contrast to [25, 26], in the present paper we study a more complicated model and analyze the numerical stability.

In this paper, we consider the following initial boundary value problem (IBVP):

$$ \textstyle\begin{cases} u_{t}(x,t)=a^{2}u_{xx}(x,t)+bu_{xx}(x,[t])+cu_{xx} (x,2 [\frac {t+1}{2} ] ), \quad t> 0, \\ u(0,t)=u(1,t)=0, \\ u(x,0)=v(x), \end{cases} $$
(1)

where \(a,b,c \in\mathbb{R}\) and \(a\neq0\), \(u: \Omega=[0,1] \times [0,\infty)\rightarrow\mathbb{R}\), \(v: [0,1]\rightarrow\mathbb{R}\), \([\cdot]\) signifies the greatest integer function.

For the sake of the coming discussion, we derive the following stability conditions of (1) by using the similar method in [27, 28].

Lemma 1

If the following conditions are satisfied:

$$ \textstyle\begin{cases} (a^{2}+b+c)((a^{2}+b-c)e^{-a^{2}\pi^{2}j^{2}}-(b-a^{2}-c))>0, \\ (a^{2}+b+c)((a^{2}+b+c)e^{-a^{2}\pi^{2}j^{2}}-(b-a^{2}+c))>0, \end{cases} $$
(2)

where

$$c\neq\frac{a^{2}}{e^{-a^{2}\pi^{2}j^{2}}-1},\quad a\neq0, $$

then the zero solution of the equation in (1) is asymptotically stable.

The stability of the numerical solution

In this section, we consider the numerical asymptotic stability of θ-methods for (1).

The difference equation

Let \(\Delta t>0\) and \(\Delta x>0\) be time and spatial stepsizes, respectively. We also assume that Δt satisfies \(\Delta t=1/m\), where \(m \geq1\) is an integer, and Δx satisfies \(\Delta x=1/p\) for \(p\in\mathbb{N}\). Define the mesh points

$$t_{n}=n\Delta t,\quad n=0,1,2,\ldots, $$

and

$$x_{i}=i\Delta x, \quad i=0,1,2,\ldots,p. $$

Applying the θ-methods to (1), we have

$$ \textstyle\begin{cases} \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t} \\ \quad = \theta\{ a^{2}\frac{u_{i+1}^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1}}{\Delta x^{2}}+b\frac {u^{h}(x_{i+1},[t_{n+1}])-2u^{h}(x_{i},[t_{n+1}])+u^{h}(x_{i-1},[t_{n+1}])}{\Delta x^{2}} \\ \qquad {}+c\frac{u^{h}(x_{i+1},2[\frac{t_{n+1}+1}{2}])-2u^{h}(x_{i},2[\frac {t_{n+1}+1}{2}])+u^{h}(x_{i-1},2[\frac{t_{n+1}+1}{2}])}{\Delta x^{2}}\} \\ \qquad {}+(1-\theta)\{a^{2}\frac{u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n}}{\Delta x^{2}}+b\frac {u^{h}(x_{i+1},[t_{n}])-2u^{h}(x_{i},[t_{n}])+u^{h}(x_{i-1},[t_{n}])}{\Delta x^{2}} \\ \qquad {}+c\frac{u^{h}(x_{i+1},2[\frac{t_{n}+1}{2}])-2u^{h}(x_{i},2[\frac {t_{n}+1}{2}])+u^{h}(x_{i-1},2[\frac{t_{n}+1}{2}])}{\Delta x^{2}}\}, \\ u_{0}^{n}=u_{p}^{n}=0, \quad n=0,1,2,\ldots, \\ u_{i}^{0}=v(x_{i}), \quad i=0,1,2,\ldots,p, \end{cases} $$
(3)

where \(u_{i}^{n}\), \(u^{h}(x_{i},2[(t_{n}+1)/2])\) and \(u^{h}(x_{i},[t_{n}])\) are approximations to \(u(x_{i},t_{n})\), \(u(x_{i},2[(t_{n}+1)/2])\) and \(u(x_{i},[t_{n}])\), respectively.

Denote \(n=km+l\), \(k=0,1,2,\ldots\) , \(l=0,1,\ldots, m-1\), by the same technique in [29], we can define \(u^{h}(x_{i},[t_{n}+\eta h])\triangleq u_{i}^{km}\), \(u^{h}(x_{i},2[(2k-1+lh+\eta h+1)/2])\triangleq u_{i}^{2km}\) and \(u^{h}(x_{i},2[(2k+lh+\eta h+1)/2])\triangleq u_{i}^{2km}\), where \(\eta\in[0,1]\). So the equation in (3) reduces to the following two recurrence relations:

$$\begin{aligned}& \frac{u_{i}^{km+l+1}-u_{i}^{km+l}}{\Delta t} \\& \quad =a^{2}\theta \biggl(\frac {u_{i+1}^{km+l+1}-2u_{i}^{km+l+1}+u_{i-1}^{km+l+1}}{\Delta x^{2}} \biggr) +a^{2}(1-\theta) \biggl(\frac {u_{i+1}^{km+l}-2u_{i}^{km+l}+u_{i-1}^{km+l}}{\Delta x^{2}} \biggr) \\& \qquad {}+(b+c) \biggl(\frac{u_{i+1}^{km}-2u_{i}^{km}+u_{i-1}^{km}}{\Delta x^{2}} \biggr), \end{aligned}$$
(4)

when k is even and

$$\begin{aligned}& \frac{u_{i}^{km+l+1}-u_{i}^{km+l}}{\Delta t} \\& \quad =a^{2}\theta \biggl(\frac {u_{i+1}^{km+l+1}-2u_{i}^{km+l+1}+u_{i-1}^{km+l+1}}{\Delta x^{2}} \biggr) +a^{2}(1-\theta) \biggl(\frac {u_{i+1}^{km+l}-2u_{i}^{km+l}+u_{i-1}^{km+l}}{\Delta x^{2}} \biggr) \\& \qquad {}+b \biggl(\frac{u_{i+1}^{km}-2u_{i}^{km}+u_{i-1}^{km}}{\Delta x^{2}} \biggr)+c \biggl(\frac{u_{i+1}^{(k+1)m}-2u_{i}^{(k+1)m}+u_{i-1}^{(k+1)m}}{\Delta x^{2}} \biggr), \end{aligned}$$
(5)

when k is odd.

Basically, in each interval \([n,n+1)\), the equation in (1) can be seen as an original PDE, so the θ-methods for (1) are convergent of \(O(\Delta t+\Delta x^{2})\) if \(\theta\neq 1/2\), of \(O(\Delta t^{2}+\Delta x^{2})\) if \(\theta= 1/2\). A more detailed analysis on the convergence of the θ-methods can be found in [30, 31].

Let \(r=\Delta t/\Delta x^{2}\), so (4) and (5) become

$$\begin{aligned}& -a^{2}\theta ru_{i+1}^{km+l+1}+\bigl(1+2a^{2} \theta r\bigr)u_{i}^{km+l+1}-a^{2}\theta ru_{i-1}^{km+l+1} \\& \quad =a^{2}(1-\theta)ru_{i+1}^{km+l}+ \bigl(1-2a^{2}(1-\theta)r\bigr)u_{i}^{km+l}+a^{2}(1- \theta )ru_{i-1}^{km+l} \\& \qquad {}+(b+c)r\bigl(u_{i+1}^{km}-2u_{i}^{km}+u_{i-1}^{km} \bigr) \end{aligned}$$
(6)

and

$$\begin{aligned}& -a^{2}\theta ru_{i+1}^{km+l+1}+\bigl(1+2a^{2} \theta r\bigr)u_{i}^{km+l+1}-a^{2}\theta ru_{i-1}^{km+l+1} \\& \quad =a^{2}(1-\theta)ru_{i+1}^{km+l}+ \bigl(1-2a^{2}(1-\theta)r\bigr)u_{i}^{km+l}+a^{2}(1- \theta )ru_{i-1}^{km+l} \\& \qquad {} +br\bigl(u_{i+1}^{km}-2u_{i}^{km}+u_{i-1}^{km} \bigr)+cr\bigl(u_{i+1}^{(k+1)m}-2u_{i}^{(k+1)m}+u_{i-1}^{(k+1)m} \bigr), \end{aligned}$$
(7)

respectively. Moreover, let \(i=1,2,\ldots,p-1\), (6) and (7) yield

$$\begin{aligned}& \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} 1+2a^{2}\theta r & -a^{2}\theta r & \cdots& 0 & 0\\ -a^{2}\theta r & 1+2a^{2}\theta r & \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& 1+2a^{2}\theta r & -a^{2}\theta r\\ 0 & 0 & \cdots& -a^{2}\theta r & 1+2a^{2}\theta r \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km+l+1}\\ u_{2}^{km+l+1}\\ \vdots\\ \vdots\\ u_{p-1}^{km+l+1} \end{array}\displaystyle \right ) \\& \quad =\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} \omega& a^{2}(1-\theta)r & \cdots& 0 & 0\\ a^{2}(1-\theta)r & \omega& \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& \omega& a^{2}(1-\theta)r\\ 0 & 0 & \cdots& a^{2}(1-\theta)r & \omega \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km+l}\\ u_{2}^{km+l}\\ \vdots\\ \vdots\\ u_{p-1}^{km+l} \end{array}\displaystyle \right ) \\& \qquad {}+(b+c)r \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} -2 & 1 & \cdots& 0 & 0\\ 1 & -2 & \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& -2 & 1\\ 0 & 0 & \cdots& 1 & -2 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km}\\ u_{2}^{km}\\ \vdots\\ \vdots\\ u_{p-1}^{km} \end{array}\displaystyle \right ) \end{aligned}$$

and

$$\begin{aligned}& \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} 1+2a^{2}\theta r & -a^{2}\theta r & \cdots& 0 & 0\\ -a^{2}\theta r & 1+2a^{2}\theta r & \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& 1+2a^{2}\theta r & -a^{2}\theta r\\ 0 & 0 & \cdots& -a^{2}\theta r & 1+2a^{2}\theta r \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km+l+1}\\ u_{2}^{km+l+1}\\ \vdots\\ \vdots\\ u_{p-1}^{km+l+1} \end{array}\displaystyle \right ) \\& \quad =\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} \omega& a^{2}(1-\theta)r & \cdots& 0 & 0\\ a^{2}(1-\theta)r & \omega& \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& \omega& a^{2}(1-\theta)r\\ 0 & 0 & \cdots& a^{2}(1-\theta)r & \omega \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km+l}\\ u_{2}^{km+l}\\ \vdots\\ \vdots\\ u_{p-1}^{km+l} \end{array}\displaystyle \right ) \\& \qquad {}+br \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} -2 & 1 & \cdots& 0 & 0\\ 1 & -2 & \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& -2 & 1\\ 0 & 0 & \cdots& 1 & -2 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{km}\\ u_{2}^{km}\\ \vdots\\ \vdots\\ u_{p-1}^{km} \end{array}\displaystyle \right ) \\& \qquad {}+cr \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} -2 & 1 & \cdots& 0 & 0\\ 1 & -2 & \cdots& 0 & 0\\ \vdots& \vdots& \ddots& \vdots& \vdots\\ 0 & 0 & \cdots& -2 & 1\\ 0 & 0 & \cdots& 1 & -2 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{@{}c@{}} u_{1}^{(k+1)m}\\ u_{2}^{(k+1)m}\\ \vdots\\ \vdots\\ u_{p-1}^{(k+1)m} \end{array}\displaystyle \right ), \end{aligned}$$

respectively, where \(\omega=1-2a^{2}(1-\theta)r\).

Introducing \(\mathbf{u}^{n}=(u_{1}^{n},u_{2}^{n},\ldots,u_{p-1}^{n})^{T}\), \(n=0,1,2,\ldots \) , \(\mathbf{v}(x)=(v(x_{1}),v(x_{2}),\ldots,v(x_{p-1}))^{T}\) and the \((p-1)\times(p-1)\) triple-diagonal matrix \(\mathbf{F}=\mathsf {diag}(-1,2,-1)\), then (3) becomes

$$ \textstyle\begin{cases} (\mathbf{I}+a^{2}\theta r\mathbf{F})\mathbf{u}^{km+l+1}=(\mathbf {I}-a^{2}(1-\theta)r\mathbf{F})\mathbf{u}^{km+l}-(b+c)r\mathbf{F}\mathbf {u}^{km}, \\ \mathbf{u}^{0}=\mathbf{v}(x), \end{cases} $$
(8)

when k is even, and

$$ \textstyle\begin{cases} (\mathbf{I}+a^{2}\theta r\mathbf{F})\mathbf{u}^{km+l+1}=(\mathbf {I}-a^{2}(1-\theta)r\mathbf{F})\mathbf{u}^{km+l}-br\mathbf{F}\mathbf {u}^{km}-cr\mathbf{F}\mathbf{u}^{(k+1)m}, \\ \mathbf{u}^{0}=\mathbf{v}(x), \end{cases} $$
(9)

when k is odd.

Stability analysis

From (8), we obtain

$$ \mathbf{u}^{km+l+1}=\mathbf{R}\mathbf{u}^{km+l}+ \mathbf{S}\mathbf{u}^{km}, $$
(10)

where

$$\begin{aligned}& \mathbf{R}=\bigl(\mathbf{I}+a^{2}\theta r\mathbf{F} \bigr)^{-1}\bigl(\mathbf {I}-a^{2}(1-\theta)r\mathbf{F}\bigr), \\& \mathbf{S}=-(b+c)r\bigl(\mathbf{I}+a^{2}\theta r\mathbf{F} \bigr)^{-1}\mathbf{F}. \end{aligned}$$

By (9), we also obtain

$$ \mathbf{u}^{km+l+1}=\mathbf{R}\mathbf{u}^{km+l}+ \mathbf{S}_{1}\mathbf {u}^{km}+\mathbf{S}_{2} \mathbf{u}^{(k+1)m}, $$
(11)

where

$$\begin{aligned}& \mathbf{R}=\bigl(\mathbf{I}+a^{2}\theta r\mathbf{F} \bigr)^{-1}\bigl(\mathbf {I}-a^{2}(1-\theta)r\mathbf{F}\bigr), \\& \mathbf{S}_{1}=-br\bigl(\mathbf{I}+a^{2}\theta r\mathbf{F} \bigr)^{-1}\mathbf{F}, \\& \mathbf{S}_{2}=-cr\bigl(\mathbf{I}+a^{2}\theta r\mathbf{F} \bigr)^{-1}\mathbf{F}. \end{aligned}$$

Iteration of (10) gives

$$ \mathbf{u}^{km+l+1}=\bigl(\mathbf{R}^{l+1}+\bigl( \mathbf{R}^{l+1}-\mathbf {I}\bigr) (\mathbf{R}-\mathbf{I})^{-1} \mathbf{S}\bigr)\mathbf{u}^{km}, $$
(12)

in the same way, from (11) we have

$$ \mathbf{u}^{km+l+1}=\bigl(\mathbf{R}^{l+1}+\bigl( \mathbf{R}^{l+1}-\mathbf {I}\bigr) (\mathbf{R}-\mathbf{I})^{-1} \mathbf{S}_{1}\bigr)\mathbf{u}^{km}+\bigl(\mathbf {R}^{l+1}-\mathbf{I}\bigr) (\mathbf{R}-\mathbf{I})^{-1} \mathbf{S}_{2}\mathbf {u}^{(k+1)m}. $$
(13)

Thus we get

$$ \mathbf{u}^{n}= \textstyle\begin{cases} (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S})\mathbf{u}^{km}, & k \mbox{ is even}, \\ (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S}_{1})\mathbf{u}^{km} \\ \quad {} +(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf{I})^{-1}\mathbf {S}_{2}\mathbf{u}^{(k+1)m}, & k \mbox{ is odd}. \end{cases} $$
(14)

So

$$ \mathbf{u}^{n}= \textstyle\begin{cases} (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S})\mathbf{u}^{2jm}, & n=2jm+l, \\ (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S}_{1})\mathbf{u}^{(2j-1)m} \\ \quad {}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf{I})^{-1}\mathbf {S}_{2}\mathbf{u}^{2jm}, & n=(2j-1)m+l. \end{cases} $$
(15)

Let \(l=m-1\) in (15) gives

$$\textstyle\begin{cases} \mathbf{u}^{(2j+1)m}=(\mathbf{R}^{m}+(\mathbf{R}^{m}-\mathbf{I})(\mathbf {R}-\mathbf{I})^{-1}\mathbf{S})\mathbf{u}^{2jm}, \quad j=0,1,\ldots, \\ \mathbf{u}^{2jm}=(\mathbf{I}-(\mathbf{R}^{m}-\mathbf{I})(\mathbf {R}-\mathbf{I})^{-1}\mathbf{S}_{2})^{-1}(\mathbf{R}^{m}+(\mathbf {R}^{m}-\mathbf{I})(\mathbf{R}-\mathbf{I})^{-1}\mathbf{S}_{1})\mathbf {u}^{(2j-1)m}, \quad j=1,2,\ldots. \end{cases} $$

Hence we have \(\mathbf{u}^{(2j+1)m}=\mathbf{M}\mathbf{u}^{(2j-1)m}\), where

$$\mathbf{M}=\bigl(\mathbf{R}^{m}+\bigl(\mathbf{R}^{m}- \mathbf{I}\bigr) (\mathbf {R}-\mathbf{I})^{-1}\mathbf{S}\bigr) \bigl( \mathbf{R}^{m}+\bigl(\mathbf{R}^{m}-\mathbf {I}\bigr) ( \mathbf{R}-\mathbf{I})^{-1}\mathbf{S}_{1}\bigr) \bigl( \mathbf{I}-\bigl(\mathbf {R}^{m}-\mathbf{I}\bigr) (\mathbf{R}- \mathbf{I})^{-1}\mathbf{S}_{2}\bigr)^{-1}. $$

Therefore

$$ \mathbf{u}^{n}= \textstyle\begin{cases} (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S})\mathbf{M}^{j}\mathbf{u}^{0}, & n=2jm+l, j=0,1,\ldots, \\ (\mathbf{R}^{l}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf {I})^{-1}\mathbf{S}_{1})\mathbf{M}^{j-1}\mathbf{u}^{1} \\ \quad {}+(\mathbf{R}^{l}-\mathbf{I})(\mathbf{R}-\mathbf{I})^{-1}\mathbf {S}_{2}\mathbf{N}\mathbf{M}^{j}\mathbf{u}^{0},& n=(2j-1)m+l, j=1,2,\ldots, \end{cases} $$
(16)

where \(\mathbf{u}^{1}=\mathbf{N}\mathbf{u}^{0}\), \(\mathbf{N}=\mathbf {R}^{m}+(\mathbf{R}^{m}-\mathbf{I})(\mathbf{R}-\mathbf{I})^{-1}\mathbf {S}\) and \(l=0,1,\ldots,m-1\).

Lemma 2

If the coefficients a, b and c satisfy

$$ \biggl\vert \frac{\beta^{m}+\frac{b}{a^{2}}(\beta^{m}-1)}{1-\frac{c}{a^{2}}(\beta ^{m}-1)} \biggr\vert < 1 $$
(17)

and

$$ \biggl\vert \beta^{m}+\frac{b+c}{a^{2}}\bigl( \beta^{m}-1\bigr) \biggr\vert < 1, $$
(18)

then the zero solution of the equation in (3) is asymptotically stable, where

$$ \beta=\frac{1-a^{2}(1-\theta)r\lambda_{\mathbf{F}}}{1+a^{2}\theta r\lambda _{\mathbf{F}}}. $$
(19)

Proof

From (16) and [25], we know that the largest eigenvalue (in modulus) of the matrix M is

$$\lambda_{\mathbf{M}}=\frac{ (\beta^{m}+\frac{b}{a^{2}}(\beta^{m}-1) ) (\beta^{m}+\frac{b+c}{a^{2}}(\beta^{m}-1) )}{1-\frac{c}{a^{2}}(\beta^{m}-1)}, $$

where β is defined in (19). The zero solution of the equation in (3) is asymptotically stable if and only if \(|\lambda_{\mathbf{M}}|<1\). So (17) and (18) are got. □

Theorem 1

Under the conditions of Lemma 2, if the conditions

$$ \bigl(a^{2}+b+c\bigr) \bigl(\beta^{m}-1\bigr) \bigl(\bigl(a^{2}+b-c\bigr)\beta^{m}-\bigl(b-a^{2}-c \bigr)\bigr)< 0 $$
(20)

and

$$ \bigl(a^{2}+b+c\bigr) \bigl(\beta^{m}-1\bigr) \bigl(\bigl(a^{2}+b+c\bigr)\beta^{m}-\bigl(b-a^{2}+c \bigr)\bigr)< 0 $$
(21)

are satisfied, where \(c\neq a^{2}/(\beta^{m}-1)\), \(a\neq0\), then the zero solution of the equation in (3) is asymptotically stable.

Proof

If \(a\neq0\), (17) and (18) are equivalent to

$$\biggl(\frac{\beta^{m}+\frac{b}{a^{2}}(\beta^{m}-1)}{1-\frac{c}{a^{2}}(\beta ^{m}-1)}+1 \biggr) \biggl(\frac{\beta^{m}+\frac{b}{a^{2}}(\beta^{m}-1)}{1-\frac {c}{a^{2}}(\beta^{m}-1)}-1 \biggr)< 0 $$

and

$$\biggl(\beta^{m}+\frac{b+c}{a^{2}}\bigl(\beta^{m}-1\bigr)+1 \biggr) \biggl(\beta^{m}+\frac {b+c}{a^{2}}\bigl(\beta^{m}-1 \bigr)-1 \biggr)< 0. $$

After some derivations we can get (20) and (21). The proof is completed. □

Definition 1

The set of all points \((a,b,c)\) which satisfies (2) is called an asymptotic stability region denoted by H.

Definition 2

The set of all points \((a,b,c)\) at which the θ-methods for (1) which satisfies (20) is asymptotically stable is called an asymptotic stability region denoted by S.

For convenience, we divide the region H into three parts:

$$\begin{aligned}& H_{0}=\bigl\{ (a,b,c)\in H: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)=0\bigr\} , \\& H_{1}=\bigl\{ (a,b,c)\in H: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)>0\bigr\} , \\& H_{2}=\bigl\{ (a,b,c)\in H: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)< 0\bigr\} . \end{aligned}$$

In the similar way, we denote

$$\begin{aligned}& S_{0}=\bigl\{ (a,b,c)\in S: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)=0\bigr\} , \\& S_{1}=\bigl\{ (a,b,c)\in S: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)>0\bigr\} , \\& S_{2}=\bigl\{ (a,b,c)\in S: \bigl(a^{2}+b+c\bigr) \bigl(a^{2}+b-c\bigr)< 0\bigr\} . \end{aligned}$$

It is easy to see that \(H=H_{0}\cup H_{1}\cup H_{2}\), \(S=S_{0}\cup S_{1}\cup S_{2}\), \(H_{i}\cap H_{j}=\Phi\), \(S_{i}\cap S_{j}=\Phi\) and \(H_{i}\cap S_{j}=\Phi\), \(i \neq j\), \(i,j=0,1,2\).

Theorem 2

Under the constraints

$$ \frac{b-a^{2}-c}{b+a^{2}-c}\leq0 $$
(22)

and

$$ \frac{b-a^{2}+c}{b+a^{2}+c}\leq0, $$
(23)

if the following conditions are satisfied:

\(r\neq1/(a^{2} \lambda_{\mathbf{F}} (1-\theta))\) and

$$ \textstyle\begin{cases} r< \min\frac{2}{a^{2} \lambda_{\mathbf{F}} (1-2\theta)},\quad 0 \leq \theta< 1/2, \\ r>0,\quad 1/2 \leq\theta\leq1, \end{cases} $$
(24)

for m is even, and

$$ r< \min\frac{1}{a^{2} \lambda_{\mathbf{F}} (1-\theta)}, $$
(25)

for m odd, then \(H_{1}\subseteq S_{1}\).

Proof

By the definition of \(H_{1}\) we know that (2) is satisfied when (22) and (23) hold. In the same way, according to the definition of \(S_{1}\) we know that (20) is satisfied when (22) and (23) hold and \(0<\beta ^{m}<1\), where β is defined in (19), then we can get \(H_{1}\subseteq S_{1}\). Therefore, (24) and (25) can be obtained from \(0<\beta^{m}<1\). The proof is completed. □

Theorem 3

Under the constraints

$$ \frac{b-a^{2}-c}{b+a^{2}-c}\geq1 $$
(26)

and

$$ \frac{b-a^{2}+c}{b+a^{2}+c}\leq0, $$
(27)

if the following conditions are satisfied:

\(r\neq1/(a^{2} \lambda_{\mathbf{F}} (1-\theta))\) and

$$\textstyle\begin{cases} r< \min\frac{2}{a^{2} \lambda_{\mathbf{F}} (1-2\theta)},\quad 0 \leq \theta< 1/2, \\ r>0, \quad 1/2 \leq\theta\leq1, \end{cases} $$

for m even, and

$$r< \min\frac{1}{a^{2} \lambda_{\mathbf{F}} (1-\theta)}, $$

for m odd, then \(H_{2}\subseteq S_{2}\).

Proof

Similar to the proof of Theorem 2, we can omit it. □

Theorem 4

Under the constraints

$$\begin{aligned}& \frac{b+a^{2}-c}{b+a^{2}+c}=0, \end{aligned}$$
(28)
$$\begin{aligned}& \frac{b-a^{2}-c}{b+a^{2}+c}< 0, \end{aligned}$$
(29)

and

$$ \frac{b-a^{2}+c}{b+a^{2}+c}\leq0, $$
(30)

if the following conditions are satisfied:

\(r\neq1/(a^{2} \lambda_{\mathbf{F}} (1-\theta))\) and

$$\textstyle\begin{cases} r< \min\frac{2}{a^{2} \lambda_{\mathbf{F}} (1-2\theta)},\quad 0 \leq \theta< 1/2, \\ r>0,\quad 1/2 \leq\theta\leq1, \end{cases} $$

for m even, and

$$r< \min\frac{1}{a^{2} \lambda_{\mathbf{F}} (1-\theta)}, $$

for m odd, then \(H_{0}\subseteq S_{0}\).

Proof

Follows directly from the proof of Theorem 2. □

Remark 1

If \(\theta=1\), then the corresponding fully implicit finite difference scheme is asymptotically stable unconditionally.

Numerical experiments

To demonstrate our theoretical result, some numerical examples are adopted in this section. Consider the following two problems:

$$\begin{aligned}& \textstyle\begin{cases} u_{t}(x,t)=u_{xx}(x,t)+\frac{1}{2} u_{xx}(x,[t])+\frac{1}{4} u_{xx} (x,2 [\frac{t+1}{2} ] ), \quad t> 0, \\ u(0,t)=u(1,t)=0, \\ u(x,0)=\sin(\pi x), \end{cases}\displaystyle \end{aligned}$$
(31)
$$\begin{aligned}& \textstyle\begin{cases} u_{t}(x,t)=u_{xx}(x,t)-2u_{xx}(x,[t])+2u_{xx} (x,2 [\frac {t+1}{2} ] ),\quad t> 0, \\ u(0,t)=u(1,t)=0, \\ u(x,0)=\sin(\pi x). \end{cases}\displaystyle \end{aligned}$$
(32)

In Tables 14 we list the absolute errors \(\operatorname{AE}(1/m,1/p)\), \(\operatorname{AE}(1/4m,1/2p)\) and \(\operatorname{AE}(1/2m, 1/2p)\) at \(x=1/2\), \(t=1\) of the θ-methods for (31) and (32), the ratio of \(\operatorname{AE}(1/m,1/p)\) over \(\operatorname{AE}(1/4m,1/2p)\) in Tables 1, 3 and the ratio of \(\operatorname{AE}(1/m,1/p)\) over \(\operatorname{AE}(1/2m,1/2p)\) in Tables 2, 4. We can see from these tables that the numerical methods conserve their orders of convergence.

Table 1 Errors of (31) with \(\theta= 0\)
Table 2 Errors of (31) with \(\theta= 1/2\)
Table 3 Errors of (32) with \(\theta= 0\)
Table 4 Errors of (32) with \(\theta= 1/2\)

In Figures 14, we draw the numerical solutions of the θ-methods. It is easy to see that the numerical solutions are asymptotically stable. In Figures 5 and 6, we draw the error figures for the numerical solutions with \(\theta=1\). It can be seen that the numerical method is of high accuracy.

Figure 1
figure1

The numerical solution of (31) with \(\theta= 0\), \(m = 6400\), \(p = 20\) and \(r = 1/16\)

Figure 2
figure2

The numerical solution of (31) with \(\theta= 0.5\), \(m = 128\), \(p = 16\) and \(r = 2\)

Figure 3
figure3

The numerical solution of (32) with \(\theta= 0\), \(m = 6400\), \(p = 20\) and \(r = 1/16\)

Figure 4
figure4

The numerical solution of (32) with \(\theta= 0.5\), \(m = 128\), \(p = 16\) and \(r = 2\)

Figure 5
figure5

Errors of (31) with \(\theta= 1\), \(m = 1024\), \(p = 32\) and \(r = 1\)

Figure 6
figure6

Errors of (32) with \(\theta= 1\), \(m = 1024\), \(p = 32\) and \(r = 1\)

References

  1. 1.

    Cavalli, F., Naimzada, A.: A multiscale time model with piecewise constant argument for a boundedly rational monopolist. J. Differ. Equ. Appl. 22, 1480–1489 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Dai, L., Fan, L.: Analytical and numerical approaches to characteristics of linear and nonlinear vibratory systems under piecewise discontinuous disturbances. Commun. Nonlinear Sci. Numer. Simul. 9, 417–429 (2004)

    Article  MATH  Google Scholar 

  3. 3.

    Dai, L., Singh, M.C.: On oscillatory motion of spring-mass systems subjected to piecewise constant forces. J. Sound Vib. 173, 217–232 (1994)

    Article  MATH  Google Scholar 

  4. 4.

    Gurcan, F., Bozkurt, F.: Global stability in a population model with piecewise constant arguments. J. Math. Anal. Appl. 360, 334–342 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Wiener, J., Lakshmikantham, V.: A damped oscillator with piecewise constant time delay. Nonlinear Stud. 1, 78–84 (2000)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Cooke, K.L., Wiener, J.: Retarded differential equations with piecewise constant delays. J. Math. Anal. Appl. 99, 265–297 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Shah, S.M., Wiener, J.: Advanced differential equations with piecewise constant argument deviations. Int. J. Math. Math. Sci. 6, 671–703 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Wiener, J.: Differential equations with piecewise constant delays. In: Lakshmikantham, V. (ed.) Trends in the Theory and Practice of Nonlinear Differential Equations, pp. 547–580. Dekker, New York (1983)

    Google Scholar 

  9. 9.

    Akhmet, M.U., Arugǎslan, D., Yılmaz, E.: Stability in cellular neural networks with a piecewise constant argument. J. Comput. Appl. Math. 233, 2365–2373 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Bereketoglu, H., Seyhan, G., Ogun, A.: Advanced impulsive differential equations with piecewise constant arguments. Math. Model. Anal. 15, 175–187 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Karakoc, F.: Asymptotic behaviour of a population model with piecewise constant argument. Appl. Math. Lett. 70, 7–13 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Muroya, Y.: New contractivity condition in a population model with piecewise constant arguments. J. Math. Anal. Appl. 346, 65–81 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Pinto, M.: Asymptotic equivalence of nonlinear and quasi linear differential equations with piecewise constant arguments. Math. Comput. Model. 49, 1750–1758 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Wiener, J.: Generalized Solutions of Functional Differential Equations. World Scientific, Singapore (1993)

    Google Scholar 

  15. 15.

    Berezansky, L., Braverman, E.: Stability conditions for scalar delay differential equations with a non-delay term. Appl. Math. Comput. 250, 157–164 (2015)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Dimbour, W.: Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space. Nonlinear Anal. 74, 2351–2357 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    El Raheem, Z.F., Salman, S.M.: On a discretization process of fractional-order logistic differential equation. J. Egypt. Math. Soc. 22, 407–412 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Muminov, M.I.: On the method of finding periodic solutions of second-order neutral differential equations with piecewise constant arguments. Adv. Differ. Equ. 2017, 336 (2017)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liu, M.Z., Song, M.H., Yang, Z.W.: Stability of Runge–Kutta methods in the numerical solution of equation \(u'(t)=au(t)+a_{0}u([t])\). J. Comput. Appl. Math. 166, 361–370 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Li, C., Zhang, C.J.: Block boundary value methods applied to functional differential equations with piecewise continuous arguments. Appl. Numer. Math. 115, 214–224 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Milosevic, M.: The Euler–Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments. J. Comput. Appl. Math. 298, 1–12 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Song, M.H., Liu, X.: The improved linear multistep methods for differential equations with piecewise continuous arguments. Appl. Math. Comput. 217, 4002–4009 (2010)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Wang, W.S., Li, S.F.: Dissipativity of Runge–Kutta methods for neutral delay differential equations with piecewise constant delay. Appl. Math. Lett. 21, 983–991 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Zhang, G.L.: Stability of Runge–Kutta methods for linear impulsive delay differential equations with piecewise constant arguments. J. Comput. Appl. Math. 297, 41–50 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Liang, H., Liu, M.Z., Lv, W.J.: Stability of θ-schemes in the numerical solution of a partial differential equation with piecewise continuous arguments. Appl. Math. Lett. 23, 198–206 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Liang, H., Shi, D.Y., Lv, W.J.: Convergence and asymptotic stability of Galerkin methods for a partial differential equation with piecewise constant argument. Appl. Math. Comput. 217, 854–860 (2010)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Wang, Q., Wen, J.C.: Analytical and numerical stability of partial differential equations with piecewise constant arguments. Numer. Methods Partial Differ. Equ. 30, 1–16 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Wiener, J., Debnath, L.: A wave equation with discontinuous time delay. Int. J. Math. Math. Sci. 15, 781–788 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Song, M.H., Liu, M.Z.: Numerical stability and oscillations of the Runge–Kutta methods for the differential equations with piecewise continuous arguments of alternately retarded and advanced type. J. Inequal. Appl. 2012, 290 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Blanco-Cocom, L., Àvila-Vales, E.: Convergence and stability analysis of the θ-method for delayed diffusion mathematical models. Appl. Math. Comput. 231, 16–25 (2014)

    MathSciNet  Google Scholar 

  31. 31.

    Zhang, Q.F., Chen, M.Z., Xu, Y.H., Xu, D.H.: Compact θ-method for the generalized delay diffusion equation. Appl. Math. Comput. 316, 357–369 (2018)

    MathSciNet  Google Scholar 

Download references

Acknowledgements

The author is grateful to the reviewers for their careful reading and useful comments. This work is supported by the National Natural Science Foundation of China (No. 11201084) and Natural Science Foundation of Guangdong Province (No. 2017A030313031).

Author information

Affiliations

Authors

Contributions

The author read and approved the final manuscript.

Corresponding author

Correspondence to Qi Wang.

Ethics declarations

Competing interests

The author declares that he has no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Wang, Q. Stability of numerical solution for partial differential equations with piecewise constant arguments. Adv Differ Equ 2018, 71 (2018). https://doi.org/10.1186/s13662-018-1514-1

Download citation

MSC

  • 65L07
  • 65L20

Keywords

  • Partial differential equation
  • Piecewise constant arguments
  • θ-methods
  • Stability